Recent Advances in Symmetry Analysis and Exact Solutions

Advances in Mathematical Physics

Recent Advances in Symmetry Analysis and Exact Solutions in Nonlinear Mathematical Physics

Guest Editors Maria Bruzoacuten Chaudry M Khalique Maria L Gandarias Rita Tracinaacute and Mariano Torrisi

Recent Advances in Symmetry Analysisand Exact Solutions in NonlinearMathematical Physics



Guest Editors Maria Bruzoacuten Chaudry M KhaliqueMaria L Gandarias Rita Tracinaacute and Mariano Torrisi

Copyright copy 2017 Hindawi All rights reserved

This is a special issue published in ldquoAdvances inMathematical Physicsrdquo All articles are open access articles distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the originalwork is properly cited

Editorial Board

G Amelino-Camelia ItalyStephen C Anco CanadaJacopo Bellazzini ItalyLuigi C Berselli ItalyRaffaella Burioni ItalyManuel Calixto SpainJoseacute F Carintildeena SpainDongho Chae Republic of KoreaJohn D Clayton USAPierluigi Contucci ItalyGiampaolo Cristadoro ItalyClaudio Dappiaggi ItalyPrabir Daripa USAPietro drsquoAvenia ItalyManuel De Leoacuten Spain

Ciprian G Gal USAAlexander Iomin IsraelGiorgio Kaniadakis ItalyBoris G Konopelchenko ItalyPavel Kurasov SwedenRemi Leacuteandre FranceXavier Leoncini FranceEmmanuel Lorin CanadaWen-Xiu Ma USAJuan C Marrero SpainNikos Mastorakis BulgariaAnupam Mazumdar UKMing Mei CanadaAndrei D Mironov RussiaTakayuki Miyadera Japan

Hagen Neidhardt GermanyAndreacute Nicolet FranceAnatol Odzijewicz PolandMikhail Panfilov FranceAlkesh Punjabi USAEugen Radu PortugalSoheil Salahshour IranAntonio Scarfone ItalyDimitrios Tsimpis FranceRicardo Weder MexicoXiao-Jun Yang ChinaValentin Zagrebnov FranceYao-Zhong Zhang Australia

Contents

Recent Advances in Symmetry Analysis and Exact Solutions in Nonlinear Mathematical PhysicsMaria Bruzoacuten Chaudry M Khalique Maria L Gandarias Rita Tracinaacute and Mariano TorrisiVolume 2017 Article ID 8325854 2 pages

The Stochastic Resonance Behaviors of a Generalized Harmonic Oscillator Subject to Multiplicative andPeriodically Modulated NoisesSuchuan Zhong KunWei Lu Zhang Hong Ma and Maokang LuoVolume 2016 Article ID 5164575 12 pages

TheRational Solutions and Quasi-Periodic Wave Solutions as well as Interactions of119873-SolitonSolutions for 3 + 1 Dimensional Jimbo-Miwa EquationHongwei Yang Yong Zhang Xiaoen Zhang Xin Chen and Zhenhua XuVolume 2016 Article ID 7241625 14 pages

Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky EquationsWeishi Yin Fei Xu Weipeng Zhang and Yixian GaoVolume 2016 Article ID 4632163 9 pages

An Efficient Numerical Method for the Solution of the Schroumldinger EquationLicheng Zhang and Theodore E SimosVolume 2016 Article ID 8181927 20 pages

Stability of the Cauchy Additive Functional Equation on Tangle Space and ApplicationsSoo Hwan KimVolume 2016 Article ID 4030658 10 pages

New Periodic Solutions for a Class of Zakharov EquationsCong Sun and Shuguan JiVolume 2016 Article ID 6219251 6 pages

EditorialRecent Advances in Symmetry Analysis and Exact Solutions inNonlinear Mathematical Physics

Maria Bruzoacuten1 Chaudry M Khalique2 Maria L Gandarias1

Rita Tracinaacute3 andMariano Torrisi3

1University of Cadiz Cadiz Spain2North-West University Vanderbijlpark South Africa3Universita di Catania Catania Italy

Correspondence should be addressed to Maria Bruzon mbruzonucaes

Received 4 April 2017 Accepted 5 April 2017 Published 16 April 2017

Copyright copy 2017 Maria Bruzon et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Partial differential equations are used to describe a wide vari-ety of physical phenomena such as fluid dynamics plasmaphysics solid mechanics and quantum field theory thatarise in physics Many of these equations are nonlinear andin general these equations are often very difficult to solveexplicitly Many systematic methods are usually employedto study the nonlinear equations these include the gener-alized symmetry method the Painleve analysis the inversescattering method the Backlund transformationmethod theconservation lawmethod theCole-Hopf transformation andthe Hirota bilinear method

Constructing exact solutions in particular travellingwave solutions of nonlinear equations plays an importantrole in soliton theory Several important direct methodshave been developed for obtaining travelling wave solutionsto nonlinear partial differential equations such as the inversescattering method the tanh-function method the extendedtanh-function method the G1015840G method the simplestmethod and the modified simplest method

The authors of this special issue had been invited tosubmit original research articles as well as review articlesin the following topics methods for obtaining solutionsof partial differential equations conservation laws generalmethods for obtaining solutions of ordinary differentialequations travelling wave solutions novel applications inphysics

However we received 22 papers in these research fieldsAfter a rigorous reviewing process six articles were finally

accepted for publicationThese articles contain somenew andinnovative techniques and ideas that may stimulate furtherresearches in several branches of theory and applications ofthe transformation groups

In ldquoThe Rational Solutions and Quasi-Periodic WaveSolutions aswell as Interactions of119873-Soliton Solutions for 3 +1 Dimensional Jimbo-Miwa Equationrdquo H Yang et al analyzethe rational solutions quasi-periodicwave solutions obtainedby the Hirota method and the theta function for the 3 + 1dimensional Jimbo-Miwa equation The knowledge of thesesolutions allows them to explain the interaction of the 119873-soliton solutions that is to show when the soliton solutioncan be changed into the resonant solution andwhen therewillappear the pursue collision that is the soliton with the fasterspeed will catch up with the soliton with the slower speed(after the collision the two solitons will continue to spreadas the previous speed and the direction)

In ldquoAnEfficientNumericalMethod for the Solution of theSchrodinger Equationrdquo L Zhang and T E Simos show theoriginal development of a new five-stage symmetric two-stepfourteenth-algebraic-order method with vanished phase-lagand its first second and third derivatives More specificallythey firstly introduce the development of the new methodand the determination of the local truncation error and suc-cessively the local truncation error analysis and the stabilityFinally the interval of periodicity analysis and the efficiencyof the new method are considered by applying it to couple ofSchrodinger equations

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 8325854 2 pageshttpsdoiorg10115520178325854

2 Advances in Mathematical Physics

In the paper titled ldquoAsymptotic Expansion of the Solutionsto Time-Space Fractional Kuramoto-Sivashinsky Equationsrdquothe attention of authors is devoted to finding the asymp-totic expansion of solutions to fractional partial differentialequations with initial conditions The residual power seriesmethod is proposed for time-space fractional partial differ-ential equations where the fractional integral and derivativeare described in the sense of Riemann-Liouville integraland Caputo derivative They apply the method to the linearand nonlinear time-space fractional Kuramoto-Sivashinskyequation with initial value and obtain asymptotic expansionof the solutions which confirm the accuracy and efficiency ofthe method

In ldquoNew Periodic Solutions for a Class of Zakharov Equa-tionsrdquo C Sun and S Ji consider a class of nonlinear Zakharovequations Inspired byAngluorsquos idea by applying the Jacobianelliptic function method they obtain new periodic solutionsfor Klein-Gordon Zakharov equations Zakharov equationsand Zakharov-Rubenchik equations

In the paper titled ldquoStability of the Cauchy AdditiveFunctional Equation on Tangle Space and Applicationsrdquo SH Kim introduces real tangles (a generalization of rationaltangles) and its operations to enumerating tangles by usingthe calculus of continued fractionMoreover he studies aboutthe analytical structure of tangles knots and links by usingnew operations between real tangles which need not havethe topological structure As for applications of the analyticalstructure he proves the generalized Hyers-Ulam stability ofthe Cauchy additive functional equation 119891(119909 oplus 119910) = 119891(119909) oplus119891(119910) in tangle space which is a set of real tangles with analyticstructure and describes the DNA recombination as the actionof some enzymes on tangle space

In their paper titled ldquoThe Stochastic Resonance Behaviorsof a Generalized Harmonic Oscillator Subject to Multi-plicative and Periodically Modulated Noisesrdquo S Zhong etal study the stochastic resonance (SR) characteristics of ageneralized Langevin linear system driven by a multiplicativenoise and a periodically modulated noise They take inconsideration a generalized Langevin equation (GLE) drivenby an internal noise with long-memory and long-rangedependence such as fractional Gaussian noise and Mittag-Leffler noise Such a model is appropriate to characterizingthe chemical and biological solutions as well as to somenanotechnological devices An exact analytic expressionof the output amplitude is obtained Based on it somecharacteristic features of stochastic resonance phenomenonare revealed On the other hand by the use of the exactexpression they obtain the phase diagram for the resonantbehaviors of the output amplitude versus noise intensityunder different values of system parameters These resultscould give the theoretical basis for practical use and controlof the SR phenomenon of this mathematical model in futureworks

Acknowledgments

We would like to thank all the authors who sent their worksand all the referees for the time spent in reviewing the

papers Their contributions and their efforts have been veryimportant for the publication of this special issue

Maria BruzonChaudry M KhaliqueMaria L Gandarias

Rita TracinaMariano Torrisi

Research ArticleThe Stochastic Resonance Behaviors of a GeneralizedHarmonic Oscillator Subject to Multiplicative and PeriodicallyModulated Noises

Suchuan Zhong1 Kun Wei2 Lu Zhang3 Hong Ma3 and Maokang Luo3

1College of Aeronautics and Astronautics Sichuan University Chengdu 610065 China2School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu 611731 China3College of Mathematics Sichuan University Chengdu 610065 China

Correspondence should be addressed to Maokang Luo makaluoscueducn

Received 21 September 2016 Revised 9 November 2016 Accepted 20 November 2016

Academic Editor Maria L Gandarias

Copyright copy 2016 Suchuan Zhong et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and aperiodically modulated noise are studied (the two noises are correlated) In this paper we consider a generalized Langevin equation(GLE) driven by an internal noise with long-memory and long-range dependence such as fractional Gaussian noise (fGn) andMittag-Leffler noise (M-Ln) Such a model is appropriate to characterize the chemical and biological solutions as well as to somenanotechnological devices An exact analytic expression of the output amplitude is obtained Based on it some characteristicfeatures of stochastic resonance phenomenon are revealed On the other hand by the use of the exact expression we obtain thephase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of systemparametersThese useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon ofthis mathematical model in future works

1 Introduction

The phenomenon of stochastic resonance (SR) characterizedthe cooperative effect between weak signal and noise ina nonlinear systems which was originally perceived forexplaining the periodicity of ice ages in early 1980s [1 2] Inthe past three decades the SR phenomenon attracted greatattentions and has been documented in a large number ofliteratures in biology physics chemistry and engineering [3ndash6] such as SR in magnetic systems [7] and tunnel diode [8]and in cancer growth models [9 10]

Nowadays there have been considerable developments inSR and the original understanding of SR is extended Firstlyin the initial stage of investigation of SR the nonlinearitysystem noise and periodic signal were thought of as threeessential ingredients for the presence of SR However in 1996Berdichevsky and Gitterman [11] found that SR phenomenoncan occur in the linear system with multiplicative colorednoise [12 13] It is because the multiplicative noise breaksthe symmetry of the potential of the linear system and this

gives rise to the SR phenomenon which has been explainedbyValenti et al in literatures [14 15] Valenti et al investigatedthe SR phenomenon in population dynamics where themul-tiplicative noise source is Gaussian or Levy type Moreoverthe appearance of SR in a trapping overdamped monostablesystem was also investigated in literature [16] recently Sec-ondly the conventional definition about SR phenomenonis the signal-to-noise ratio (SNR) versus the noise intensityexhibits a peak [17 18] whereas the generalized SR [19]implies the nonmonotonic behaviors of a certain function ofthe output signal (such as the first and second moments theautocorrelation function) on the system parameters Thirdlyanother interesting extension of SR lies in the fact that outputamplitude 119860 attains a maximum value by increasing thedriving frequency Ω Such phenomenon indicates the bonafide SR which was introduced by Gammaitoni et al appearsat some value ofΩ [20 21]

The SR phenomenon driven by Gaussian noise has beeninvestigated both theoretically and experimentally However

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 5164575 12 pageshttpdxdoiorg10115520165164575


the Gaussian noise is just an ideal model for actual fluctua-tions and not always appropriate to describe the real noisyenvironment For instance in the nonequilibrium situationthe stochastic processes describing the interactions of a testparticle with the environment exhibit a heavy tailed non-Gaussian distribution Nowadays Dybiec et al investigatedthe resonant behaviors of a stochastic dynamics systemwith Levy stable noise and an isotropic 120572-stable noise withheavy tails and jumps in literatures [22 23] Dubkov et alinvestigated the Levy flight superdiffusion as a self-similarLevy process and derive the fractional Fokker-Planck equa-tion (FFPE) for probability distribution from the Langevinequation with Levy stable noise [24ndash26]

Recently more and more scholars began to pay atten-tion to another important class of non-Gaussian noise thebounded noise It should be emphasized is that the boundednoise is a more realistic and versatile mathematical model ofstochastic fluctuations in applications and it is widely appliedin the domains of statistical physics biology and engineeringin the last 20 years Furthermore the well-known telegraphnoise such as dichotomous noise (DN) and trichotomousnoise that are widely used in the studied of SR phenomenais a special case of bounded noise The deepening anddevelopment of theoretical studies on bounded noise led tothe fact that lots of scholars investigated the effect of boundednoise on the stochastic resonant behaviors in the specialmodel in physics biology and engineering [27]

Since Richardsonrsquos work in literature [28] a large numberof observations related to anomalous diffusion [29ndash33] havebeen reported in several scientific fields for example brainstudies [34 35] social systems [36] biological cells [37 38]animal foraging behavior [39] nanoscience [40 41] andgeophysical systems [42] One of the main aspects of thesesituations is the correlation functions of the above anomalousdiffusion phenomena which may be related to the non-Markovian characteristic of the stochastic process [43] Thetypical characteristic of anomalous diffusion lies in themean-square displacement (MSD) which satisfies ⟨119909(119905)2⟩ sim 119905120572where the diffusion exponents 0 lt 120572 lt 1 and 120572 gt 1 indicatesubdiffusion and superdiffusion respectively When 120572 = 1the normal diffusion is recovered [44]

It is well-known that the normal diffusion can bemodeledby a Langevin equation where a Brownian particle subjectedto a viscous drag from the surrounding medium is charac-terized by a friction force and it also subjected to a stochasticforce that arises from the surrounding environmentThe fric-tion constant determines how quickly the system exchangesenergy with the surrounding environment For a realisticdescription of the surrounding environment it is difficult tochoose a universal value of the friction constant Indeed inorder to depict the real situation more effectively a differentvalue of the friction constant should be adopted Hence ageneralization of the Langevin equation is needed leading tothe so-called generalized Langevin equation (GLE) [45] TheGLE is an equation ofmotion for a non-Markovian stochasticprocess where the particle has a memory effect to its velocity

Nowadays the GLE driven by a fractional Gaussian noise(fGn) [46]with a power-law friction kernel is extensively used

for modeling anomalous diffusion processes For instance inthe study on dynamics of single-molecule when the electrontransfer (ET) was used to probe the conformational fluctu-ations of single-molecule enzyme the distance between theET donor and acceptor can be modeled well through a GLEdriven by an fGn [47ndash54] Besides Vinales and Despositohave introduced a novel noise whose correlation function isproportional to a Mittag-Leffler function which is called theMittag-Leffler noise (M-Ln) [55ndash57] The correlation func-tion behaves as a power-law for large times but is nonsingularat the origin due to the inclusion of a characteristic time

Theoverwhelmingmajority of previous studies of SRhaverelated to the case where the external noise and the weakperiodic force are introduced additively However Dykmanet al [58] studied the case where the signal is multiplied tonoise namely the noise is modulated by a signal They foundthatwhen an asymmetric bistable system is driven by a signal-modulated noise stochastic resonance appeared in contrastto the additive noise new characteristics emerge and theirresults were in agreement with experiments FurthermoreCao and Wu [59] studied the SR characteristics of a linearsystem driven by a signal-modulated noise and an additivenoise It seems that a periodically modulated noise is notuncommon and arises for example at the output of anyamplifier (optics or radio astronomy) whose amplificationfactor varies periodically with time

Due to the synergy of generalized friction kernel of a GLEand the periodically modulated noise the stochastic resonantbehaviors of a GLE can be influenced In contrast to thecase that has been investigated before new dynamic char-acteristics emerge Motivated by the above discussions wewould like to explore the stochastic resonance phenomenonin a generalized harmonic oscillator with multiplicative andperiodically modulated noises Moreover we consider theGLE is driven by a fractional Gaussian noise and a Mittag-Leffler noise respectively in this paper We focus on thevarious nonmonotonic behaviors of the output amplitude 119860with the systemparameters and the parameters of the internaldriven noise

The physical motivations of this paper are as follows (1)in view of the importance of stochastic generalized harmonicoscillator (linear oscillator) with memory in physics chem-istry and biology and due to the periodicallymodulated noisearising at the output of the amplifier of the optics deviceand radio astronomy device to establish a physical model inwhich the SR can contain the effects of the two factors thelinearity of the system and the periodical modulation of thenoise (2) The second one is to give a theoretical foundationfor the study of SR characteristic features of a generalizedharmonic oscillator subject to multiplicative periodicallymodulated noises and external periodic force Our studyshows that such a model leads to stochastic resonancephenomenon Meanwhile an exact analytic expression of theoutput amplitude is obtained Based on it some characteristicfeatures of SR are revealed

The paper is organized as follows Section 2 gives theintroductions of the generalized Langevin equation the frac-tional Gaussian noise and the Mittag-Leffler noise Section 3gives analytical expression of the output amplitude of the

Advances in Mathematical Physics 3

systemrsquos steady response Section 4 presents the simulationresults and gives the discussions Section 5 gives conclusion

2 System Model

21 The Generalized Langevin Equation The generalizedLangevin equation (GLE) is an equation of motion for thenon-Markovian stochastic process where the particle has amemory effect to its velocity Anomalous diffusion in physicaland biological systems can be formulated in the framework ofaGLE that reads asNewtonrsquos law for a particle of the unitmass(119898 = 1) [11 47 60ndash66]

(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)

where 119909(119905) is the displacement of the Brownian particle attime 119905 120574 gt 0 is the friction constant 119870(119905) represents thememory kernel of the frictional force and 119889119880(119909)119889119909 is theexternal force under the potential 119880(119909) The random force120578(119905) is zero-centered and stationary Gaussian that obeys thegeneralized second fluctuation-dissipation theorem [67]

⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)

where 119896119861 is the Boltzmann constant and 119879 is the absolutetemperature of the environment

22TheFractional GaussianNoise FractionalGaussian noise(fGn) and fractional Brownian motion (fBm) were originallyintroduced by Mandelbrot and Van Ness [46] for modelingstochastic fractal processes The fGn 120576119867(119905) = 120576119867(119905) 119905 gt 0with a constant Hurst parameter 12 lt 119867 lt 1 can beused to more accurately characterize the long-range depen-dent process [the autocorrelation function 119903(119896) satisfiessuminfin119896=minusinfin 119903(119896) = infin] than traditional short-range dependentstochastic process [the autocorrelation function 119903(119896) satisfiessuminfin119896=minusinfin 119903(119896) lt infin] The short-range dependent stochasticprocess is for example Markov Poisson or autoregressivemoving average (ARMA) process

Now consider the one-side normalized fBm which is aGaussian process 119861119867(119905) = 119861119867(119905) 119905 gt 0 which shows theproperties below [68]

(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)

The fGn 120576119867(119905) = 120576119867(119905) 119905 gt 0 given by [47ndash49]

120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)

is a stationary Gaussian process with ⟨120576119867(119905)⟩ = 0 Thereforeaccording to (3) and (4) and the LrsquoHospitalrsquos rule theautocorrelation function 119862(119905) of fGn can be derived as

119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+

⟨[119861119867 (0 + 119904) minus 119861119867 (0)]119904 [119861119867 (119905 + 119904) minus 119861119867 (119905)]119904 ⟩= 2119896119861119879 sdot lim

119904rarr0+

|119905 + 119904|2119867 + |119905 minus 119904|2119867 minus 2 |119905|211986721199042 = 2119896119861119879sdot lim119904rarr0+

2119867 |119905 + 119904|2119867minus1 minus 2119867 |119905 minus 119904|2119867minus14119904 = 2119896119861119879sdot lim119904rarr0+

2119867 (2119867 minus 1) |119905 + 119904|2119867minus2 + 2119867 (2119867 minus 1) |119905 minus 119904|2119867minus24 = 2119896119861119879sdot 2119867 (2119867 minus 1) |119905 + 0|2119867minus2 + 2119867 (2119867 minus 1) |119905 minus 0|2119867minus24= 2119896119861119879 sdot 119867 (2119867 minus 1) 1199052119867minus2 119905 gt 0

(5)

23 The Mittag-Leffler Noise It is well-known that the phys-ical origin of anomalous diffusion is related to the long-timetail correlationsThus in order tomodel anomalous diffusionprocess a lot of different power-law correlation functions areemployed in (1) and (2)

Vinales and Desposito have introduced a novel noisewhose correlation function is proportional to aMittag-Lefflerfunction which is called Mittag-Leffler noise [55ndash57] Thecorrelation function of Mittag-Leffler noise behaves as apower-law for large times but is nonsingular at the origindue to the inclusion of a characteristic time The correlationfunction of Mittag-Leffler noise is given by

119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)

where 120591 is called characteristicmemory time and thememoryexponent 120572 can be taken as 0 lt 120572 lt 2 The 119864120572(sdot) denotes theMittag-Leffler function that is defined through the series

119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)

which behaves as a stretched exponential for short times andas inverse power-law in the long-time regime when 120572 = 1Meanwhile when 120572 = 1 the correlation function equation(4) reduces to the exponential form

119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)

which describes a standard Ornstein-Uhlenbeck process


24 The System Model In this paper we consider a periodi-cally driven linear system with multiplicative noise and peri-odically modulated additive noise described by the followinggeneralized Langevin equation

(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)

where 1205960 is the intrinsic frequency of the harmonic oscillator119880(119909) = 1205962011990922 The fluctuations of 12059620 in (9) are modeled asa Markovian dichotomous noise 1205851(119905) [69] which consists ofjumps between two values minus119886 and 119886 119886 gt 0 with stationaryprobabilities 119875119904(minus119886) = 119875119904(119886) = 12 The statistical propertiesof 1205851(119905) are ⟨1205851 (119905)⟩ = 0

⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)

where 1198862 is the noise intensity and V is the correlation ratewith 1205910 = 1V being the correlation time1205852(119905) is a zero mean signal-modulated noise with cou-pling strength119863 with noise 1205851(119905) [70 71] that is⟨1205851 (119905) 1205852 (119904)⟩ = 119863120575 (119905 minus 119904) (11)

In (9) 1198601 and Ω are the amplitude and frequency of theexternal periodic force1198601 sin(Ω119905) Meanwhile1198602 andΩ arethe amplitude and frequency of the periodically modulatedadditive noise 1198602 sin(Ω119905)1205852(119905) respectively

In this paper we assume that the external noise 1205851(119905) 1205852(119905)and the internal noise 120578(119905) satisfy ⟨120578(119905)1205851(119904)⟩ = ⟨120578(119905)1205852(119904)⟩ =0with different origins In the next section we will obtain theexact expression of the first moment of the output signal

3 First Moment

31 The Analytical Expression of the Output Amplitude of aGLE First of all we should transfer the stochastic equation(9) to the deterministic equation for the average value ⟨119909⟩For this purpose we use the well-known Shapiro-Loginov[72] procedure which yields for exponentially correlatednoise (10)

⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)

Equation (9) depicted the motion of 119909(119905) is bounded by anoisy harmonic force field by averaging realization of thetrajectory of the stochastic equation (9) and applying thecharacteristics of the noises 1205851(119905) 1205852(119905) and 120578(119905) we obtainthe equation of the particlersquos average displacement ⟨119909⟩1198892 ⟨119909⟩1198891199052 + 120574int119905

0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)

It can be found that (13) shows the synthetic affections ofthe particlersquos average displacement ⟨119909⟩ and the multiplicative

noise coupling term ⟨1205851119909⟩ In order to deal with the newmultiplicative noise coupling term ⟨1205851119909⟩ we multiply bothsides of (9) with 1205851(119905) and then average to construct a closedequations of ⟨119909⟩ and ⟨1205851119909⟩

⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)

Using the Shapiro-Loginov formula (12) and the charac-teristics of the generalized integration (14) turns to be

[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)

To summarize for the linear generalized Langevin equa-tion (9) to be investigated in this paper it is a stochasticdifferential equation driven by an internal noise 120578(119905) Whenwe want to obtain the particlersquos average displacement ⟨119909⟩from (9) we average (9) and obtain the traditional classicaldifferential equation (13) for ⟨119909⟩ and the new multiplicativenoise coupling term ⟨1205851119909⟩ In order to deal with the newcoupling term ⟨1205851119909⟩ we do some mathematical calculationsand have another ordinary differential equation (15) Finallywe obtain two linear closed equations (13) and (15) for 1199091 =⟨119909⟩ and 1199092 = ⟨1205851119909⟩

In order to solve the closed equations (13) and (15) weuse the Laplace transform technique 119883119894 = LT[119909119894(119905)] ≜int+infin0

119909119894(119905)119890minus119904119905119889119905 119894 = 1 2 [73] under the long time limit 119905 rarrinfin condition we obtain the following equations

[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)

(119904) = LT[119870(119905)]means performing Laplace transformThe solutions of (16) can be represented as

119883(119886119904)1 (119904) = 11 (119904) sdot LT [1198601 sin (Ω119905)] + 21 (119904)sdot LT [1198602119863 sin (Ω119905)]


(17)


where 119896119894(119904) = LT[119867119896119894(119905)] 119896 119894 = 1 2 and 119896119894(119904) can beobtained from (16) Particularly we have

11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)

with (119904) = [12059620 + 1199042 +120574119904 sdot (119904)][12059620 + (119904+ V)2 +120574(119904+ V) sdot (119904 +V)] minus 1198862

Applying the inverse Laplace transform technique by thetheory of ldquosignals and systemsrdquo the product of the Laplacedomain functions corresponding to the convolution of thetime domain functions we can obtain the solutions 119909(119886119904)1 (119905) =⟨119909(119905)⟩119886119904 ≜ lim119905rarrinfin⟨119909(119905)⟩ and 119909(119886119904)2 (119905) = ⟨1205851(119905)119909(119905)⟩119886119904 ≜lim119905rarrinfin⟨1205851(119905)119909(119905)⟩ through (17)

119909(119886119904)1 (119905) = 11986711 (119905) lowast [1198601 sin (Ω119905)] + 11986721 (119905)lowast [1198602119863 sin (Ω119905)]


(19)

where lowast is the convolution operatorEquations (13) and (15) with 1199091 = ⟨119909⟩ and 1199092 = ⟨1205851119909⟩ can

be regarded as a linear system and the forces 1198601 sin(Ω119905) and1198602119863 sin(Ω119905) can be regarded as the input periodic signalsBy the theory of ldquosignals and systemsrdquo when we put periodicsignals 1198601 sin(Ω119905) and 1198602119863 sin(Ω119905) into a linear systemwith system functions 11986711(119905) and 11986721(119905) the output signalsare still periodic signals meanwhile the frequency of theoutput signals are the same as the frequency of the inputsignals We denote the output signals as 11986011 sin(Ω119905 + 12059311)and 11986021 sin(Ω119905 + 12059321) respectively which can be shown inFigure 1

Moreover the amplitudes 11986011 and 11986021 of the outputsignals are proportion to the amplitudes 1198601 and 1198602119863 of theinput signals and the proportion constants are the amplitudesof the frequency response functions 11986711(119890119895Ω) and 11986721(119890119895Ω)that is 11986011 = 1198601|11986711(119890119895Ω)| and 11986021 = 1198602119863|11986721(119890119895Ω)|

Thus from (19) we obtain the following expression ofparticlersquos average displacement 119909(119886119904)1 (119905)119909(119886119904)1 (119905) = 11986711 (119905) lowast [1198601 sin (Ω119905)] + 11986721 (119905) lowast [1198602119863

sdot sin (Ω119905)] = 11986011 sin (Ω119905 + 12059311) + 11986021 sin (Ω119905+ 12059321) = radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321)sdot sin[[[

Ω119905

+ arcsin (11986011 sin12059311 + 11986021 sin12059321)radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321)]]]

(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)

Figure 1 The relationships of the input periodic signals and theoutput signals by the theory of ldquosignals and systemsrdquo

where11986011 11986021 and 12059311 12059321 are the amplitude and phase shiftof the long-time behaviors of the output signals respectively

In addition we can obtain the expressions of frequencyresponse functions 11986711(119890119895Ω) and 11986721(119890119895Ω) through (18)Through the derivation we can suppose that the expressionsof11986711(119890119895Ω) and11986721(119890119895Ω) can be simplified as follows

11986711 (119890119895Ω) ≜ 1198601 + 1198611 sdot 1198951198621 + 1198631 sdot 119895 11986721 (119890119895Ω) ≜ 1198602 + 1198612 sdot 1198951198622 + 1198632 sdot 119895

(21)

where 119860 119894 119861119894 119862119894 119863119894 119894 = 1 2 are real numbersThen the amplitudes of frequency response functions11986711(119890119895Ω) and11986721(119890119895Ω) are1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = radic11986711 (119890119895Ω)119867lowast11 (119890119895Ω) = radic11986021 + 1198612111986221 + 11986321 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = radic11986721 (119890119895Ω)119867lowast21 (119890119895Ω) = radic11986022 + 1198612211986222 + 11986322

(22)

where 119867lowast1198941(119890119895Ω) 119894 = 1 2 is the conjugation of 1198671198941(119890119895Ω) 119894 =1 2Meanwhile the amplitudes of the output signals are

11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)

In this paper with the expression of the particlersquos averagedisplacement 119909(119886119904)1 (119905) in (20) we mainly discuss the resonantbehaviors of the output amplitude 119860 which is defined as

119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)

It should be emphasized that the following inequalitymust hold for the sake of the stability of solutions [69]

0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)

In this paper we assume the stability condition (25) issatisfied


32 The Output Amplitude of a GLE with Fractional Gaus-sian Noise When the internal noise 120578(119905) in the generalizedLangevin equation (9) is fractional Gaussian noise withcorrelation function (5) from the fluctuation-dissipationtheorem (2) we derive the power-law memory kernel 119870(119905)presented by Hurst exponent119867 0 lt 119867 lt 1

119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)

Performing the Laplace transform we obtain the related(119904) = LT[119870(119905)] (119904) = Γ (2119867 + 1) sdot 1199041minus2119867 0 lt 119867 lt 1 (27)

From (18) (20) (24) and (27) we get the output amplitude119860expressed by (24) with

11986011 = 1198601radic11989112 + 1198912211989132 + 11989142 12059311 = arctan(11989121198913 minus 1198911119891411989111198913 + 11989121198914) 11986021 = 1198602119863radic 111989132 + 11989142

(28)

12059321 = arctan(minus11989141198913) (29)

where

1198911 = 12059620 + 1198872 cos (2120579) + 1205741198872minus2119867sdot Γ (2119867 + 1) cos [(2 minus 2119867) 120579] 1198912 = 1198872 sin (2120579) + 1205741198872minus2119867sdot Γ (2119867 + 1) sin [(2 minus 2119867) 120579] 1198913 = 1198720 +1198721 cos (2120579) + 1198722 cos [(2 minus 2119867) 120579]

+1198723 cos [(2 minus 2119867) 1205872 + 2120579]+1198724 cos [(1205872 + 120579) (2 minus 2119867)]+1198725 cos [(2 minus 2119867) 1205872 ]

1198914 = 1198721 sin (2120579) + 1198722 sin [(2 minus 2119867) 120579]+1198723 sin [(2 minus 2119867) 1205872 + 2120579]+1198724 sin [(1205872 + 120579) (2 minus 2119867)]+1198725 sin [(2 minus 2119867) 1205872 ]

119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059640 minus 1198862 minus Ω212059620 1198721 = (12059620 minus Ω2) 11988721198722 = 120574 (12059620 minus Ω2) 1198872minus2119867Γ (2119867 + 1) 1198723 = 120574Ω2minus21198671198872Γ (2119867 + 1) 1198724 = 1205742Ω2minus21198671198872minus2119867Γ2 (2119867 + 1) 1198725 = 12057412059620Ω2minus2119867Γ (2119867 + 1)

(30)

33The Output Amplitude of a GLE withMittag-Leffler NoiseWhen the internal noise 120578(119905) in the generalized Langevinequation (9) is Mittag-Leffler noise with correlation function(6) from the fluctuation-dissipation theorem (2) we derivetheMittag-Lefflermemory kernel119870(119905) presented bymemorytime 120591 and memory exponent 120572

119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)

Performing the Laplace transform we obtain the related(119904) = LT[119870(119905)] (119904) = 119904120572minus11 + (120591119904)120572 0 lt 120572 lt 2 120591 gt 0 (32)

From (18) (20) (24) and (32) we obtain the output ampli-tude 119860 expressed by (24) with

11986011 = 1198601radic11989212 + 1198922211989232 + 11989242 12059311 = arctan(11989221198923 minus 1198921119892411989211198923 + 11989221198924) 11986021 = 1198602119863radic ℎ12 + ℎ2211989232 + 11989242 12059321 = arctan(ℎ21198923 minus ℎ11198924ℎ11198923 + ℎ21198924)

(33)

where

1198921 = 12059620 +1198727 cos (120572120579) +1198722 cos (2120579)+ 11987221198725 cos [(2 + 120572) 120579]+ 119872412059620 cos(1205872 120572)+11987241198727 cos [(1205872 + 120579) 120572]


+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]

1198922 = 1198727 sin (120572120579) + 1198722 sin (2120579)+ 11987221198725 sin [(2 + 120572) 120579] + 119872412059620 sin(1205872 120572)+11987241198727 sin [(1205872 + 120579) 120572]+11987221198724 sin(2120579 + 1205872 120572)+119872211987241198725 sin [(2 + 120572) 120579 + 1205872 120572]

ℎ1 = minus1 minus1198724 cos(1205872 120572) minus1198725 cos (120572120579)minus11987241198725 cos [(1205872 + 120579) 120572]

ℎ2 = minus1198724 sin(1205872 120572) minus1198725 sin (120572120579)minus11987241198725 sin [(1205872 + 120579) 120572]

1198923 = 1198728 +11987201198722 cos (2120579) + 1198729 cos (120572120579)+119872011987221198725 cos [(2 + 120572) 120579]+ 11987210 cos(1205872 120572) +11987211 cos(2120579 + 1205872 120572)+11987212 cos [(1205872 + 120579) 120572]+11987213 cos [(2 + 120572) 120579 + 1205872 120572]

1198924 = 11987201198722 sin (2120579) + 1198729 sin (120572120579)+119872011987221198725 sin [(2 + 120572) 120579]+ 11987210 sin(1205872 120572) +11987211 sin(2120579 + 1205872 120572)+11987212 sin [(1205872 + 120579) 120572]+11987213 sin [(2 + 120572) 120579 + 1205872 120572]

119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)

4 Stochastic Resonance Behaviors ofa GLE with Multiplicative and PeriodicallyModulated Noises

In this section we will perform the numerical simulationson the above analytical expression in (24) with the internalnoise 120578(119905) modeled as fractional Gaussian noise and Mittag-Leffler noise in Sections 41 and 42 respectively It can beseen from the analytical expression in (24) the behaviorsof 119860 are fully determined by the combination of the systemparameters 120574 12059620 1198862 V 1198601 1198602 119863 and Ω and the parametersof 120578(119905) Based on it some characteristic features of stochasticresonance behaviors are revealed

41 The Stochastic Resonance Behaviors of GLE with a Frac-tional Gaussian Noise From (25) and (27) we obtain thestability condition ofGLEwith fractional Gaussian noise 120578(119905)

0 lt 1198862 le 1198862crfGn= 12059620 [12059620 + V2 + 120574 sdot Γ (2119867 + 1) sdot V2minus2119867] (35)

with 0 lt 119867 lt 1It can be seen from the stability condition (35) that the

critical noise intensity 1198862crfGn is determined by 12059620 V 120574 and119867 Besides from (24) the behaviors of output amplitude 119860are fully determined by the combination of the parameters120574 12059620 1198862 V 1198601 1198602 119863Ω and119867 Thus in Figure 2 we depictthe phase diagram in the 119863 minus Ω plane for the emergence ofthe stochastic resonance behaviors of 119860 versus 1198862 at 12059620 = 11198601 = 1119867 = 055 1198602 = 1 120574 = 03 and V = 001

In the unshaded region [see the domain (0) of level= 0 which corresponds to Figure 2] the output amplitude119860(1198862) varies monotonically as the noise intensity 1198862 varieswhich means the SR phenomenon is impossible Meanwhilein the shaded regions it corresponds to the traditional SR


X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1

Figure 2 The phase diagram for the stochastic resonance behaviors of the output amplitude 119860(1198862) at 12059620 = 1 1198601 = 1 119867 = 055 1198602 = 1120574 = 03 and V = 001 and the values of Level in Figure 2 reflect the number of SR peaks for different combination of119863 andΩ

05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)

Figure 3 The output amplitude 119860 versus the noise intensity 1198862 with various (119863Ω) be chose through Figure 2 with parameters 12059620 = 11198601 = 1 1198602 = 1 V = 001119867 = 055 and 120574 = 03 and (a)119863 = 03 and Ω = 095 (b)119863 = 09 and Ω = 02 (c)119863 = 13 and Ω = 06phenomenon taking place and two phases can be discernedin the resonant domain

(1) The light shaded region (i) corresponds to the single-peak SR phenomenon [see the domain of level = 1which corresponds to Figure 2]

(2) The dark shaded region (ii) corresponds to thedouble-peaks SR phenomenon [see the domain oflevel = 2 which corresponds to Figure 2]

From Figure 2 we can find that when the drivingfrequency Ω is large enough [Ω gt 095] or small enough[Ω lt 001] it is impossible to induce the SR phenomenon

In Figure 3 we plot the curves given by (24) and (28)in which the dependence of the output amplitude 119860 on thenoise intensity 1198862 for different values of the systemparameters(119863 Ω) can be chosen from Figure 2 to verify the correctnessof the results shown in Figure 2 As shown in Figure 3(a)when 119863 = 03 and Ω = 095 which belongs to the unshadeddomain in Figure 2 the output amplitude 119860(1198862) monotonicbehavior decreased with the increasing of 1198862 which meansthe SR phenomenon does not take place In Figure 3(b) when119863 = 09 and Ω = 02 which corresponds to the lightgrey domain in Figure 2 the curve shows that the outputamplitude 119860(1198862) attains a maximum value at some values of1198862 that is the single-peak SR phenomenon takes place by


X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D

Figure 4The phase diagram for the stochastic resonance behaviors of the output amplitude119860(1198862) at12059620 = 11198601 = 1 120572 = 06 120591 = 021198602 = 1120574 = 03 and V = 005 and the values of level in Figure 3 reflect the number of SR peaks for different combination of 119863 andΩ

increasing 1198862 Furthermore one can see from Figure 3(c) thatthe double-peaks SR phenomenon happens for the reasonthat the parameter combination of 119863 = 13 and Ω = 06belongs to the dark grey domain in Figure 2 It should beemphasized that the double-peaks SR phenomenon happensbecause of the presence of two types of noise external andinternal noise sources The double-peaks SR phenomenoncan take place in biological systems such as neuronal systems[74] in which the internal noise is due to signals coming fromall other neurons and external noise is the environmentalnoise due to its interaction with the neuronal system

The main contribution of this section is as follows withthe help of phase diagram for the SR phenomenon we caneffectively control the SR phenomenon of this generalizedharmonic system in a certain range and further broaden theapplication scope of the SR phenomenon in physics biologyand engineering such as the detection of weak stimuli byspiking neurons in the presence of certain level of noisybackground neural activity [75]

42The Stochastic Resonance Behaviors of GLE with aMittag-Leffler Noise From (25) and (32) we obtain the stabilitycondition of GLE with a Mittag-Leffler noise 120578(119905)

0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)

with 0 lt 120572 lt 2 120591 gt 0It is found from the stability condition (36) that the

critical noise intensity 1198862crMLn is determined by 12059620 V 120574 120591 and120572 In Figure 4 the phase diagram in the 119863 minus Ω plane for theemergence of SR phenomenon of 119860(1198862) at 12059620 = 1 1198601 = 1120572 = 06 120591 = 02 1198602 = 1 120574 = 03 and V = 005 is shownThe same as Figure 2 when parameters (119863Ω) belong tothe unshaded regions (0) the SR phenomenon is impossiblewhen (119863Ω) belong to the light grey regions (i) the single-peak SR phenomenon happens when (119863Ω) belong to the

dark grey regions (ii) the double-peaks SR phenomenontakes place Moreover the sufficiently large driving frequencyΩ [Ω gt 098] or small enough Ω [Ω lt 001] cannot inducethe system to produce SR phenomenon

In Figure 5 we also show the curves given by (24) and (33)in which the dependence of the output amplitude 119860 on thenoise intensity 1198862 for different values of the systemparameters(119863Ω) can be chosen from Figure 4 to verify the correctnessof the results shown in Figure 4

As shown in Figure 5(a) when 119863 = 01 and Ω = 005which belongs to the unshaded domain in Figure 4 theoutput amplitude 119860(1198862) monotonic increase occurs with theincreasing of 1198862 which means the SR phenomenon does nottake place In Figure 5(b) when 119863 = 11 and Ω = 025which corresponds to the light grey domain in Figure 4the curve shows that the output amplitude 119860(1198862) attains amaximum value at some values of 1198862 that is the single-peakSR phenomenon takes place by increasing 1198862 Furthermoreone can see from Figure 5(c) that the double-peaks SRphenomenon happens for the reason that the parametercombination of 119863 = 04 and Ω = 07 belongs to the darkgrey domain in Figure 4

5 Conclusions

To summarize in this paper we explore the SR phenomenonin a generalized Langevin equation with multiplicativeperiodically modulated noises and external periodic forceMoreover the system internal noise is modeled as a frac-tional Gaussian noise and aMittag-Leffler noise respectivelyWithout loss of generality the fluctuations of system intrinsicfrequency aremodeled as amultiplicative dichotomous noiseBy the use of the stochastic averagingmethod and the Laplacetransform technique we obtain the exact expression of theoutput amplitude 119860 given by (24)


0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)

Figure 5The output amplitude119860 versus the noise intensity 1198862 with various (119863Ω) chosen through Figure 4 with parameters12059620 = 11198601 = 1120572 = 06 120591 = 02 1198602 = 1 120574 = 03 and V = 005 and (a)119863 = 01 andΩ = 005 (b)119863 = 11 andΩ = 025 (c)119863 = 04 and Ω = 07

We focus on the various nonmonotonic behaviorsof the output amplitude 119860 with the system parameters120574 12059620 1198862 V 1198601 1198602 119863 and Ω and the parameters of theinternal driven noiseWith the exact expression of the outputamplitude 119860 we find the conventional SR takes place withthe increases of the noise intensity 1198862 for fractional Gaussiannoise and Mittag-Leffler noise respectively Moreover wegive the phase diagram in119863minusΩ plane for the emergence of SRphenomenon of 119860(1198862) and find the single-peak and double-peaks SR phenomena

We believe all the results in this paper not only supplythe theoretical investigations of the generalized harmonicoscillator subject to multiplicative periodically modulatednoises and external periodic force but also can suggest someexperimental anomalous diffusion results in physical andbiological applications in the future [76]

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Key Program of NationalNatural Science Foundation of China (Grant nos 11171238and 11601066) and Natural Science Foundation for the Youth(Grant nos 11501386 and 11401405)

References

[1] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[2] R Benzi G Parisi and A Vulpiani ldquoStochastic resonance inclimatic changerdquo Tellus vol 34 article 10 1982

[3] R Benzi ldquoStochastic resonance from climate to biologyrdquoNonlinear Processes in Geophysics vol 17 no 5 pp 431ndash4412010

[4] M D McDonnell and D Abbott ldquoWhat is stochastic reso-nance Definitions misconceptions debates and its relevanceto biologyrdquo PLoS Computational Biology vol 5 no 5 Article IDe1000348 2009

[5] L Gammaitoni P Hanggi P Jung and FMarchesoni ldquoStochas-tic resonance a remarkable idea that changed our perception ofnoiserdquo European Physical Journal B vol 69 no 1 pp 1ndash3 2009

[6] T Wellens V Shatokhin and A Buchleitner ldquoStochastic reso-nancerdquo Reports on Progress in Physics vol 67 no 1 pp 45ndash1052004

[7] R N Mantegna B Spagnolo L Testa and M TrapaneseldquoStochastic resonance in magnetic systems described bypreisach hysteresis modelrdquo Journal of Applied Physics vol 97no 10 Article ID 10E519 2005

[8] R N Mantegna and B Spagnolo ldquoStochastic resonance in atunnel diode in the presence of white or coloured noiserdquo IlNuovo Cimento D vol 17 no 7-8 pp 873ndash881 1995

[9] A Behera and S F C OrsquoRourke ldquoThe effect of correlatednoise in a Gompertz tumor growth modelrdquo Brazilian Journalof Physics vol 38 no 2 pp 272ndash278 2008

[10] A Fiasconaro A Ochab-Marcinek B Spagnolo and EGudowska-Nowak ldquoMonitoring noise-resonant effects in can-cer growth influenced by external fluctuations and periodic


treatmentrdquo European Physical Journal B vol 65 no 3 pp 435ndash442 2008

[11] V Berdichevsky and M Gitterman ldquoStochastic resonance inlinear systems subject to multiplicative and additive noiserdquoPhysical Review E vol 60 no 2 pp 1494ndash1499 1999

[12] M Gitterman ldquoHarmonic oscillator with fluctuating dampingparameterrdquo Physical Review E vol 69 no 4 Article ID 0411012004

[13] J-H Li and Y-X Han ldquoPhenomenon of stochastic resonancecaused by multiplicative asymmetric dichotomous noiserdquo Phys-ical Review E vol 74 no 5 Article ID 051115 2006

[14] D Valenti A Fiasconaro and B Spagnolo ldquoStochastic reso-nance andnoise delayed extinction in amodel of two competingspeciesrdquo Physica A Statistical Mechanics and Its Applicationsvol 331 no 3-4 pp 477ndash486 2004

[15] A La Cognata D Valenti A A Dubkov and B SpagnololdquoDynamics of two competing species in the presence of Levynoise sourcesrdquo Physical Review E vol 82 no 1 Article ID 0111212010

[16] N V Agudov A V Krichigin D Valenti and B SpagnololdquoStochastic resonance in a trapping overdamped monostablesystemrdquo Physical Review E vol 81 no 5 Article ID 051123 8pages 2010

[17] J K Douglass L Wilkens E Pantazelou and F Moss ldquoNoiseenhancement of information transfer in crayfish mechanore-ceptors by stochastic resonancerdquo Nature vol 365 no 6444 pp337ndash340 1993

[18] K Wiesenfeld and F Moss ldquoStochastic resonance and thebenefits of noise from ice ages to crayfish and SQUIDsrdquoNaturevol 373 no 6509 pp 33ndash36 1995

[19] M Gitterman ldquoClassical harmonic oscillator with multiplica-tive noiserdquo Physica A vol 352 no 2ndash4 pp 309ndash334 2005

[20] L Gammaitoni F Marchesoni and S Santucci ldquoStochasticresonance as a bona fide resonancerdquoPhysical Review Letters vol74 no 7 pp 1052ndash1055 1995

[21] L Gammaitoni P Hanggi P Jung and FMarchesoni ldquoStochas-tic resonancerdquoReviews ofModern Physics vol 70 no 1 pp 223ndash287 1998

[22] B Dybiec and E Gudowska-Nowak ldquoLevy stable noise-induced transitions stochastic resonance resonant activationand dynamic hysteresisrdquo Journal of StatisticalMechanicsTheoryand Experiment vol 2009 Article ID P05004 16 pages 2009

[23] B Dybiec K Sczepaniec and I M Sokolov ldquoNon-equilibriumescape problems under bivariate 120572-stable noisesrdquo Acta PhysicaPolonica B vol 47 no 5 pp 1327ndash1339 2016

[24] G M Viswanathan E P Raposo and M G E da LuzldquoLevy flights and superdiffusion in the context of biologicalencounters and random searchesrdquo Physics of Life Reviews vol5 no 3 pp 133ndash150 2008

[25] A A Dubkov B Spagnolo and V V Uchaikin ldquoLevy flightsuperdiffusion an introductionrdquo International Journal of Bifur-cation and Chaos in Applied Sciences and Engineering vol 18no 9 pp 2649ndash2672 2008

[26] A Dubkov and B Spagnolo ldquoLangevin approach to Levyflights in fixed potentials exact results for stationary probabilitydistributionsrdquo Acta Physica Polonica B vol 38 no 5 pp 1745ndash1758 2007

[27] A drsquoOnofrio Bounded Noises in Physics Biology and Engineer-ing Springer New York NY USA 2013

[28] L F Richardson ldquoAtmospheric diffusion shown on a distanceneighbor graphrdquo Proceedings of the Royal Society A vol 110 no765 pp 709ndash737 1926

[29] B OrsquoShaughnessy and I Procaccia ldquoAnalytical solutions fordiffusion on fractal objectsrdquo Physical Review Letters vol 54 no5 pp 455ndash458 1985

[30] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A vol 211 no1 pp 13ndash24 1994

[31] L Acedo and S Bravo Yuste ldquoShort-time regime propagator infractalsrdquo Physical Review E - Statistical Physics Plasmas Fluidsand Related Interdisciplinary Topics vol 57 no 5 pp 5160ndash51671998

[32] DCamposVMendez and J Fort ldquoDescription of diffusive andpropagative behavior on fractalsrdquo Physical Review E vol 69 no3 Article ID 031115 2004

[33] D H N Anh P Blaudeck K H Hoffmann J Prehl and STarafdar ldquoAnomalous diffusion on random fractal compositesrdquoJournal of Physics A Mathematical and Theoretical vol 40 no38 pp 11453ndash11465 2007

[34] F Santamaria S Wils E De Schutter and G J AugustineldquoAnomalous diffusion in purkinje cell dendrites caused byspinesrdquo Neuron vol 52 no 4 pp 635ndash648 2006

[35] X J Zhou Q Gao O Abdullah and R L Magin ldquoStudies ofanomalous diffusion in the human brain using fractional ordercalculusrdquo Magnetic Resonance in Medicine vol 63 no 3 pp562ndash569 2010

[36] S Hapca J W Crawford and I M Young ldquoAnomalous dif-fusion of heterogeneous populations characterized by normaldiffusion at the individual levelrdquo Journal of the Royal SocietyInterface vol 6 no 30 pp 111ndash122 2009

[37] D V Nicolau Jr J F Hancock and K Burrage ldquoSources ofanomalous diffusion on cell membranes a Monte Carlo studyrdquoBiophysical Journal vol 92 no 6 pp 1975ndash1987 2007

[38] G Guigas and M Weiss ldquoSampling the cell with anomalousdiffusionmdashthe discovery of slownessrdquo Biophysical Journal vol94 no 1 pp 90ndash94 2008

[39] A M Edwards R A Phillips N W Watkins et al ldquoRevisitingLevy flight search patterns of wandering albatrosses bumble-bees and deerrdquo Nature vol 449 no 7165 pp 1044ndash1048 2007

[40] N Yang G Zhang and B Li ldquoViolation of Fourierrsquos law andanomalous heat diffusion in silicon nanowiresrdquo Nano Todayvol 5 no 2 pp 85ndash90 2010

[41] S Ozturk Y A Hassan and V M Ugaz ldquoInterfacial complexa-tion explains anomalous diffusion in nanofluidsrdquo Nano Lettersvol 10 no 2 pp 665ndash671 2010

[42] V R Voller and C Paola ldquoCan anomalous diffusion describedepositional fluvial profilesrdquo Journal of Geophysical ResearchEarth Surface vol 115 no F2 2010

[43] A A Tateishi E K Lenzi L R da Silva H V Ribeiro S Picoliand R S Mendes ldquoDifferent diffusive regimes generalizedLangevin and diffusion equationsrdquo Physical Review E vol 85no 1 Article ID 011147 6 pages 2012

[44] J Klafter and I M Sokolov ldquoAnomalous diffusion spreads itswingsrdquo Physics World vol 18 no 8 pp 29ndash32 2005

[45] H Ness L Stella C D Lorenz and L Kantorovich ldquoApplica-tions of the generalized Langevin equation towards a realisticdescription of the bathsrdquo Physical Review B vol 91 no 1 ArticleID 014301 pp 1ndash15 2015

[46] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968

[47] S C Kou and X S Xie ldquoGeneralized Langevin equation withfractional Gaussian noise subdiffusion within a single protein


moleculerdquo Physical Review Letters vol 93 no 18 Article ID180603 4 pages 2004

[48] W Min B P English G Luo B J Cherayil S C Kou andX S Xie ldquoFluctuating enzymes lessons from single-moleculestudiesrdquo Accounts of Chemical Research vol 38 no 12 pp 923ndash931 2005

[49] W Min G Luo B J Cherayil S C Kou and X S XieldquoObservation of a power-law memory Kernel for fluctuationswithin a single protein moleculerdquo Physical Review Letters vol94 no 19 Article ID 198302 2005

[50] WMin andX S Xie ldquoKramersmodel with a power-law frictionkernel dispersed kinetics and dynamic disorder of biochemicalreactionsrdquo Physical Review E vol 73 no 1 Article ID 010902pp 1ndash4 2006

[51] H Yang G Luo P Karnchanaphanurach et al ldquoProteinconformational dynamics probed by single-molecule electrontransferrdquo Science vol 302 no 5643 pp 262ndash266 2003

[52] S Zhong K Wei S Gao and H Ma ldquoStochastic resonancein a linear fractional Langevin equationrdquo Journal of StatisticalPhysics vol 150 no 5 pp 867ndash880 2013

[53] S Zhong K Wei S Gao and H Ma ldquoTrichotomous noiseinduced resonance behavior for a fractional oscillator withrandom massrdquo Journal of Statistical Physics vol 159 no 1 pp195ndash209 2015

[54] S Zhong H Ma H Peng and L Zhang ldquoStochastic resonancein a harmonic oscillator with fractional-order external andintrinsic dampingsrdquo Nonlinear Dynamics vol 82 no 1-2 pp535ndash545 2015

[55] A D Vinales andM A Desposito ldquoAnomalous diffusion exactsolution of the generalized Langevin equation for harmonicallybounded particlerdquo Physical Review E vol 73 no 1 Article ID016111 4 pages 2006

[56] A D Vinales and M A Desposito ldquoAnomalous diffusioninduced by a Mittag-Leffler correlated noiserdquo Physical ReviewE vol 75 no 4 Article ID 042102 2007

[57] M A Desposito and A D Vinales ldquoMemory effects in theasymptotic diffusive behavior of a classical oscillator describedby a generalized Langevin equationrdquo Physical Review E vol 77no 3 Article ID 031123 2008

[58] M I Dykman D G Luchinsky P V E McClintock N DStein and N G Stocks ldquoStochastic resonance for periodicallymodulated noise intensityrdquo Physical Review A vol 46 no 4 ppR1713ndashR1716 1992

[59] L Cao and D J Wu ldquoStochastic resonance in a linear systemwith signal-modulated noiserdquo Europhysics Letters vol 61 no 5pp 593ndash598 2003

[60] D S Grebenkov andM Vahabi ldquoAnalytical solution of the gen-eralized Langevin equation with hydrodynamic interactionssubdiffusion of heavy tracersrdquo Physical Review E vol 89 no 1Article ID 012130 18 pages 2014

[61] S Burov and E Barkai ldquoCritical exponent of the fractionalLangevin equationrdquo Physical Review Letters vol 100 no 7Article ID 070601 2008

[62] G Y Hu and R F OrsquoConnell ldquo1f noise a nonlinear-generalized-Langevin-equation approachrdquo Physical Review Bvol 41 no 9 pp 5586ndash5594 1990

[63] H-Y Yu DM Eckmann P S Ayyaswamy and R Radhakrish-nan ldquoComposite generalized Langevin equation for Brownianmotion in different hydrodynamic and adhesion regimesrdquoPhysical Review E vol 91 no 5 Article ID 052303 pp 1ndash11 2015

[64] S C Kou ldquoStochastic modeling in nanoscale biophysics subd-iffusion within proteinsrdquoThe Annals of Applied Statistics vol 2no 2 pp 501ndash535 2008

[65] K G Wang and M Tokuyama ldquoNonequilibrium statisticaldescription of anomalous diffusionrdquo Physica A vol 265 no 3pp 341ndash351 1999

[66] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[67] R Kubo ldquoThe fluctuation-dissipation theoremrdquo Reports onProgress in Physics vol 29 no 1 article no 306 pp 255ndash2841966

[68] Y S Mishura Stochastic Calculus for Fractional BrownianMotion and Related Processes vol 1929 of Lecture Notes inMathematics Springer Berlin Germany 2008

[69] K Laas R Mankin and E Reiter ldquoInfluence of memory timeon the resonant behavior of an oscillatory system describedby a generalized langevin equationrdquo International Journal ofMathematical Models and Methods in Applied Sciences vol 5no 2 pp 280ndash289 2011

[70] Z-Q Huang and F Guo ldquoStochastic resonance in a fractionallinear oscillator subject to random viscous damping and signal-modulated noiserdquo Chinese Journal of Physics vol 54 no 1 pp69ndash76 2016

[71] F Guo C-Y Zhu X-F Cheng andH Li ldquoStochastic resonancein a fractional harmonic oscillator subject to random mass andsignal-modulated noiserdquo Physica A StatisticalMechanics and ItsApplications vol 459 pp 86ndash91 2016

[72] V E Shapiro and V M Loginov ldquolsquoFormulae of differentiationrsquoand their use for solving stochastic equationsrdquo Physica AStatistical Mechanics and Its Applications vol 91 no 3-4 pp563ndash574 1978

[73] A V Oppenheim A S Willsky and S H Nawab Signals andSystems Prentice Hall Beijing China 2005

[74] J M G Vilar and J M Rubı ldquoStochastic multiresonancerdquoPhysical Review Letters vol 78 no 15 pp 2882ndash2885 1997

[75] J F Mejias and J J Torres ldquoEmergence of resonances in neuralsystems the interplay between adaptive threshold and short-term synaptic plasticityrdquo PLoS ONE vol 6 no 3 Article IDe17255 pp 869ndash880 2011

[76] K Laas R Mankin and A Rekker ldquoConstructive influenceof noise flatness and friction on the resonant behavior of aharmonic oscillatorwith fluctuating frequencyrdquoPhysical ReviewE vol 79 no 5 Article ID 051128 2009

Research ArticleThe Rational Solutions and Quasi-Periodic WaveSolutions as well as Interactions of119873-Soliton Solutions for3 + 1 Dimensional Jimbo-Miwa Equation

Hongwei Yang1 Yong Zhang1 Xiaoen Zhang1 Xin Chen1 and Zhenhua Xu23

1College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China2Key Laboratory of Ocean Circulation and Waves Institute of Oceanology Chinese Academy of Sciences Qingdao 266071 China3Function Laboratory for Ocean Dynamics and Climate Qingdao National Laboratory for Marine Science and TechnologyQingdao 266237 China

Correspondence should be addressed to Zhenhua Xu xzh0532sinacom

Received 7 September 2016 Revised 25 October 2016 Accepted 10 November 2016

Academic Editor Rita Tracina

Copyright copy 2016 Hongwei Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The exact rational solutions quasi-periodic wave solutions and119873-soliton solutions of 3 + 1 dimensional Jimbo-Miwa equation areacquired respectively by using theHirotamethod whereafter the rational solutions are also called algebraic solitarywaves solutionsand used to describe the squall lines phenomenon and explained possible formation mechanism of the rainstorm formation whichoccur in the atmosphere so the study on the rational solutions of soliton equations has potential application value in the atmospherefield the soliton fission and fusion are described based on the resonant solution which is a special form of the119873-soliton solutionsAt last the interactions of the solitons are shown with the aid of119873-soliton solutions

1 Introduction

Some natural phenomena in the physics and in the biol-ogy can be depicted by multitudinous nonlinear partialdifferential equations Therefore the solutions of the partialdifferential equations become the focus points with whichwe are concerned [1] There are various ways to get thesolutions such as the Darboux transformation Backlundtransformation [2 3] Inverse scattering transformationHomogeneous balance method and Traveling wave solution[4ndash6] However the methods mentioned above cannot beexpressing the periodicity of the partial differential equations

Unlike the above method the Hirota method [7 8] playsa crucial role during obtaining the119873-soliton solutions by theperturbation and the quasi-periodic wave solutions based onthe Riemann theta functions [9] Hence it is important torewrite the partial differential equation into the bilinear formswith the help of the variate transformation In 2008 Lambertand Springael [10] proposed an explicit way to construct thebilinear forms for the constant coefficient equation

It is well known that the Hirota method has been widelyapplied to the 1 + 1 dimensional equations and the 2 +1 dimensional equations [11ndash23] but the method is rarelyused to the 3 + 1 dimensional equation As for the 3 +1 dimensional equation the quasi-periodic wave solutionshappening during the arbitrary two spatial variables 119909 119910 119911at one time 119905 or between one spatial variable and the timevariable under the other two spatial variables are constantsOn the other hand the rational solutions have attractedmore and more attention recently [24ndash27] because of theirgraceful structure and potential application value in applieddisciplines The author also applied this kind of rationalsolutions (also called algebraic solitary wave solutions) todiscuss the algebraic Rossby solitary waves and explain theblocking phenomenon which happen in the real atmosphereand ocean [28 29]

In this paper we first introduce the well-known 3 +1 dimensional Jimbo-Miwa equation which has significantefforts in science it was investigated by Jimbo and Miwa in[30] and its one-soliton solutions were studied by Wazwaz



[31 32] according to the Tanh-Coth method and its trav-eling wave solutions were discussed by Ma and Lee [33]by using rational function transformations The methodprovides more systematical and convenient handling of thesolution process of nonlinear equations Lately we presenta brief introduction about the approach and the propertiesof the Bell-polynomial Then the bilinear form of Jimbo-Miwa equation is gained by applying the Bell-polynomialits rational solutions quasi-periodic wave solutions and 119873-soliton solutions are obtained based on the Hirota methodand Riemann theta function Finally the resonant solutionand the interactions of the 119873-soliton solutions are givenunder the Hirota method

2 The Bell-Polynomial

In order to get the 119873-soliton solutions of the nonlinearevolutions equations (NLEES) we must get the bilinear formof the NLEES Lambert et al connected the Bell-polynomialwith the Hirota 119863 operator and give rise to an explicit wayto construct the bilinear form to the NLEES Firstly we arebriefly devoted to the notations of the Bell-polynomial

The definition of the multidimensional Bell-polynomialis as follows

1198841198991119909111989921199092 sdotsdotsdot119899119903119909119903 (119892)equiv 1198841198991 1198992119899119903 (11989211989711199091 119897119903119909119903 (1 le 119897119894 le 119899119894 0 le 119894 le 119903))= 119890minus1198921205971198991119909112059711989921199092 sdot sdot sdot 120597119899119903119909119903119890119892

(1)

where 119892 is a 119862infin multivariablesrsquo functionAs for a special function 119892 with the variables 119909 119911 we give

rise to the following several initial value under the definitionof the multivariables Bell-polynomial

119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)

Then we provide the redefinition of the binary Bell-pol-ynomial as

12059111989911199091119899119903119909119903 (120592 120596)equiv 1198841198991119899119903 (119892)1003816100381610038161003816100381611989211989711199091119897119903119909119903= 12059211989711199091119897119903119909119903 sum119903119894=1 119897119894 is odd12059611989711199091119897119903119909119903 sum119903119894=1 119897119894 is even

(3)

where 120592 and 120596 both are the 119862infin functions with the variables1199091 1199092 119909119903 We set out some initial expressions dependingon (3) as

120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)

The link between the binary Bell-polynomial (3) and theHirota 119863-operator can be presented through a transforma-tional identity

12059111989911199091 119899119903119909119903 (120592 = ln 119865119866 120596 = ln119865119866)= (119865 sdot 119866)minus111986311989911199091 sdot sdot sdot 119863119899119903119909119903119865 sdot 119866

(5)

where the Hirota operator is defined by

11986311989911199091 sdot sdot sdot 119863119899119903119909119903119865 sdot 119866= (1205971199091 minus 12059711990910158401)1198991 sdot sdot sdot (120597119909119903 minus 1199091015840119903) 119865 (1199091 119909119903)times 119866 (11990910158401 1199091015840119903)1003816100381610038161003816100381611990910158401=1199091 1199091015840119903=119909119903

(6)

In particular when 119865 = 119866 (5) can be read as

119865minus211986311989911199091 sdot sdot sdot 1198631198991199031199091199031198652 = 12059111989911199091119899119903119909119903 (120592 = 0 120596 = 2 ln119865)

=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)

In (7) the Bell-polynomial is equal to the119875-polynomial whensum119903119894=1 119899119894 is even and then we give the first lower order 119875-polynomial

1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)

As for the NLEES we can rewrite them as bilinear from withthe aid of the119875-polynomial and show the119873-soliton solutionsand the quasi-periodic wave solutions

3 The Bilinear Form of the 3 + 1 DimensionalJimbo-Miwa Equation

The 3 + 1 dimensional Jimbo-Miwa equation is

119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)

Letting 119906 = 119902119909119909 and substituting it into (9) and integratingtwice with respect to 119909 yield

119902119909119909119909119910 + 3119902119909119909119902119909119910 + 2119902119910119905 minus 3119902119909119911 minus 120582 = 0 (10)where 120582 is an arbitrary integral constant So in terms of the119875-polynomial (10) can be written as

1198753119909119910 (119902) + 2119875119910119905 (119902) minus 3119875119909119911 (119902) minus 120582 = 0 (11)Giving a change of dependent variables

119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)


then we can acquire the bilinear form of (9) as

119869 (119863119909 119863119910 119863119911 119863119905) equiv (1198633119909119863119910 + 2119863119910119863119905 minus 3119863119909119863119911) 119865sdot 119865 minus 1205821198652 = 0 (13)

where the definition of the generalized bilinear119863 operator is

119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)120572119894119901120572119895119901 120597119898+119899minus119894minus119895119865 (119909 119905)120597119909119898minus119894120597119905119899minus119895

120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)

where 120572119904119901 is calculated as follows

120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901

11986321199051198632119910119865 sdot 119865 = 2119865119910119905119865 minus 211986511990511986511991011986321199091198632119911119865 sdot 119865 = 2119865119909119911119865 minus 2119865119909119865119911119863321199091198632119910119865 sdot 119865 = 21198653119909119910119865 minus 21198653119909119865119910 minus 61198652119909119910119865119909

+ 61198651199091199101198652119909

(15)

Consequently let 120582 = 0 then the linear combination of (13)constructs the 3 + 1 dimensional Jimbo-Miwa equation as

(1198633119909119863119910 + 2119863119910119863119905 minus 3119863119909119863119911) 119865 sdot 119865= 21198653119909119910119865 minus 21198653119909119865119910 minus 61198652119909119910119865119909 + 61198651199091199101198652119909+ 4119865119910119905119865 minus 4119865119905119865119910 minus 6119865119909119911119865 + 6119865119909119865119911

(16)

4 The Rational Solutions of the 3 + 1Dimensional Jimbo-Miwa Equation

In this section we use the symbolic computation with Mapleand obtain polynomial solutions whose degree of 119909 is lessthan 3 and degrees of 119910 119911 and 119905 are less than 2 to the 3 +1 dimensional Jimbo-Miwa equation

119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)

where 119888119894119895rsquos are constants and we acquire 36 classes of polyno-mial solutions to (16) Among the 36 classes of solutions we

enumerate 13 classes of solutions and see Appendix A wherethe involved constants 119888119894119904119897119895rsquos are arbitrary provided that thesolutions are meaningful We can confirm that there are 13distinct classes of rational solutions generated from (12) to the3 + 1 dimensional Jimbo-Miwa equation (9) by consideringthe transformation of the coefficient 119888119894119904119897119895 for the detailedexpression of rational solutions see Appendix B

These above-mentioned solutions are very tedious anddifficult to apply in other subjects here we obtain somereduced form solutions 1199061 can be reduced to

1198801 = minus (14711990541199094 + 16811990531199094 + 8411990531199093 + 9011990521199094minus 421199054119909 + 4811990521199093 + 91199054 + 241199051199094 minus 481199053119909 + 1811990521199092+ 121199051199093 + 31199094 minus 601199052119909 + 61199053 minus 24119905119909 + 41199052 minus 2119905minus 6119909 + 1) (711990521199093 + 41199051199093 + 31199052119909 + 31199051199092 + 1199093 + 1199052+ 2119905119909 + 119905 + 119909 + 1)minus2

(18)

when 119888119894119904119897119895 = 1 + 119894119904119897119895 The picture of the solution (18) ispresented in Figure 1

In the same way we obtain reduced form solution of 1199065 asfollows

1198805 = (minus038881199094 minus 051841199093119910 minus 0259211990921199102minus 021601199091199102 + 0435456119909 + 0577152119910+ 10368119909119910 + 008281199102 minus 05184) (0361199093+ 0361199092119910 + 030119909119910 + 02016 minus 072119909 + 024119910)minus2

(19)

when 119888119894119904119897119895 = (1 + 119894 + 119904 + 119897 + 119895)100 and 119905 = 001 The pictureof the solution (19) is presented in Figure 2

For the solution 11990611 we have11988011 = minus (6119905 + 5)2

(6119909119905 + 5119909 + 4)2 (20)

when 119888119894119904119897119895 = 1 + 119894 + 119904 + 119897 + 119895 The picture of the solution (20)is presented in Figure 3

For the solution 11990612 we get11988012 = minus 1

(119909 + 119910)2 (21)

when 119888119894119904119897119895 = 1 + 119894119904 + 119897119895 The picture of the solution (21) ispresented in Figure 4

Remark 1 In fact these above-mentioned rational solutionsare greatly different from those common soliton solutions asthe form ldquosech2rdquo The latter describes that from the balancebetween nonlinearity and dispersion it is possible to havesteady waves of a permanent form which are called classicalsolitary waves while as we know these solutions which arederived in the paper can be used to describe algebraic solitarywaves [28 29] During propagation this kind of solitary


minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)

minus40 minus20 0 20 40minus40

minus30

minus20

minus10

0

10

20

30

40

x

t

(b)

Figure 1 Pictures of (18) 3D plot (a) and density plot (b)

minus40minus2002040

minus40minus2002040minus60

minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)


waves will have fission and form an interesting phenomenonsolitary waves in a line that the big amplitude solitary waveis at the front and the small amplitude solitary wave is insubsequent which is deserved in the real atmosphere inthe process of thunderstorm and called squall lines So therational solutions of soliton equations are used to explainpossible formation mechanism of the rainstorm formationSo the study on the rational solutions of soliton equations haspotential application value in the atmosphere field

5 The Quasi-Periodic Wave Solutions of the3 + 1 Dimensional Jimbo-Miwa Equation

In this section we want to get the quasi-periodic wavesolution of the 3 + 1 dimensional Jimbo-Miwa equation

by applying the Hirota method Long time ago the thetafunctions have been systematically used to constructmultiplequasi-periodic solutions [9 34] Hence we let the Riemannfunction of (9) be

119865 = 119899=infinsum119899=minusinfin

1198902120587119894119899120585+1205871198941198992120591 (22)

where 119899 isin 119885 120591 isin 119862 Im 120591 gt 0 and 120585 = 119896119909 + 119897119910 + 119886119911 +119908119905 in addition the parameters 119896 119897 119886 119908 are constant to bedetermined Inserting (22) into (13) we have

119869119865 sdot 119865 = 119869 (119863119909 119863119910 119863119911 119863119905)119899=infinsum119899=minusinfin

1198902120587119894119899120585+1205871198941198992120591

sdot 119898=infinsum119898=minusinfin

1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)


= 119899=infinsum119899=minusinfin

119898=infinsum119898=minusinfin

119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]

sdot 119890120587119894(1198992+(119902minus119899)2)120591 1198902120587119894119902120585 = 119902=infinsum119902=minusinfin

119869 (119902) 1198902120587119894119902120585(23)

Under the calculation of (23) we can denote that

119869 (119902) = 119899=infinsum119899=minusinfin

119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591

= ℎ=infinsumℎ=minusinfin

119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)

sdot 119890120587119894(1198992+(119902minusℎminus2)2)120591 sdot 1198902120587119894(119902minus1)120591

(24)

where 119861 = 2ℎ minus 119902 + 2 119902 = 119898 + 119899 and ℎ = 119899 minus 1From the characters of (24) we can get the following

recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)

If we set 119869(0) = 0 and 119869(1) = 0 it can satisfy (13) that is


(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum

119899=minusinfin(16 (2119899 minus 1)4 12058741198963119897 minus 81205872 (2119899 minus 1)2 119897119908

+ 121205872 (2119899 minus 1)2 119886119896 minus 120582) 119890120587119894(21198992minus2119899+1)120591 = 0

(26)

With the purpose of computational convenience we can set

11990211 = minus119899=infinsum119899=minusinfin

321205872119899211989711989021205871198941198992120591

11990212 =119899=infinsum119899=minusinfin

(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)


Then (26) can be written as

11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)

The parameters 119908 120582 can be given by (28) as

119908 = 1199021211990231 minus 11990221119902131199022111990211 minus 1199023111990222 120582 = 1199021211990211 minus 11990222119902131199022111990211 minus 1199023111990222

(29)

Therefore we can obtain the quasi-periodic wave solution of(9) as

119906 = 2 (ln119865)119909119909 (30)

where 119865 satisfies (22) meanwhile 119908 and 120582 accord with (28)The picture of the quasi-periodic wave solutions of (9) can beshown in Figure 5

6 The119873-Soliton Solutions of the 3 + 1Dimensional Jimbo-Miwa Equation

In [35] a multiple Exp-function method was proposed toobtain the multiwave solutions and multisoliton solutionsHere we can give rise to the 119873-soliton solutions of (9)by applying the Hirota direct method and perturbationapproach With the variable transformation 119906 = 2(ln119865)119909119909(9) has been changed into the bilinear formWe let 120582 = 0 and

119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)

Substitute (31) into (13) and compare the power of 120576 basingthe traditional Hirota method we can present the119873-solitonsolution as follows

119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905

119890119860119894119895 = 2 (119897119895 minus 119897119894) (119908119894 minus 119908119895) + 3 (119886119894 minus 119886119895) (119896119894 minus 119896119895) + (119896119894 minus 119896119895)3 (119897119895 minus 119897119894)

2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)

and then we list the one-soliton solution and the two-solitonsolution

(1) One-Soliton Solution

119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)

where the coefficients 119896 119897 119886 119908 satisfy 2119897119908+1198963119897minus3119886119896 = 0 andthen the one-soliton solution is

119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)

Figure 5 One quasi-periodic wave solution to (9) with the parameters 119896 = 01 119897 = 01 119886 = 0 and 120591 = 119894 Perspective view of the wave (a)overhead view of the wave (b) with contour plot shown (The red lines are crests and the blue lines are troughs)

(2) Two-Soliton Solution

119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where

1205781 = 1198961119909 + 1198971119910 + 1198861119911 + 11990811199051205782 = 1198962119909 + 1198972119910 + 1198862119911 + 119908211990511989011986012 = 2 (1198972 minus 1198971) (1199081 minus 1199082) + 3 (1198861 minus 1198862) (1198961 minus 1198962) + (1198961 minus 1198962)

3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)

Similarly the coefficients 119896119894 119897119894 119886119894 119908119894 (119894 = 1 2) satisfy 2119897119894119908119894 +1198963119894 119897119894 minus 3119886119894119896119894 = 0 and then the two-soliton solution is

119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)

As for the two-soliton solutionwhen the coefficient 119890119860119894119895 =0 the soliton solution can be changed into the resonantsolution whose propagation will be shown in Figure 6

From Figure 6 we can see that in Figure 6(a) therehappened resonance collision to the two-soliton solution andappeared the third soliton in Figure 6(b) occurs the solitonfusionwhose amplitude has been changed after the resonancecollision

When the coefficient 119890119860119894119895 = 0 there will happen thesoliton pursue collision due to the different soliton speed aswe all know that the wave with the faster speed will catch upwith the slower speed wave then collision of the two-solitoncan be happening and the picture of the pursue collision willbe shown in Figure 7

7 Conclusion

In this paper we first introduce a 3 + 1 dimensional Jimbo-Miwa equation and get its bilinear form rational solutionsquasi-periodic wave solutions and119873-soliton solutions basedon the Hirota method and the theta function Afterwardswe analyze the rational solutions and quasi-periodic wavesolutions and draw the propagation picture Finally weexplain the interaction of the 119873-soliton solutions and get aconclusion that when the coefficient 119890119860119894119895 = 0 the solitonsolution can be turned into the resonant solution after theresonance collision and there will appear the soliton fusionphenomenon when the coefficient 119890119860119894119895 = 0 there willappear the pursue collision that is the soliton with the fasterspeed will catch up with the soliton with the slower speedafter the collision and the two-soliton solution will continuespreading in the previous speed and the direction Althoughwe acquire the solutions of the 3 + 1 dimensional Jimbo-Miwa equation the integrability [36] of this equation is notdiscussed the problem is worthy of exploring


minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)

Figure 6 The resonant solution to (9) with the parameters 1198961 = 06 1198962 = 08 1198971 = 1198972 = 1 and 1198861 = 1198862 = 0

minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02

Figure 7 The pursue collision of (35) with 1198961 = 06 1198962 = 08 1198971 = 1 1198972 = 2 and 1198861 = 1198862 = 0


Appendix

A The Expression of 1198651198651 = 119905211990931199101199111198883112 + 11990511990931199101199111198883111 + 11990521199091199101199111198881112 + 11990511990921199101199111198882111+ 11990931199101199111198883110 + 1199051199091199101199111198881111 + 1199051199101199111198880111 + 1199091199101199111198881110+ 1199101199111198880110 + 11990521199101199111198880112

1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052

1198882111minus 1198881011 (11988801121198882111 minus 11988811111198881112) 11991111990511988811121198882111minus (1198880112

21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)

B The Rational Solutions of (3 + 1)-Dimensional Jimbo-Miwa Equation

1199061 = minus (31199054119909411988831122 + 61199053119909411988831111198883112+ 41199053119909311988821111198883112 + 61199052119909411988831101198883112+ 3119909411988831102 minus 6119905411990911988801121198883112+ 41199052119909311988821111198883111 + 6119905119909411988831101198883111 + 119905411988811122minus 6119905311990911988801111198883112 minus 6119905311990911988801121198883111+ 2119905311990911988811121198882111 + 21199052119909211988821112+ 4119905119909311988821111198883110 + 31199052119909411988831112minus 2119905311988801121198882111 + 2119905311988811111198881112minus 6119905211990911988801101198883112 minus 6119905211990911988801111198883111 + 119905211988811112+ 2119905211990911988811111198882111 minus 2119905211988801111198882111+ 2119905211988811101198881112 minus 6119905211990911988801121198883110 + 11988811102minus 611990511990911988801101198883111 minus 611990511990911988801111198883110+ 211990511990911988811101198882111 + 211990511988811101198881111 minus 611990911988801101198883110minus 211990511988801101198882111) (119905211990931198883112 + 11990511990931198883111+ 11990521199091198881112 + 11990511990921198882111 + 11990931198883110 + 11990521198880112+ 1199051199091198881111 + 1199051198880111 + 1199091198881110 + 1198880110)minus2

1199062 = minus119888311011988821113 (2119905211990921199102119888211151198883110+ 4119905119909311991021198882111411988831102+ 2119905211990911991021198881111119888211141198883110 + 3119909411991021198882111311988831103minus 2119905211991031198880211119888211141198883110minus 4119905119909119910311988802111198882111311988831102 minus 211990521199102119888111011988821115minus 1199052119910211988811112119888211131198883110 minus 12119905211991021198882111411988831102minus 411990511990911991021198881110119888211141198883110minus 61199051199091199102119888111121198882111211988831102


minus 3611990511990911991021198882111311988831103+ 41199051199103119888021111988811111198882111211988831102+ 6119909119910311988802111198881111119888211111988831103+ 119910411988802112119888211111988831103 minus 21199052119910119888101011988821115minus 41199051199091199101198881010119888211141198883110+ 4119905119910211988811101198881111119888211131198883110+ 2119905119910211988811113119888211111988831102+ 18119905119910211988811111198882111211988831103+ 61199091199102119888111011988811111198882111211988831102+ 611990911991021198881111311988831103 + 5411990911991021198881111119888211111988831104+ 21199103119888021111988811101198882111211988831102+ 411990511991011988810101198881111119888211131198883110+ 6119909119910119888101011988811111198882111211988831102+ 11988810102119888211131198883110+ 21199102119888021111988810101198882111211988831102+ 119910211988811102119888211131198883110+ 211991011988810101198881110119888211131198883110) (1199051199092119910119888211141198883110+ 11990931199101198882111311988831102 + 1199051199091199101198881111119888211131198883110+ 11990511991021198880211119888211131198883110 + 119909119910211988802111198882111211988831102+ 119905119910119888111011988821114 + 11990511991011988811112119888211121198883110+ 61199051199101198882111311988831102 + 1199091199101198881110119888211131198883110minus 119910211988802111198881111119888211111988831102 + 119905119888101011988821114minus 1199101198881111311988831102 minus 11991011988811101198881111119888211121198883110minus 91199101198881111119888211111988831103 minus 11988810101198881111119888211121198883110+ 1199091198881010119888211131198883110)minus2

1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112

+ 1211991011991111988811123119888121121198882111 + 411991121198881112411988812112minus 1811990511988801121198881112211988821113 + 1811990511988811111198881112311988821112minus 54119888111211988821115 + 1811990911988811111198881112211988821113minus 1811991011988801121198881112119888121111988821113+ 18119910119888111111988811122119888121111988821112minus 12119911119888011211988811122119888121111988821112+ 1211991111988811111198881112311988812111198882111+ 9119888111121198881112211988821112minus 1811988801121198881111119888111211988821113 + 181198880112211988821114)sdot (3119905119909119888111231198882111 + 2119909119911119888111231198881211+ 311990921198881112211988821112 + 31199091199101198881112211988812111198882111+ 31199051198880112119888111221198882111 minus 31198880112211988821112+ 31199091198881111119888111221198882111+ 31199101198880112119888111211988812111198882111+ 31198880112119888111111988811121198882111 + 21199111198880112119888111221198881211+ 9119888111211988821113)minus2

1199064 = minus1198882111211988811122 (21199051199091199102119888111231198882111+ 2119909211991021198881112211988821112 + 211990911991031198881112211988812111198882111+ 11991041198881112211988812112 minus 211990511991021198880112119888111221198882111+ 21199051199102119888111111988811123 + 211991021198880112211988821112+ 211990911991021198881111119888111221198882111+ 211991031198880112119888111211988812111198882111+ 21199091199101198881011119888111221198882111 + 1199052119910211988811124minus 211991021198880112119888111111988811121198882111+ 211991021198881011119888111221198881211 + 11991021198881111211988811122+ 61199102119888111211988821113 + 21199101198880112119888101111988811121198882111+ 1198881011211988811122) (119905119909119910119888111231198882111+ 1199051199102119888111231198881211 + 11990921199101198881112211988821112+ 11990911991021198881112211988812111198882111 + 1199051199101198880112119888111221198882111+ 1199091199101198881111119888111221198882111minus 11991021198880112119888111211988812111198882111 + 119905119888101111988811123


minus 1199101198880112211988821112 + 11991021198881111119888111221198881211+ 1199091198881011119888111221198882111 minus 3119910119888111211988821113+ 1198881011119888111111988811122 minus 1198880112119888101111988811121198882111+ 1199101198880112119888111111988811121198882111)minus2

1199065 = (minus311990941198882111211988830112 minus 41199093119910119888211131198883011minus 21199091199102119888111111988821113 + 361199051199091198882111211988830112minus 21199092119910211988821114 + 12119905119910119888211131198883011+ 121199091199101198880111119888211121198883011 + 21199102119888011111988821113minus 11991021198881111211988821112 + 61199091198880011119888211121198883011+ 2119910119888001111988821113 + 611990511988821111198883011minus 91198880111211988830112) (119909311988821111198883011 + 119909211991011988821112+ 61199101198880111119888111111988821111198883011 minus 311990911988801111198883011+ 11991011988801111198882111 + 11988800111198882111 + 11990911991011988811111198882111)minus2

1199066 = minus (27119909411988830114 + 361199093119910119888211111988830113+ 18119909211991021198882111211988830112 minus 1811991031198880211119888211111988830112+ 61199091199103119888211131198883011 + 119910411988821114minus 541199091199102119888021111988830113 + 16211990511990911988830114+ 54119905119910119888211111988830113 minus 54119909119910119888011111988830113+ 108119909119911119888211111988830113 minus 1811991021198880111119888211111988830112+ 361199101199111198882111211988830112 minus 54119909119888001111988830113minus 181199101198880011119888211111988830112) (minus3119909311988830112minus 119909119910211988821112 + 611991111988821111198883011 minus 3119910211988802111198883011+ 911990511988830112 minus 311991011988801111198883011 minus 311988800111198883011minus 3119909211991011988821111198883011)minus2

1199067 = 31198883111 (minus9119909411991021198882111211988831113minus 12119909311991021198882111311988831112+ 18119909119910211988801111198882111211988831112+ 10811990511990911991021198882111211988831113 minus 611990921199102119888211141198883111minus 2711991041198880211211988831113 + 3611990511991021198882111311988831112minus 611990911991021198881111119888211131198883111minus 18119910311988802111198881111119888211111988831112

+ 611991031198880211119888211131198883111 + 611991021198880111119888211131198883111minus 18119910211988802111198881011119888211111988831112minus 3119910211988811112119888211121198883111minus 611991011988810111198881111119888211121198883111 + 2119910119888101111988821114minus 311988810112119888211121198883111) (119888101111988821112+ 3119909119888101111988821111198883111 + 91199091199102119888021111988831112+ 18119905119910119888211111988831112 + 3119909119910119888111111988821111198883111+ 31199092119910119888211121198883111 + 31199102119888021111988821111198883111+ 3119910119888011111988821111198883111 + 31199093119910119888211111988831112)minus2

1199068 = 31198883111 (minus121199093119888211111988831112 minus 61199092119888211121198883111minus 31199102119888121121198883111 minus 6119909119888111111988821111198883111+ 10811990511990911988831113 minus 9119909411988831113 + 36119905119888211111988831112+ 18119909119888011111988831112 minus 6119910119888111111988812111198883111+ 2119910119888121111988821112 + 6119888011111988821111198883111minus 3119888111121198883111) (3119909311988831112 + 3119909211988821111198883111+ 311990911991011988812111198883111 + 1811990511988831112 + 311990911988811111198883111+ 11991011988812111198882111 + 311988801111198883111)minus2

1199069 = minus119910 (3119909411991011988831112 + 4119909311991011988821111198883111+ 1811990511990911991011988831112 minus 6119909119910211988802111198883111+ 611990511991011988821111198883111 + 2119909211991011988821112 minus 611990911991011988801111198883111+ 211990911991011988811111198882111 minus 2119910211988802111198882111minus 611990911988800111198883111 minus 211991011988801111198882111 + 11991011988811112minus 211988800111198882111) (minus11990931199101198883111 minus 11990921199101198882111+ 31199051199101198883111 minus 1199091199101198881111 minus 11991021198880211 minus 1199101198880111minus 1198880011)minus2

11990610 = 31198883111 (minus91199094119910211988831113 minus 181199094119910119888301111988831112minus 1211990931199102119888211111988831112 + 108119905119909119910211988831113minus 91199094119888301121198883111 minus 611990921199102119888211121198883111+ 216119905119909119910119888301111988831112 + 61199102119888011111988821111198883111+ 361199051199102119888211111988831112 + 181199091199102119888011111988831112minus 61199091199102119888111111988821111198883111 minus 3119888101121198883111


+ 36119905119910119888211111988830111198883111 + 36119909119910119888011111988830111198883111minus 121199093119910119888211111988830111198883111 minus 31199102119888111121198883111+ 18119909119888011111988830112 + 6119909119888101111988821111198883011+ 6119910119888011111988821111198883011 minus 6119910119888101111988811111198883111+ 2119910119888101111988821112 + 108119905119909c301121198883111)sdot (3119909311991011988831112 + 3119909211991011988821111198883111 + 1811990511991011988831112+ 311990911991011988811111198883111 + 1811990511988830111198883111+ 311990911988810111198883111 + 311991011988801111198883111 + 311988801111198883011+ 3119909311988830111198883111 + 11988810111198882111)minus2

11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2

11990612 = minus11988810222 (1199091198881022 + 1199101198880122)minus2 11990613 = minus (11991021198881222 + 1199101198881122 + 1198881022)2 (11990911991021198881222+ 1199091199101198881122 + 11991021198880222 + 1199091198881022 + 1199101198880122)minus2

(B1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by NSFC Shandong Joint Fund forMarine Science Research Centers (no U1406401) CAS Inter-disciplinary Innovation Team ldquoOcean Mesoscale DynamicalProcesses and Ecological Effectrdquo Youth Innovation Promo-tion Association CAS Open Fund of the Key Laboratory ofOcean Circulation andWaves Chinese Academy of Sciences(no KLOCAW1401) and Young Teachers Support Programof SDUST (no BJ162)

References

[1] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 CambridgeUniversityPress New York NY USA 1991

[2] C Rogers and W F Shadwick Backlund Transformations andTheir Applications Academic Press London UK 1982

[3] R M Miura Backlund Transformations Springer Berlin Ger-many 1978

[4] J B Li and Z J Qiao ldquoBifurcation and traveling wave solutionsfor the Fokas equationrdquo International Journal of Bifurcation andChaos vol 25 no 10 Article ID 1550136 13 pages 2015

[5] A Chen W Zhu Z Qiao and W Huang ldquoAlgebraic travelingwave solutions of a non-local hydrodynamic-type modelrdquo

Mathematical Physics Analysis and Geometry vol 17 no 3-4pp 465ndash482 2014

[6] L Zhang and C M Khalique ldquoTraveling wave solutions andinfinite-dimensional linear spaces of multiwave solutions toJimbo-Miwa equationrdquoAbstract and Applied Analysis vol 2014Article ID 963852 7 pages 2014

[7] RHirota ldquoExact solution of the Korteweg-deVries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[8] R Hirota ldquoExact N-soliton solutions of the wave equation oflong waves in shallow-water and in nonlinear latticesrdquo Journalof Mathematical Physics vol 14 pp 810ndash814 1973

[9] A Nakamura ldquoA direct method of calculating periodic wavesolutions to nonlinear evolution equations I Exact two-periodic wave solutionrdquo Journal of the Physical Society of Japanvol 47 no 5 pp 1701ndash1705 1979

[10] F Lambert and J Springael ldquoSoliton equations and simplecombinatoricsrdquoActa ApplicandaeMathematicae vol 102 no 2-3 pp 147ndash178 2008

[11] W-X Ma and Y C You ldquoSolving the Korteweg-de Vries equa-tion by its bilinear form Wronskian solutionsrdquo Transactions ofthe American Mathematical Society vol 357 no 5 pp 1753ndash1778 2005

[12] H Dong Y Zhang Y Zhang and B Yin ldquoGeneralized bilineardifferential operators binary bell polynomials and exact peri-odic wave solution of Boiti-Leon-Manna-Pempinelli EquationrdquoAbstract and Applied Analysis vol 2014 Article ID 738609 6pages 2014

[13] H H Dong and Y F Zhang ldquoExact periodic wave solutionof extended (2 + 1)-dimensional shallow water wave equationwith generalized D119901 operatorrdquo Communications in TheoreticalPhysics vol 63 no 4 p 401 2015

[14] W X Ma ldquoGeneralized bilinear differential equationsrdquo Studiesin Nonlinear Sciences vol 2 no 4 pp 140ndash144 2011

[15] E G Fan ldquoThe integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomialsrdquo PhysicsLetters A vol 375 no 3 pp 493ndash497 2011

[16] E G Fan and Y C Hon ldquoSuper extension of Bell polynomialswith applications to supersymmetric equationsrdquo Journal ofMathematical Physics vol 53 no 1 Article ID 013503 18 pages2012

[17] Y H Wang and Y Chen ldquoIntegrability of the modified gener-alised Vakhnenko equationrdquo Journal of Mathematical Physicsvol 53 no 12 Article ID 123504 20 pages 2012

[18] Y H Wang and Y Chen ldquoBinary bell polynomials bilinearapproach to exact periodicwave solutions of (2+1)-dimensionalnonlinear evolution equationsrdquo Communications in TheoreticalPhysics vol 56 no 4 p 672 2011

[19] Y-H Wang and Y Chen ldquoBacklund transformations andsolutions of a generalized Kadomtsev-Petviashvili equationrdquoCommunications in Theoretical Physics vol 57 no 2 pp 217ndash222 2012

[20] E Fan ldquoNew bilinear Backlund transformation and Lax pairfor the supersymmetric two-Boson equationrdquo Studies in AppliedMathematics vol 127 no 3 pp 284ndash301 2011

[21] Y L Song and M A Han ldquoPeriodic solutions of periodic delaypredator-prey systemwith nonmonotonic functional responserdquoJournal of Shanghai Jiaotong University vol E-8 pp 107ndash1102003

[22] H Wang and T C Xia ldquoBell polynomial approach to anextended Korteweg-de Vries equationrdquo Mathematical Methodsin the Applied Sciences vol 37 no 10 pp 1476ndash1487 2014


[23] YHWang andYChen ldquoBinaryBell polynomialmanipulationson the integrability of a generalized (2 + 1)-dimensionalKortewegode Vries equationrdquo Journal of Mathematical Analysisand Applications vol 400 no 2 p 624 2013

[24] C-G Shi B-Z Zhao and W-X Ma ldquoExact rational solutionsto a Boussinesq-like equation in (1 + 1)-dimensionsrdquo AppliedMathematics Letters vol 48 pp 170ndash176 2015

[25] Y Zhang and W-X Ma ldquoRational solutions to a KdV-likeequationrdquo Applied Mathematics and Computation vol 256 pp252ndash256 2015

[26] X Lu W-X Ma S-T Chen and C M Khalique ldquoA note onrational solutions to a Hirota-Satsuma-like equationrdquo AppliedMathematics Letters vol 58 pp 13ndash18 2016

[27] Y F Zhang and W X Ma ldquoA study on rational solutions to aKP-like equationrdquo Zeitschrift fur Naturforschung A vol 70 no4 p 263 2015

[28] H W Yang D Z Yang Y L Shi S S Jin and B S YinldquoInteraction of algebraic Rossby solitary waves with topographyand atmospheric blockingrdquo Dynamics of Atmospheres andOceans vol 71 pp 21ndash34 2015

[29] D H Luo ldquoOn the Benjamin-Ono equation and its generaliza-tion in the atmosphererdquo Science in China Series B vol 32 no10 pp 1233ndash1245 1989

[30] M Jimbo and T Miwa ldquoSolitons and infinite-dimensional Liealgebrasrdquo Publications of the Research Institute for MathematicalSciences vol 19 no 3 pp 943ndash1001 1983

[31] A-M Wazwaz ldquoNew solutions of distinct physical structuresto high-dimensional nonlinear evolution equationsrdquo AppliedMathematics and Computation vol 196 no 1 pp 363ndash3702008

[32] A-M Wazwaz ldquoMultiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff Jimbo-Miwa and YTSF equationsrdquoApplied Mathematics and Computation vol 203 no 2 pp592ndash597 2008

[33] W-X Ma and J-H Lee ldquoA transformed rational functionmethod and exact solutions to the 3 + 1 dimensional Jimbo-Miwa equationrdquo Chaos Solitons amp Fractals vol 42 no 3 pp1356ndash1363 2009

[34] W-X Ma R Zhou and L Gao ldquoExact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (1 + 2)dimensionsrdquoModern Physics Letters A vol 24 no 21 pp 1677ndash1688 2009

[35] W-XMa TW Huang and Y Zhang ldquoAmultiple exp-functionmethod for nonlinear differential equations and its applicationrdquoPhysica Scripta vol 82 no 6 Article ID 065003 2010

[36] X Wang X Zhang and P Zhao ldquoBinary nonlinearization forAKNS-KN coupling systemrdquoAbstract and Applied Analysis vol2014 Article ID 253102 12 pages 2014

Research ArticleAsymptotic Expansion of the Solutions to Time-Space FractionalKuramoto-Sivashinsky Equations

Weishi Yin1 Fei Xu2 Weipeng Zhang2 and Yixian Gao3

1School of Science Changchun University of Science and Technology Jilin 130022 China2School of Mathematics and Statistics Northeast Normal University Changchun Jilin 130024 China3School of Mathematics and Statistics Center for Mathematics and Interdisciplinary Sciences Northeast Normal UniversityChangchun Jilin 130024 China

Correspondence should be addressed to Yixian Gao gaoyx643nenueducn

Received 13 July 2016 Revised 9 November 2016 Accepted 23 November 2016

Academic Editor Maria Bruzon

Copyright copy 2016 Weishi Yin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initialconditions A new method the residual power series method is proposed for time-space fractional partial differential equationswhere the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative Weapply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtainasymptotic expansion of the solutions which demonstrates the accuracy and efficiency of the method

1 Introduction

The Kuramoto-Sivashinsky (KS) equation in one spacedimension

119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)

has attracted a great deal of interest as a model for complexspatiotemporal dynamics in spatially extended systems andas a paradigm for finite-dimensional dynamics in a partialdifferential equation 1198632119909119906 term in (1) is responsible foran instability at large scales dissipative term 1198634119909119906 providesdamping at small scales and the nonlinear term 119906119863119909119906stabilizes by transferring energy between large and smallscales The KS equation dates back to the mid-1970s andwas first introduced by Kuramoto [1] in the study of phaseturbulence in the Belousov-Zhabotinsky reaction-diffusionsystems An extension of this equation to two or more spatialdimensionswas given by Sivashinsky [2 3] inmodelling smallthermal diffusive instabilities in laminar flame fronts andin small perturbations from a reference Poiseuille flow of a

flame layer on an inclined plane In one space dimension itis also used as model for the problem of Benard convectionin an elongated box and it may be used to describe longwaves on the interface between two viscous fluids andunstable drift wave in plasmas As a dynamical system KSequation is known for its chaotic solutions and complicatedbehavior due to the terms that appear Because of this factKS equation was studied extensively as a paradigm of finitedynamics in a partial differential equation Its multimodaloscillatory and chaotic solutions have been investigated [4ndash8] its nonintegrability was established via Painleve analysis[9] and due to its bifurcation behavior a connection to lowfinite-dimensional dynamical systems is established [10 11]TheKS equation is nonintegrable therefore the exact solutionof this equation is not obtainable and only numerical schemeshave been proposed [12 13]

Partial differential equations (PDEs) which arise in real-world physical problems are often too complicated to besolved exactly and even if an exact solution is obtainable therequired calculations may be too complicated to be practicalor difficult to interpret the outcome Very recently somepractical approximate analytical solutions are proposed to



solve KS equation such as Chebyshev spectral collocationscheme [14] lattice Boltzmann technique [15] local discon-tinuous Galerkin method [16] tanh function method [17]variational iteration method [18] perturbation methods [19]classical and nonclassical symmetries method for the KSequation dispersive effects [20] Riccati expansion method[21] and Lie symmetry method [22] and see also [23ndash27]In the last few decades fractional-order models are found tobe more adequate than integer-order models for some real-world problems Fractional derivatives provide an excellenttool for the description of memory and hereditary propertiesof variousmaterials and processesThis is themain advantageof fractional differential equations in comparison with clas-sical integer-order models Fractional differential equationsarise in many engineering and scientific disciplines as themathematical modelling of systems and processes in thefields of physics chemistry aerodynamics electrodynamicsof complex medium polymer rheology and so forth involvesderivatives of fractional order In particular for the construc-tion of the approximate solutions of the fractional PDEsvarious methods are proposed finite difference method [28]the finite element method [29] the differential transforma-tion method [30] the fractional subequation method [31]the fractional complex transform method [32] the modi-fied simple equation method [33] the variational iterationmethod [34] the Lagrange characteristic method [35] theiteration method [36] and so on For the time-fractional KSequation in [37] the authors constructed the analytical exactsolutions via fractional complex transform and they obtainednew types of exact analytical solutions In [38] Rezazadehand Ziabary found travelling wave solutions by the generaltime-space KS equation by a subequation method

In this work we apply residual power series (RPS)method to construct the asymptotic expansion of the solutionto the more general linear KS equation

119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)

and the nonlinear KS equation

119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)

where 0 lt 120572 120578 le 1 34 lt 120591 le 1 12 lt 120590 le 1 120573 120574and 120575 are any arbitrary constants and (119909 119905) isin R times RThegeneral response expression contains different parametersdescribing the order of the fractional derivative that canbe varied to obtain various responses The fractional powerseries solutions can be obtained by the RPS method Particu-larly if we take special parameters 120572 = 1 120591 = 1 120590 = 1 120578 = 1120573 = 1 120574 = 1 and 120575 = 1 (here the equation is integer-order)and special initial condition 1198860(119909) = 119909 the exact solution oflinear KS equation is

119906 (119909 119905) = 119909 minus 119905 (5)

and the exact solution of nonlinear KS equation also can beobtained

119906 (119909 119905) = 1199091 + 119905 (6)

Here the solution of nonlinear KS equation we obtained isdifferent from Porshokouhi and Ghanbarirsquos work in [39] forthe integer-order KS equation They take the travel waveinitial condition

1198860 (119909) = 119888 + 519radic1119 (11 tanh3 (119896 (119909 minus 1199090))minus 9 tanh (119896 (119909 minus 1199090)))

(7)

and get the exact traveling wave solution

119906 (119909 119905) = 119888 + 519radic1119 (11 tanh3 (119896 (119909 minus 119888119905 minus 1199090))minus 9 tanh (119896 (119909 minus 119888119905 minus 1199090)))

(8)

of (2) with the same special parameters 120572 = 1 120591 = 14120590 = 1 120578 = 1 120573 = 1 120574 = 1 and 120575 = 1 Theirskills mainly depend on the variational iteration method andobtain the different numerical examples with different para-meters To the best of information of the authors no previousresearch work has been done using proposed technique forsolving time-space fractional KS equation Our method canbe applied to the time-space linear and nonlinear fractionalKS equations The main advantage of the RPS method is thatit can be applied directly for all types of differential equa-tion because it depends on the recursive differentiation oftime-fractional derivative and uses given initial conditionsto calculate coefficients of the multiple fractional powerseries solutionwithminimal calculations Another importantadvantage is that this method does not require linearizationperturbation or discretization of the variables it is notaffected by computational round-off errors and does notrequire large computer memory and extensive time

The rest of this paper is organised as follows In Section 2some necessary concepts on the theory of fractional calculusare presented The main steps of the PRS method are pro-posed in Section 3 Section 4 is the application of RPSmethodto construct analytical solution of linear and nonlinear time-space fractional KS equation with initial value The paper isconcluded with some general remarks in Section 5

2 Some Concepts on the Theory ofFractional Calculus

There are several definitions of the fractional integral andfractional derivative which are not necessarily equivalentto each other (see [40ndash42]) Riemann-Liouville integral andCaputo derivative are the two most used forms which havebeen introduced in [43ndash45] In this section we give somenotions we need in this paper

Definition 1 (see [40 41 43]) A real function 119891(119905) 119905 gt 0 issaid to be in the space 119862120583 120583 isin R if there exists a real number


119901 gt 120583 such that 119891(119905) = 1199051199011198911(119905) where 1198911(119905) isin 119862[0 +infin) andit is said to be in the space 119862119899120583 if 119891(119899)(119905) isin 119862120583 119899 isin NDefinition 2 (see [40 41 43]) The Riemann-Liouville frac-tional integral operator of order 120572 ⩾ 0 of a function 119891 isin 119862120583120583 ⩾ minus1 is defined as

1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050

(119905 minus 120591)120572minus1 119891 (120591) 119889120591 120572 gt 0 119905 gt 120591 gt 1199050119891 (119905) 120572 = 0

(9)

where the symbol 1199050119868120572119905 represents the 120572th Riemann-Liouvillefractional integral of 119891 of 119905 between the limits 1199050 and 119905Property 1 (see [40 41 43]) Here the properties of theoperator 1199050119868120572119905 are given for 119891 isin 119862120583 120583 ⩾ minus1 120572 120573 119862 isin Rand 120574 ⩾ minus1(Pro1) 1199050119868120572119905 1199050119868120573119905 119891 (119905) = 1199050119868120572+120573119905 119891 (119905) = 1199050119868120573119905 1199050119868120572119905 119891 (119905) (Pro2) 1199050119868120572119905 119862 = 119862Γ (120572 + 1) (119905 minus 1199050)120572 (Pro3) 1199050119868120572119905 119905120574 = Γ (120574 + 1)Γ (120574 + 120572 + 1) 119905120574+12057210038161003816100381610038161199051199050

(10)

Definition 3 (see [40 41 43]) The Caputo fractional deriva-tive of order 120572 gt 0 of 119891 isin 119862119899minus1 119899 isin N is defined as

1199050119863120572119905119891 (119905) fl

1199050119868119899minus120572119905119891(119899) (119905) 119899 minus 1 lt 120572 lt 119899 119905 gt 1199050119889119899119891 (119905)11988911990511989910038161003816100381610038161003816100381610038161003816119905=1199050 120572 = 119899 (11)

where the symbol 1199050119863120572119905 119891(119905) represents the 120572th Caputo frac-tional derivative of 119891 with respect to 119905 at 1199050Property 2 (see [40 41 43]) Here the properties of theoperator 1199050119863120572119905 are given for 120574 gt minus1 119905 gt 119904 ⩾ 0 and 119862 isin R

(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)

(Pro2) 1199050119863120572119905 119862 = 0(Pro3) 1199050119863120572119905 119905120574 = Γ (120574 + 1)Γ (120574 minus 120572 + 1) 119905120574minus1205721003816100381610038161003816119905=1199050

(12)

Remark 4 (see [41 46]) The Caputo time-fractional deriva-tive operator of order 120572 of function 119906(119909 119905)with respect to 119905 at1199050 is defined as

1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905

120597119899119906 (119909 119905)120597119905119899 119899 minus 1 lt 120572 lt 119899 119905 gt 1199050119889119899119891 (119905)11988911990511989910038161003816100381610038161003816100381610038161003816119905=1199050 120572 = 119899 isin N

(13)

where 119909 isin R and 119905 gt 0

Remark 5 (see [41 46]) The Caputo space-fractional deriva-tive operator of order 120573 of function 119906(119909 119905) with respect to 119909at 1199090 is defined as

1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909

120597119898119906 (119909 119905)120597119905119898 119898 minus 1 lt 120572 lt 119898 119909 gt 1199090120597119898119906 (119909 119905)12059711990911989810038161003816100381610038161003816100381610038161003816119909=1199090 120572 = 119898 isin N

(14)

where 119909 isin R and 119905 gt 0Definition 6 (see [41 46]) A power series representation ofthe forminfinsum119899=0

119888119899 (119905 minus 1199050)119899120572 fl 1198880 + 1198881 (119905 minus 1199050)120572 + 1198882 (119905 minus 1199050)2120572+ sdot sdot sdot

(15)

where119898 minus 1 lt 120572 le 119898 and 119905 ⩾ 1199050 is called a fractional powerseries (FPS) about 1199050 where 119905 is a variable and 119888119899 are constantscalled the coefficients of the series

Theorem 7 Suppose that 119891 has a FPS representation at 1199050 ofthe form

119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)

and119877 is the radius of convergence of the FPS If119891(119905) isin 119862[1199050 1199050+119877) and 1199050119863119899120572119905 119891(119905) isin 119862(1199050 1199050 + 119877) for 119899 = 0 1 2 then thecoefficients 119888119899 will take the form of

119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)

where 1199050119863119899120572119905 = 1199050119863120572119905 sdot 1199050119863120572119905 sdot sdot 1199050119863120572119905 (119899-times)

This result is similar to [46 Theorem 22] and [47Theorem 34] It is convenient to give the details for thefollowing applications thus we write the process of the proofin the form of function with one variable

Proof First of all notice that if we put 119905 = 1199050 into (16) it yields1198880 = 119891 (1199050) = 1199050119863

0120572

119905119891 (119905)

Γ (0120572 + 1) (18)

Applying the operator 1199050119863120572119905 one time on (16) leads to

1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)

Again by applying the operator 1199050119863120572119905 two times on (16) onecan obtain

1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)


By now the pattern is clearly found if we continue applyingrecursively the operator 1199050119863120572119905 119899-times on (16) then it is easyto discover the following form for 119888119899

119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)

that is it has the same formas (17) which completes the proof

Following the similar result as [46 Theorem 23] or [48Remark 8] we can obtain the following corollary

Corollary 8 If 119891(119905) = 119906(119909 119905) then we have

119906 (119909 119905) = infinsum119899=0

119888119899 (119909) (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 119909 isin R 1199050 ⩽ 119905 lt 1199050 + 119877

(22)

and 119877 is the radius of convergence of the FPS if 119863119899120572119905 isin 119862(R times(1199050 1199050 + 119877)) 119899 = 0 1 2 then the coefficients are given by119888119899(119909) = 1199050119863119899120572119905 119906(119909 119905)Γ(119899120572 + 1)3 Algorithm of RPS Method

Let us consider the higher-order time-space fractional differ-ential equation with initial values as follows

119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)

where 119909 isin R 119905 isin R and119866 (119909 119905) = 119865 (119906119863120572119905 1199061198632120572119905 119906 119863(119898minus1)120572119905 1199061198631205731119909 1199061198631205732119909 119906 119863120573119897119909 119906) (24)

119906(119909 sdot) and119866(119909 sdot) are analytical function about 119905 (119898minus1)119898 lt120572 ⩽ 1 119898 isin N 119901 minus 1 lt 120573119901 ⩽ 119901 and 119901 = 1 2 119897 119886119902(119909) isin119862infin(R) 119902 = 0 1 2 119898 minus 1Assume that 119906(119909 119905) is analytical about 119905 the solution of

the system can be written in the form of

119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)

where 119906119899(119909 119905) are terms of approximations and are given as

119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)

where 119877 is the radius of convergence of above series Obvi-ously when 119894 = 0 1 2 119898 minus 1 using the terms 119863119894120572119905 119906(119909 119905)which satisfy the initial condition we can get

119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)

So we have the initial guess approximation of 119906(119909 119905) in thefollowing form

119906initial (119909 119905) fl 1199060 (119909 119905) + 1199061 (119909 119905) + sdot sdot sdot + 119906119898minus1 (119909 119905)= 1198860 (119909) + 1198861 (119909)Γ (120572 + 1) 119905120572 + sdot sdot sdot+ 119886119898minus1 (119909)Γ ((119898 minus 1) 120572 + 1) 119905(119898minus1)120572

(28)

On the other aspect as well if we choose 119906initial(119909 119905) as initialguess approximation 119906119894(119909 119905) for 119894 = 119898119898 + 1119898 + 2 theapproximate solutions of 119906(119909 119905) of (23) by the 119896th truncatedseries are

119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)

Before applying RPS method for solving (23) we first givesome notations

Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)

Substituting the 119896th truncated series 119906119896(119909 119905) into (23) we canobtain the following definition for 119896th residual function

Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)

Then we have following facts

(1) lim119896rarrinfin

119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim

119896rarr+infinRes119896 (119909 119905) = Res (119909 119905) = 0

119909 isin R |119905| lt 119877(33)


These show that the residual function Resinfin(119909 119905) is infinitelymany times differentiable at 119905 = 0 On the other hand we canshow that

0 = 119863(119896minus119898)120572119905 Resinfin (119909 119905)1003816100381610038161003816119905=0= 119862119896 (119909) Γ (119896120572 + 1) + 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 997904rArr

119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)

Since minus119863(119896minus119898)120572119905 119866119896(119909 119905)|119905=0 is not dependent on 119905 denoting itby 119891119896(119909) by Theorem 7 (34) can be written as

119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)

In fact the relation of (35) is a fundamental rule in the RPSmethod and its applications So the fractional power seriessolution of (23) is

119906 (119909 119905) = 119906initial (119909 119905) + infinsum119894=119898

119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)

where 119886119894(119909) (119894 = 0 1 2 119898 minus 1) are given by the initialconditions and 119891119894(119909) (119894 = 119898119898 + 1119898 + 2 ) have beenconstructed by RPS method in the form in (35)

4 Application of RPS Method to Time-SpaceFractional KS Equation

In this section we apply the RPS method to the linearand nonlinear time-space fractional KS equation with theinitial conditionsThe fractional power series solutions can beobtained by the recursive equation (35) with time-fractionalderivative while it will use the given initial conditions Andthe fractional power series solutions we consider in the fol-lowing examples are all in the convergence radius of theseries

41 Linear Time-Space Fractional KS Equation Consider thelinear time-space fractional KS equation

119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)

In (37) if 119906(119909 119905) is analytical about 119905 then 119906(119909 119905) can bewritten as the fractional power series

119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)

If we can take the initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)

According to the initial condition it is obvious that 1198910(119909) =1198860(119909) Before getting the coefficients 119891119899(119909) (119899 = 1 2 ) wefirst present some notations as in Section 3

119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)

It follows from the facts in (33) that the residual functionResinfin(119909 119905) is infinitely many times differentiable at 119905 = 0 Onthe other hand it follows from Definition 3 and (40) that

0 = 119863(119896minus1)120572119905 Resinfin (119909 119905)1003816100381610038161003816119905=0 = 119863(119896minus1)120572119905 Res119896 (119909 119905)10038161003816100381610038161003816119905=0= 119863(119896minus1)120572119905 (119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905))10038161003816100381610038161003816119905=0= 119863119896120572119905 119906119896 (119909 119905)10038161003816100381610038161003816119905=0 + 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0= 119891119896 (119909) + 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)

Equation (42) gives the iterative formula for the coefficients119891119899(119909) (119899 = 1 2 ) so we can obtain

1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)


For general 119896 isin Z we have119891119896 (119909)= minus (1205731198634120591119909 119886119896minus1 (119909) + 1205741198632120590119909 119886119896minus1 (119909) + 120575119863120578119909119886119896minus1 (119909))≜ 119886119896 (119909)

(44)

So the 119896th approximate solution of (37) with initial value (39)is

119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)

where 119886119894(119909) (119894 = 0 1 2 119896) are given by (39) and (44)Specifically if taking 120572 = 1 120591 = 1 120590 = 1 120578 = 1 120573 = 1120574 = 1 and 120575 = 1 (the equation becomes integer order) and

special initial value 1198860(119909) = 119909 we can obtain

1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)

So we can obtain the 119896th approximate power series solutionof integer-order equation (37)

119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)

Thus the exact solution is

119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)

when 120572 = 1 120591 = 1 120590 = 1 120578 = 1 120573 = 1 120574 = 1 120575 = 1 and1198860(119909) = 11990942 Nonlinear Time-Space Fractional KS Equation Let usrewrite the nonlinear time-space fractional KS equation

119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)

If 119906(119909 119905) is analytical about 119905 of (49) then 119906(119909 119905) can bewritten as the fractional power series

119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)

Here the initial value satisfies

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)

According to the initial condition it is obvious that 1198910(119909) =1198860(119909)

Before getting the coefficients119891119899(119909) (119899 = 1 2 ) we alsopresent some symbols

119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)

It also follows from the facts in (33) that the residual functionResinfin(119909 119905) is infinitely many times differentiable at 119905 = 0

On the other hand it follows from Definition 3 and thenotations above that


(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)


For general 119896 isin Z we have119891119896 (119909) = minus(1205731198634120591119909 119886119896minus1 (119909) + 1205741198632120590119909 119886119896minus1 (119909)

+ 120575119896minus1sum119895=0

Γ ((119896 minus 1) 120572 + 1)Γ (119895120572 + 1) Γ ((119896 minus 1 minus 119895) 120572 + 1)119886119895 (119909)

sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)

where 119886119894(119909) (119894 = 0 1 2 119896) are given by (51) and (56)In particular if taking 120572 = 1 120591 = 1 120590 = 1 120578 = 1 120573 = 1120574 = 1 120575 = 1 and 1198860(119909) = 119909 we can obtain

1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909

119886119896minus1 (119909) = (minus1)119896minus1 (119896 minus 1)119909119886119896 (119909) = 1198634119909119886119896minus1 (119909) + 1198632119909119886119896minus1 (119909)+ 119896minus1sum119895=0


sdot 119863119909119886119896minus1minus119895 (119909) = 119896minus1sum119895=0

Γ ((119896 minus 1) + 1)Γ (119895 + 1) Γ ((119896 minus 1 minus 119895) + 1)sdot (minus1)119895 119895119909 (minus1)119896minus1minus119895 (119896 minus 1 minus 119895)= 119896minus1sum119895=0

(119896 minus 1)119895 (119896 minus 1 minus 119895) (minus1)119896minus1 119895119909 (119896 minus 1 minus 119895)= (minus1)119896 (119896 minus 1) sdot 119896119909 = (minus1)119896 119896119909 119896 = 1 2 3

(58)

So the 119896th approximate FPS solution of (49) with initial value(51) is

119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0

(minus119905)119894 = 1199091 minus (minus119905)1198961 + 119905 (59)

When 119905 isin (minus1 1) it shows119906 (119909 119905) = lim

119896rarrinfin119906119896 (119909 119905) = lim

119896rarrinfin1199091 minus (minus119905)1198961 + 119905 = 1199091 + 119905 (60)

which is the exact solution for (49) with special parameters120572 = 1 120591 = 1 120590 = 1 120578 = 1 120573 = 1 120574 = 1 and 120575 = 1 and specialinitial value 1198860(119909) = 1199095 Concluding Remarks

In this paper we have used a new method the residualpower series method for the general fractional differentialequations The asymptotic expansion of the solutions can beobtained successfully with respect to initial conditions whichare infinitely differentiable by the RPS method We applyRPS method to linear and nonlinear Kuramoto-Sivashinskyequation with infinitely differential initial conditions andobtain the asymptotic expansion of the solutions Particularlyif taking special parameters and special initial value theanalytical solutions are obtained These applications showthat thismethod is efficient and does not require linearizationor perturbation it is not affected by computational round-off errors and does not require large computer memory andextensive time

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The research is supported in part by NSFC grant (51278221and 51378076) NIETPCS (201610200032) and Jilin Scienceand Technology Development Project (20160520094JH)

References

[1] Y Kuramoto ldquoDiffusion-induced chaos in reaction systemsrdquoProgress ofTheoretical Physics Supplements vol 64 pp 346ndash3671978

[2] G I Sivashinsky ldquoNonlinear analysis of hydrodynamic insta-bility in laminar flames I Derivation of basic equationsrdquo ActaAstronautica vol 4 no 11-12 pp 1177ndash1206 1977

[3] G I Sivashinsky ldquoOn flame propagation under conditions ofstoichiometryrdquo SIAM Journal on Applied Mathematics vol 39no 1 pp 67ndash82 1980

[4] J M Hyman and B Nicolaenko ldquoThe Kuramoto-Sivashinskyequation a bridge between PDEs and dynamical systemsrdquoPhysica D Nonlinear Phenomena vol 18 no 1ndash3 pp 113ndash1261986

[5] J M Hyman B Nicolaenko and S Zaleski ldquoOrder and com-plexity in the Kuramoto-Sivashinskymodel of weakly turbulentinterfacesrdquoPhysicaDNonlinear Phenomena vol 23 no 1ndash3 pp265ndash292 1986

[6] I G Kevrekidis B Nicolaenko and J C Scovel ldquoBack inthe saddle again a computer assisted study of the Kuramoto-Sivashinsky equationrdquo SIAM Journal on Applied Mathematicsvol 50 no 3 pp 760ndash790 1990


[7] V S Lrsquovov V LebedevM Paton and I Procaccia ldquoProof of scaleinvariant solutions in the Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations in 1+1 dimensions analytical and numer-ical resultsrdquo Nonlinearity vol 6 no 1 pp 25ndash47 1993

[8] D Michelson ldquoSteady solutions of the Kuramoto-Sivashinskyequationrdquo Physica D Nonlinear Phenomena vol 19 no 1 pp89ndash111 1986

[9] R Conte and M Musette ldquoPainleve analysis and Backlundtransformation in the Kuramoto-Sivashinsky equationrdquo Journalof Physics A Mathematical and General vol 22 no 2 pp 169ndash177 1989

[10] R Temam Infinite-Dimensional Dynamical Systems inMechan-ics and Physics vol 68 of Applied Mathematical SciencesSpringer New York NY USA 2nd edition 1997

[11] R W Wittenberg and P Holmes ldquoScale and space localizationin the Kuramoto-Sivashinsky equationrdquo Chaos vol 9 no 2 pp452ndash465 1999

[12] S Benammou and K Omrani ldquoA finite element method for theSivashinsky equationrdquo Journal of Computational and AppliedMathematics vol 142 no 2 pp 419ndash431 2002

[13] SMomani ldquoA numerical scheme for the solution of Sivashinskyequationrdquo Applied Mathematics and Computation vol 168 no2 pp 1273ndash1280 2005

[14] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Dordrecht Netherlands 1994

[15] S-J Liao ldquoA kind of approximate solution techniquewhich doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997

[16] R A Van Gorder and K Vajravelu ldquoAnalytic and numericalsolutions to the Lane-Emden equationrdquo Physics Letters A vol372 no 39 pp 6060ndash6065 2008

[17] J Awrejcewicz I V Andrianov and L I ManevitchAsymptoticapproaches in nonlinear dynamics Springer Series in Synerget-ics Springer-Verlag Berlin Germany 1998

[18] M Sajid and T Hayat ldquoComparison of HAM and HPM meth-ods in nonlinear heat conduction and convection equationsrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2296ndash2301 2008

[19] M Van Dyke Perturbation methods in fluid mechanics TheParabolic Press Stanford Calif Annotated edition 1975

[20] M S Bruzon M L Gandarias and J C Camacho ldquoClassicaland nonclassical symmetries for aKuramotondashSivashinsky equa-tion with dispersive effectsrdquo Mathematical Methods in theApplied Sciences vol 30 no 16 pp 2091ndash2100 2007

[21] M Chen H Hu and H Zhu ldquoConsistent riccati expansion andexact solutions of the Kuramoto-Sivashinsky equationrdquoAppliedMathematics Letters vol 49 pp 147ndash151 2015

[22] R Kumar R K Gupta and S S Bhatia ldquoLie symmetry ana-lysis and exact solutions for a variable coefficient generalisedKuramoto-Sivashinsky equationrdquo Romanian Reports in Physicsvol 66 no 4 pp 923ndash928 2014

[23] A Alizadeh-Pahlavan V Aliakbar F Vakili-Farahani and KSadeghy ldquoMHD flows of UCM fluids above porous stretch-ing sheets using two-auxiliary-parameter homotopy analysismethodrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 2 pp 473ndash488 2009

[24] D Baldwin U Goktas W Hereman L Hong R S Martinoand J C Miller ldquoSymbolic computation of exact solutionsexpressible in hyperbolic and elliptic functions for nonlinear

PDEsrdquo Journal of Symbolic Computation vol 37 no 6 pp 669ndash705 2004

[25] N A Kudryashov ldquoQuasi-exact solutions of the dissipativeKuramoto-Sivashinsky equationrdquo Applied Mathematics andComputation vol 219 no 17 pp 9213ndash9218 2013

[26] K Yabushita M Yamashita and K Tsuboi ldquoAn analytic solu-tion of projectile motion with the quadratic resistance law usingthe homotopy analysis methodrdquo Journal of Physics A Mathe-matical and Theoretical vol 40 no 29 pp 8403ndash8416 2007

[27] S-P Zhu ldquoA closed-form analytical solution for the valuation ofconvertible bonds with constant dividend yieldrdquoThe ANZIAMJournal vol 47 no 4 pp 477ndash494 2006

[28] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Compu-tational Physics vol 225 no 2 pp 1533ndash1552 2007

[29] Q Huang G Huang and H Zhan ldquoA finite element solutionfor the fractional advection-dispersion equationrdquo Advances inWater Resources vol 31 no 12 pp 1578ndash1589 2008

[30] S Momani and Z M Odibat ldquoFractional Green function forlinear time-fractional inhomogeneous partial differential equa-tions in fluid mechanicsrdquo Journal of Applied Mathematics ampComputing vol 24 no 1-2 pp 167ndash178 2007

[31] S Zhang and H-Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[32] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article no 97 2013

[33] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputers ampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[34] M Inc ldquoThe approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions byvariational iteration methodrdquo Journal of Mathematical Analysisand Applications vol 345 no 1 pp 476ndash484 2008

[35] J Guy ldquoLagrange characteristic method for solving a classof nonlinear partial differential equations of fractional orderrdquoApplied Mathematics Letters vol 19 no 9 pp 873ndash880 2006

[36] F Xu Y Gao and W Zhang ldquoConstruction of analytic solu-tion for time-fractional Boussinesq equation using iterativemethodrdquo Advances in Mathematical Physics vol 2015 ArticleID 506140 7 pages 2015

[37] S Sahoo and S Saha Ray ldquoNew approach to find exact solutionsof time-fractional Kuramoto-Sivashinsky equationrdquo Physica Avol 434 pp 240ndash245 2015

[38] H Rezazadeh and B P Ziabarya ldquoSub-equation method forthe conformable fractional generalized kuramoto sivashinskyequationrdquo Computational Research Progress in Applied Scienceamp Engineering vol 2 no 3 pp 106ndash109 2016

[39] M G Porshokouhi and B Ghanbari ldquoApplication of Hersquos varia-tional iteration method for solution of the family of Kuramoto-Sivashinsky equationsrdquo Journal of King Saud University-Sciencevol 23 no 4 pp 407ndash411 2011

[40] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[41] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993


[42] Y Zhou Basic Theory of Fractional Differential Equations vol6 World Scientific 2014

[43] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011

[44] R Gorenflo and FMainardi Fractional Calculus Springer 1997[45] K B Oldham and J SpanierThe Fractional Calculus Academic

Press New York NY USA 1974[46] A El-Ajou O Abu Arqub and S Momani ldquoApproximate ana-

lytical solution of the nonlinear fractional KdV-Burgers equa-tion a new iterative algorithmrdquo Journal of Computational Phy-sics vol 293 pp 81ndash95 2015

[47] A El-Ajou O Abu Arqub Z Al Zhour and S Momani ldquoNewresults on fractional power series theories and applicationsrdquoEntropy vol 15 no 12 pp 5305ndash5323 2013

[48] F Xu Y Gao X Yang and H Zhang ldquoConstruction of frac-tional power series solutions to fractional Boussinesq equationsusing residual power series methodrdquoMathematical Problems inEngineering vol 2016 Article ID 5492535 15 pages 2016

Research ArticleAn Efficient Numerical Method for the Solution ofthe Schroumldinger Equation

Licheng Zhang1 and Theodore E Simos23

1School of Information Engineering Changrsquoan University Xirsquoan 710064 China2Department of Mathematics College of Sciences King Saud University PO Box 2455 Riyadh 11451 Saudi Arabia3Laboratory of Computational Sciences Department of Informatics and Telecommunications Faculty of EconomyManagement and Informatics University of Peloponnese 221 00 Tripolis Greece

Correspondence should be addressed toTheodore E Simos tsimosconfgmailcom

Received 27 May 2016 Accepted 5 July 2016


Copyright copy 2016 L Zhang and T E Simos This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The development of a new five-stage symmetric two-step fourteenth-algebraic order method with vanished phase-lag and its firstsecond and third derivatives is presented in this paper for the first time in the literature More specifically we will study (1) thedevelopment of the new method (2) the determination of the local truncation error (LTE) of the new method (3) the localtruncation error analysis which will be based on test equation which is the radial time independent Schrodinger equation (4)the stability and the interval of periodicity analysis of the new developed method which will be based on a scalar test equation withfrequency different than the frequency of the scalar test equation used for the phase-lag analysis and (5) the efficiency of the newobtained method based on its application to the coupled Schrodinger equations

1 Introduction

The approximate solution of the close-coupling differentialequations of the Schrodinger type is studied in this paperTheabove-mentioned problem has the following form

[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)

where 1 le 119894 le 119873 and119898 = 119894 and the boundary conditions areas follows

119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)

where 119895119897(119909) and 119899119897(119909) are the spherical Bessel and Neumannfunctions We will examine the case in which all channels areopen (see [1])

Defining a matrix1198701015840 and diagonal matrices119872119873 by (see[1])

1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)

we obtain a new form of the asymptotic condition (3)

y sim M + NK1015840 (5)

In several scientific areas (eg quantum chemistrytheoretical physics material science atomic physics andmolecular physics) there exists a real problem which isthe rotational excitation of a diatomic molecule by neutralparticle impact The mathematical model of this problemcan be expressed with close-coupling differential equationsarising from the Schrodinger equation Denoting as in [1]



the entrance channel by the quantum numbers (119895 119897) the exitchannels by (1198951015840 1198971015840) and the total angular momentum by 119869 =119895 + 119897 = 1198951015840 + 1198971015840 we find that

[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)

119864 is the kinetic energy of the incident particle in the center-of-mass system 119868 is the moment of inertia of the rotator and120583 is the reduced mass of the system

The above-described problem will be solved numericallyvia finite difference method of the form of special multistepmethod

The multistep finite difference method has the generalform

119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)

where 119909119894119898119894=minus119898 are distinct points within the integration areaand ℎ given by ℎ = |119909119894+1 minus 119909119894| 119894 = 1 minus 119898(1)119898 minus 1 is the stepsize or step length of the integration

Remark 1 A method (8) is called symmetric multistepmethod or symmetric 2119898-step method if 119888minus119894 = 119888119894 and 119887minus119894 =119887119894 119894 = 0(1)119898

If we apply the symmetric 2119898-stepmethod (119894 = minus119898(1)119898)to the scalar test equation

11991010158401015840 = minus1206012119910 (9)

we obtain the difference equation

119860119898 (V) 119910119899+119898 + sdot sdot sdot + 1198601 (V) 119910119899+1 + 1198600 (V) 119910119899+ 1198601 (V) 119910119899minus1 + sdot sdot sdot + 119860119898 (V) 119910119899minus119898 = 0 (10)

and the associated characteristic equation

119860119898 (V) 120582119898 + sdot sdot sdot + 1198601 (V) 120582 + 1198600 (V) + 1198601 (V) 120582minus1 + sdot sdot sdot+ 119860119898 (V) 120582minus119898 = 0 (11)

where V = 120601ℎ ℎ is the step length and 119860119895(V) 119895 = 0(1)119896 arepolynomials of V

We give some definitions

Definition 2 (see [2]) For a symmetric 2119898-step method withcharacteristic equation given by (11) one will say that it has an

interval of periodicity (0 V20) if for all V isin (0 V20) the roots120582119894 119894 = 1(1)2119898 of (11) satisfy

1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)

where 120579(V) is a real function of V

Definition 3 (see [2]) One calls P-stable method a multistepmethod if its interval of periodicity is equal to (0infin)Definition 4 One calls singularly P-stablemethod amultistepmethod if its interval of periodicity is equal to (0infin) minus 119878(where 119878 is a set of distinct points)Definition 5 (see [3 4]) For a symmetric 2119898-step methodwith the characteristic equation given by (11) one defines asthe phase-lag the leading term in the expansion of

119905 = V minus 120579 (V) (13)

Then if the quantity 119905 = 119874(V119902+1) as V rarr infin the order of thephase-lag is 119902Definition 6 (see [5]) One calls symmetric 2119898-step methodphase-fitted if its phase-lag is equal to zero

Theorem 7 (see [3]) The symmetric 2119898-step method withcharacteristic equation given by (11) has phase-lag order 119902 andphase-lag constant 119888 given by

minus 119888V119902+2 + 119874 (V119902+4)= 2119860119898 (V) cos (119898V) + sdot sdot sdot + 2119860119895 (V) cos (119895V) + sdot sdot sdot + 1198600 (V)21198982119860119898 (V) + sdot sdot sdot + 21198952119860119895 (V) + sdot sdot sdot + 21198601 (V) (14)

2 The New Five-Stage Fourteenth-AlgebraicOrder P-Stable Two-Step Method withVanished Phase-Lag and Its First Secondand Third Derivatives

We consider the following family of five-stage symmetrictwo-step methods

119899 = 119910119899 minus 1198860ℎ2 (119891119899+1 minus 2119891119899 + 119891119899minus1) 119899 = 119910119899 minus 1198861ℎ2 (119891119899+1 minus 2119899 + 119891119899minus1) minus 21198862ℎ2119899119899+12 = 12 (119910119899 + 119910119899+1) minus ℎ2 [1198863119899 + (18 minus 1198863)119891119899+1] 119899minus12 = 12 (119910119899 + 119910119899minus1) minus ℎ2 [1198863119899 + (18 minus 1198863)119891119899minus1] 119910119899+1 + 1198864119910119899 + 119910119899minus1= ℎ2 [1198871 (119891119899+1 + 119891119899minus1) + 1198870119891119899 + 1198872 (119899+12 + 119899minus12)]

(15)


where 1198860 = 45469862066800 1198861 = minus279326878564 1198862 =minus8691913439282 1198863 = 671964152720800 119891119899+119894 = 11991010158401015840(119909119899+119894119910119899+119894) 119894 = minus1(12)1 119899 = 11991010158401015840(119909119899 119899) 119899 = 11991010158401015840(119909119899 119899) and1198864 119887119895 119895 = 0(1)2 are free parametersApplication of the above-mentioned method (15) to the

scalar test equation (9) leads to the difference equation (10)and the characteristic equation (11) with119898 = 1 and

1198601 (V) = 1 + V2 (1198871 + 121198872) minus V41198872 (1198863 minus 18)+ 2V6119887211988611198863 + 4V8119887211988601198863 (1198862 minus 1198861)

1198600 (V) = 1198864 + V2 (1198870 + 1198872) + 2V411988721198863+ 4V611988721198863 (1198862 minus 1198861)minus 8V8119887211988601198863 (1198862 minus 1198861)

(16)

If we request the new method (15) to have vanishedphase-lag and its first second and third derivatives then thefollowing system of equations is obtained

Phase-Lag (PL) = 1198790119879denom = 0First Derivative of the Phase-Lag = 11987911198792denom = 0

Second Derivative of the Phase-Lag = 11987921198793denom = 0

Third Derivative of the Phase-Lag = 11987931198794denom = 0

(17)

where 119879119895 119895 = 0(1)3 are given in Appendix ANow solving the system of (17) we determine the other

coefficients of the new obtained three-stage two-stepmethod

1198870 = 1136407 1198794119879denom1 1198871 = 1136407 1198795119879denom1 1198872 = 145469 1198796119879denom1 1198864 = 1198797119879denom2

(18)

where the formulae 119879119895 119895 = 4(1)7 119879denom1 and 119879denom2 aregiven in Appendix B

Additionally to the above formulae for the coefficients1198864 119887119895 119895 = 0(1)2 we give also the following Taylorseries expansions of these coefficients for the case of heavy

cancelations for some values of |V| in the formulae given by(18)1198870

= 7320563882 + 268231049V10262060331977892448minus 473946402833V12224533292438658249446400minus 841146961608850447V1426876635104907392458734080000minus 34711252155550520483V1623840650403457053406595478323200minus 450175574700292070703419V187016973932030008047188109768345600000+ sdot sdot sdot

1198871= 51911383292 + 268231049V101572361991867354688

minus 61738277963993V121347199754631949496678400minus 559868876269811659V14161259810629444354752404480000minus 602760578937670659833V163576097560518558010989321748480000minus 526438577422138492034251V1884203687184360096566257317220147200000+ sdot sdot sdot

1198872= minus1997095823 minus 268231049V10393090497966838672

+ 15790029293123V12336799938657987374169600+ 181541151070183V1440314952657361088688101120+ 21664254981643052111V1689402439012963950274733043712000+ 8364955394669318397937V18842036871843600965662573172201472000+ sdot sdot sdot

1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)

The behavior of the coefficients is given in Figure 1


40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1

Behavior of the coefficient b0b0

Behavior of the coefficient b2

b2

Behavior of the coefficient a4a4


b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08

Figure 1 Behavior of the coefficients of the new obtained method given by (18) for several values of V = 120601ℎ

The local truncation error of the new developed five-stagetwo-step method (15) with the coefficients given by (18)-(19)which is indicated as NM2S5S3DV is given by

LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)

3 Analysis of the Method

31 Error Analysis The test equation used for the local trun-cation error (LTE) analysis is given by

11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)

where 119881(119909) is a potential function 119881119888 is a constant valueapproximation of the potential for the specific 119909 and 119866 =119881119888minus119864 and 119864 is the energyThis is the radial time independentSchrodinger equation

We will study the following methods

311 Classical Method (ie Method (15) withConstant Coefficients)

LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)


312 The Five-Stage Two-Step Method with VanishedPhase-Lag and Its First Second andThird Derivatives Developed in Section 3

LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)

If we substitute the higher order derivatives requested inthe LTE formulae which are obtained using the test problem(21) into the LTE expressions we produce new formulae ofLTE which have the general form

LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)

where 119886119894 are constant numbers (classical methods) or formu-lae of 120601 (fitted methods) and 119901 is the algebraic order of thespecific method

Two cases of the parameter 119866 are studied

(i) TheEnergy and the Potential Are Closed to Each OtherConsequently 119866 = 119881119888 minus 119864 asymp 0 rArr 119866119894 = 0 119894 = 1 2 Therefore the local truncation error for the classicalmethod (constant coefficients) and the local trunca-tion error for the five-stage two-step method witheliminated phase-lag and its first second and thirdderivatives developed in Section 3 are the same sincethe formulae of the LTE are free from 119866 (ie LTE =ℎ1199011198860 in (24)) and the free from119866 terms (ie the termsof the formulae which do not have the quantity 119866) inthe local truncation errors in this case are the sameand are given in Appendix C From the above it is easyto see that for these values of 119866 the methods are ofcomparable accuracy

(ii) The Energy and the Potential Have Big DifferenceConsequently 119866 ≫ 0 or 119866 ≪ 0 and |119866| is a largenumber For this case the most accurate method isthe method with the minimum power of 119866 in theformulae of LTE (ie the method with the highestaccuracy is the method with minimum 119894 in (24))

We give now the asymptotic expansions of the localtruncation errors

313 Classical Method

LTECL = 134317551343236245401600ℎ16 (119910 (119909) 1198668 + sdot sdot sdot)+ 119874 (ℎ18)

(25)

314 The Four-Stage Two-Step P-StableMethod with VanishedPhase-Lag and Its First Second and Third DerivativesDeveloped in Section 3

LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)

The above leads us to the following theorem

Theorem 8 (i) Classical method (ie method (15) withconstant coefficients) for this method the error increases as theeighth power of 119866

(ii) Fourteenth-algebraic order five-stage two-step methodwith vanished phase-lag and its first second and third deriva-tives developed in Section 3 for this method the error increasesas the fifth power of 119866

Therefore for large values of |119866| = |119881119888 minus 119864| thenew developed fourteenth-algebraic order five-stage two-stepmethod with vanished phase-lag and its first second and thirdderivatives developed in Section 3 is the most accurate methodfor the numerical solution of the radial Schrodinger equation

32 Stability and Interval of Periodicity Analysis Let us definethe scalar test equation for the stability and interval ofperiodicity analysis

11991010158401015840 = minus1205962119910 (27)

Remark 9 Comparing the test equations (9) and (27) wehave that the frequency 120601 is not equal to the frequency120596 thatis 120596 = 120601

The difference equation which is produced after applica-tion of the new method (15) with the coefficients given by(18)-(19) to the scalar test equation (27) is given by

1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500

Figure 2 119904-V plane of the new obtained five-stage symmetric two-step fourteenth-algebraic ordermethodwith vanished phase-lag andits first second and third derivatives

minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)

119904 = 120596ℎ and V = 120601ℎTaking into account the coefficients 119886119894 119894 = 0 1 and119887119894 119894 = 0(1)2 and their substitution into formulae (29) the

new stability polynomials are produced

1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)

where the formulae 119879119896 119896 = 8 9 and 119879denom3 are given inAppendix D119904-V plane of the new produced method is shown inFigure 2

Remark 10 Observing 119904-V region we can define two areas

(i) The shadowed area which defines the area where themethod is stable

(ii) The white area which defines the area where themethod is unstable

For problems of the form of the coupled equations arisingfrom the Schrodinger equation and related problems (inquantum chemistry material science theoretical physicsatomic physics astronomy astrophysics physical chemistryand chemical physics) the area of the region which playscritical role in the stability of the numerical methods is thesurroundings of the first diagonal of the 119904-V plane where 119904 =V Studying this case we found that the interval of periodicityis equal to (0 24)

The above development leads to the following theorem

Theorem 11 The five-stage symmetric two-step method devel-oped in Section 3

(i) is of fourteenth algebraic order

(ii) has vanished the phase-lag and its first second andthird derivatives

(iii) has an interval of periodicity equal to (0 24) (when 119904 =V)

4 Numerical Results

We will apply the new produced method to the approximatesolution for coupled differential equations of the Schrodingertype

41 Error Estimation For the numerical solution of the pre-viously referred problem we will use variable step methodsAn algorithm is called variable step when it is based onthe change of the step size of the integration using a localtruncation error estimation (LTEE) procedure In the pastdecades several methods of constant or variable steps havebeen developed for the approximation of coupled differentialequations of the Schrodinger type and related problems (seeeg [1ndash55])

In our numerical tests we use local estimation procedurewhich is based on an embedded pair and on the fact that wehave better approximation for the problems with oscillatoryandor periodical solutions having maximal algebraic orderof the methods

The local truncation error in 119910119871119899+1 is estimated by

LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)

where 119910119871119899+1 gives the lower algebraic order solution which isobtained using the twelfth-algebraic ordermethod developedin [54] and 119910119867119899+1 gives the higher order solution which isobtained using the five-stage symmetric two-step method offourteenth algebraic order with vanished phase-lag and itsfirst second and third derivatives developed in Section 3

In our numerical tests the changes of the step sizes arereduced on duplication of step sizes We use the followingprocedure

(i) If LTE lt acc then the step size is duplicated that isℎ119899+1 = 2ℎ119899(ii) If acc le LTE le 100 acc then the step size remains

stable that is ℎ119899+1 = ℎ119899(iii) If 100 acc lt LTE then the step size is halved and the

step is repeated that is ℎ119899+1 = (12)ℎ119899In the above ℎ119899 is the step length used for the 119899th step ofthe integration and acc is the requested accuracy of the localtruncation error (LTE)

Remark 12 The local extrapolation technique is also usedthat is while for a local truncation error estimation less thanacc we use the lower algebraic order solution 119910119871119899+1 it is thehigher algebraic order solution 119910119867119899+1 which is accepted at eachpoint as integration

42 Coupled Differential Equations We will present thenumerical solution of the coupled Schnrodinger equations


(1) using the boundary conditions (2) and (3) Mathematicalmodels of this form are observed in many real problemsin quantum chemistry material science theoretical physicsatomic physics physical chemistry and chemical physics andso forth The methodology fully described in [1] will be usedfor the approximate solution for this problem

Equation (1) contains the potential 119881 which can bepresented as (see for details [1])

119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)

Based on (32) the coupling matrix element is given by

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)

where 1198912 coefficients are obtained from formulas given byBernstein et al [56] and k1198951015840119895 is a unit vector parallel to thewave vector k1198951015840119895 and 119875119894 119894 = 0 2 are Legendre polynomials(see for details [57]) The above leads to the new form ofboundary conditions

11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840

(119909) sim 12057511989511989510158401205751198971198971015840 exp [minus119894 (119896119895119895119909 minus 12 119897120587)] minus( 119896119894119896119895)

12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where

S = (I + iK) (I minus iK)minus1 (36)

We will use the variable step method described in Sec-tion 41 in order to compute the cross sections for rotationalexcitation of molecular hydrogen by impact of various heavyparticles

We use the following parameters in our example

2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)

For our test we take 119869 = 6 and we will consider excitationof the rotator from 119895 = 0 state to levels up to 1198951015840 = 2 4 and 6which is equivalent to sets of four nine and sixteen coupleddifferential equations respectively Using the methodologyfully described by Bernstein [57] and Allison [1] we consider

Table 1 Coupled differential equations Real time of computation(in seconds) (RTC) andmaximumabsolute error (MErr) to calculate|119878|2 for the variable step methods Method IndashMethod X acc = 10minus6We note that ℎmax is themaximum step size119873 indicates the numberof equations of the set of coupled differential equations

Method 119873 ℎmax RTC MErr

Method I4 0014 325 12 times 10minus39 0014 2351 57 times 10minus216 0014 9915 68 times 10minus1

Method II4 0056 155 89 times 10minus49 0056 843 74 times 10minus316 0056 4332 86 times 10minus2

Method III4 0007 4515 90 times 100916

Method IV4 0112 039 11 times 10minus59 0112 348 28 times 10minus416 0112 1931 13 times 10minus3

Method V4 0448 014 34 times 10minus79 0448 137 58 times 10minus716 0448 958 82 times 10minus7

Method VI4 0448 009 29 times 10minus79 0448 110 45 times 10minus716 0448 857 74 times 10minus7

Method VII4 0448 006 13 times 10minus79 0448 104 17 times 10minus716 0448 758 29 times 10minus7

Method VIII4 0448 004 88 times 10minus89 0448 102 92 times 10minus816 0448 748 89 times 10minus8

Method IX4 0448 004 97 times 10minus89 0448 101 12 times 10minus716 0448 715 23 times 10minus7

Method X4 0896 002 70 times 10minus89 0896 045 57 times 10minus816 0896 611 65 times 10minus8

the potential infinite for119909 lt 1199090Thewave functions then tendto zero and then the boundary conditions (34) are given by

11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)

For comparison purposes the following variable stepmethods are used

(i) Method I The iterative Numerov method of Allison[1]

(ii) Method II The variable step method of Raptis andCash [47]

(iii) Method III The embedded Runge-Kutta Dormandand Prince method 5(4) [49]

(iv) Method IV The embedded Runge-Kutta methodERK4(2) developed in Simos [41]


(v) Method V The embedded symmetric two-stepmethod developed in [50]

(vi) Method VI The embedded symmetric two-stepmethod developed in [52]

(vii) Method VII The embedded symmetric two-stepmethod developed in [52]

(viii) Method VIII The embedded symmetric two-stepmethod developed in [53]

(ix) Method IX The embedded symmetric two-stepmethod developed in [54]

(x) Method X The developed embedded symmetric two-step method developed in this paper

The results are presented in Table 1 More specifically wepresent (1) the requested real time of computation by thevariable step methods mentioned above in order to calculatethe square of the modulus of S matrix for sets of 4 9 and 6coupled differential equations and (2) the maximum error inthe computation of the square of the modulus of Smatrix

5 Conclusions

In this paper we introduce for the first time in the literaturea new five-stage symmetric two-step fourteenth-algebraicorder family of methods and we produced a method of the

family with vanished phase-lag and its first second and thirdderivatives In this paper

(1) we introduced the new family of methods(2) we developed the newmethod of the new family with

vanished phase-lag and its first second and thirdderivatives

(3) we analyzed the local truncation error(4) we analyzed the interval of periodicity and the stabil-

ity of the new developed method(5) finally we analyzed the efficiency of the new obtained

method on the numerical solution for coupledSchrodinger equations

From the analysis presented above we conclude thatthe new developed method is much more efficient thanthe known ones for the numerical solution for the coupledSchrodinger equations

All computations were carried out on IBM PC-AT com-patible 80486 using double precision arithmetic with 16significant digitsrsquo accuracy (IEEE standard)

Appendix

A Formulae of 119879119895 119895 = 0(1)3 and 119879denom

1198790 = (45469V81198872 + 70383601198872V6 + 6528866401198872V4 minus 265712832000+ (minus2657128320001198871 minus 1328564160001198872) V2) cos (V) minus 45469V81198872+4310334001198872V6 minus 338669906401198872V4 + (minus1328564160001198870 minus 1328564160001198872) V2

minus13285641600011988641198791 = minus2067429961(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469

+ (minus265712832000119887145469 minus 132856416000119887245469 ) V2)2 sin (V) minus 39837369710880

sdot (V1211988722 minus 4393400V101198872228973 minus 26360400V8119887228973 (1198870 + 21198871 + 15322671703311988724994087615 )

minus176699033280001198872V6188196191 (1198870 + 4546911988643519180 minus 162901198871133 minus 80121198872133 + 454691759590)

minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872

+33353793031680000011988641198871188196191 minus 3335379303168000001198870188196191 ) V1198792 = minus94003972896709 (minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469+ (minus265712832000119887145469 minus 132856416000119887245469 ) V2)3 cos (V) + 5434096090152008160V2011988723

minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722

sdot (1198870 + 4546911988645278770 minus 162901198871133 minus 70091304265581981198872125536380378255 + 454692639385) V14

minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12

minus481539279148300038758400000001198872 (5411220546641198872224970438075+(72734780969119887099881752300 + 1277044854711988641198581027600 + 2191635189987119887149940876150 + 63965810599599290513800)sdot 1198872 + 1198871 (1198870 + 21198871) ) V10 minus 29815890393773595731558400000001198872

sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800

minus8073535624019119887173624597200 +141363876421114527151200) 1198872 minus 1629011988712133 + (1198870 minus 45469119886414076720 + 5910977038360) 1198871

+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541

+(121198870 minus 126689681991198864682950515280 + 135688231198871129541 minus 2869992664019341475257640 ) 1198872 + 1343928211988712129541+ (1198870 minus 111721198864129541 minus 2907765129541 ) 1198871 + 698251198870259082 + 45469119886423317380 + 4546911658690) V6


minus5531514869315897407320883200000001198872 ((minus381198864 + 7237805518164 ) 1198872 + (minus341198864 + 6719641259082 ) 1198871

+1198870 + 1396511988641036328 minus 855225518164) V4 + ((minus140701539050488631027957760000000001198864+28140307810097726205591552000000000) 11988722 + ((minus562806156201954524111831040000000001198864+56280615620195452411183104000000000) 1198871 + 281403078100977262055915520000000001198870minus2765757434657948703660441600000001198864 minus 28693459297029315946323640320000000) 1198872+562806156201954524111831040000000001198871 (minus11988641198871 + 1198870)) V2

+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469

+(minus265712832000119887145469 minus 132856416000119887245469 ) V2)4 sin (V) minus 988331660492486636108160



minus1218751529857909344000V18119887238557092608579 (1198870 + 82546696051198864385284849578 minus 6631668296621198871192642424789minus20527731296910925487651198872192414629552462777647 minus394861571517192642424789)

minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600


+ (17737276339168999119887030128731290781200 + 45625677327228191198864632703357106405200 minus 560674584742708191198871792861349757400 + 170062002887713063316351678553202600) 1198872minus 105220119887121197 + (1198870 + 45469119886412669048 + 59109715836310) 1198871 + 4546911988703016440)minus 24444125578632065770546790400000011988722V12389082443819478551 (minus18787681229658547119887227985635789403360+ (8969884197456053119887015971271578806720 + 34225482321665711198864383310517891361280 minus 13616047664809292711988717985635789403360 minus 280627602486327149191655258945680640) 1198872minus 222153379571119887129088218452 + (1198870 + 380984751119886472705747616 minus 15440194784736352873808 ) 1198871+1650933921119887036352873808 + 206742996111988648724689713920 + 20674299614362344856960) minus 4431265402194779258880000000001198872V108557092608579sdot (402858773949437389711988723438820490554820000 + (4349124035232111471198870877640981109640000 + 292609958613292911988641755281962219280000+ 153137123831206738471198871438820490554820000 minus 144221396158735377877640981109640000) 11988722 + (1550086250211988712454007965+ (72734780969119887049940876150 minus 752827801911988641997635046000 minus 539180954813499408761500) 1198871+ 9856693825311988701997635046000 + 49343119886487868000 minus 1003872196700) 1198872 + 11988712 (1198870 + 21198871) )minus 623776823123832930890612736000000000V81198872389082443819478551 (minus6651725624691198872373624597200+ (535931692811198870147249194400 minus 255436772783415111988641811388909895488000 minus 54798649217311988718180510800 + 267706836641709133143004387623328000) 11988722

+ (minus58559406136191198871236812298600 + (90405467881119887073624597200 + 23902069571198864309223308240 + 1205462023201386529135300 ) 1198871+ 284403838931198870220873791600 + 1277044854711988647421359397760 minus 1098284384033710679698880) 1198872 + (minus1629011988712133 + (1198870 + 4546911988641407672 + 45469185220) 1198871 + 454691198870502740 ) 1198871)minus 14217419993770846548447510528000000001198872V6389082443819478551 ((232819851571198864167808706810 minus 268397223503383904353405 ) 11988722

+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872

minus119886411988712 + (1198870 + 27931198864129541 + 513135259082) 1198871 minus 195511198870518164 minus 454691198864186539040 minus 4546993269520)


+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712

+(6002944955979456984694203875328000000000389082443819478551 minus 235488815872158762498373189632000000000001198870389082443819478551+ 3471736775264637667046224035840000000001198864389082443819478551 ) 1198871 + 382004678522076633240905318400000000000389082443819478551minus31188841156191646544530636800000000001198864389082443819478551 minus 2893113979387198055871853363200000000001198870389082443819478551 ) 1198872

minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871

minus6002944955979456984694203875328000000000389082443819478551 + 117744407936079381249186594816000000000001198870389082443819478551minus 578622795877439611174370672640000000001198864389082443819478551 ) 1198872 + 235488815872158762498373189632000000000001198871 (minus11988641198871 + 1198870)389082443819478551 ) V

119879denom = 45469V81198872 + 70383601198872V6 + 6528866401198872V4 minus 265712832000 + (minus2657128320001198871 minus 1328564160001198872) V2(A1)

B Formulae for the Coefficients of theProduced Method 119879119895 119895 = 4(1)7 119879denom1and 119879denom2

1198794 = (15277584V6 + 844603200V4+ 15669279360V2) (cos (V))3 + ((minus2182512V7minus 153643056V5 minus 4378489920V3

+ 15669279360V) sin (V) + 363752V8minus 2593839192V6 + 148398964560V4+ 672116719680V2 + 797138496000) (cos (V))2+ ((minus4365024V7 + 20674325616V5minus 490147842240V3 minus 812807775360V) sin (V)minus 30555168V6 minus 1689206400V4


minus 31338558720V2) cos (V) + (13095072V7+ 1013523840V5 + 31338558720V3) sin (V)+ 727504V8 minus 5149484424V6 + 206280915120V4+ 937829551680V2 minus 797138496000

1198795 = (minus181876V8 minus 17295684V6 minus 1200197880V4+ 128939096160V2 + 398569248000) (cos (V))2+ ((minus2182512V7 minus 176559432V5 minus 5645394720V3minus 7834639680V) sin (V) minus 7638792V6+ 25862004000V4 minus 406403887680V2) cos (V)+ (minus1091256V7 + 5180039592V5minus 161329966560V3 + 406403887680V) sin (V)minus 363752V8 minus 38410764V6 minus 2294820360V4+ 277464791520V2 minus 398569248000

1198796 = (minus88570944000V2 minus 265712832000) (cos (V))2minus 177141888000V2 + 265712832000

119879denom1 = V3 (V(V6 + 1191127V445469 + 6478080V245469minus 457020648045469 ) (cos (V))2 + ((16V6

+ 50850600V445469 + 1487650080V245469 minus 261154656045469 )sdot sin (V) + 8V5 + 1724133600V345469minus 135467962560V45469 ) cos (V) + (8V6

minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)

sdot (cos (V))3 + ((727504V6 + 67219440V4+ 2247792960V2 + 5223093120) sin (V) minus 90938V7+ 581713426V5 minus 48439997760V3+ 474137868960V) (cos (V))2 + ((1455008V6minus 6907446960V4 + 211658354880V2minus 270935925120) sin (V) + 1455008V5minus 112613760V3 minus 10446186240V) cos (V)+ (minus4365024V6 minus 337841280V4minus 10446186240V2) sin (V) minus 181876V7+ 1141692670V5 minus 22742250720V3minus 203201943840V

119879denom2 = (45469V7 + 1191127V5 + 6478080V3minus 4570206480V) (cos (V))2 + ((727504V6+ 50850600V4 + 1487650080V2 minus 2611546560)sdot sin (V) + 363752V5 + 1724133600V3minus 135467962560V) cos (V) + (363752V6minus 1718677320V4 + 29638785120V2+ 135467962560) sin (V) + 90938V7 + 12521841V5+ 674562000V3 + 7181753040V

(B1)

C Formulae of the AsymptoticForm of the Local Truncation ErrorsLTECL = LTENM2S5S3DV = 1198860

Formulae of the asymptotic form of the local truncationerrors are as follows


LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)

D Formulae for the Stability Polynomials119879119896 119896 = 8 9 and 119879denom3

1198798 = 159532850484000V21199042+ 265888084140001199044V2+ 2855080338541199046V2minus 302368851199048V2 + 18819619101199046V4minus 74069001001199042V8minus 7665846917401199042V6minus 471582353970001199042V4+ 17776560240 sin (V) V9+ 17776560240 cos (V) V8minus 83991760628400 sin (V) V7+ 84258409032000 cos (V) V6+ 1448447428814400 sin (V) V5minus 6620319330307200 cos (V) V4+ 6620319330307200 sin (V) V3+ 33124136824800V6+ 239299275726000V4+ 641047557915V8minus 17776560240 sin (V) 1199042V7minus 124435921680 cos (V) 1199042V6

+ 84382844953680 sin (V) 1199042V5+ 421292045160000 cos (V) 1199042V4minus 2628065155262400 sin (V) 1199042V3minus 6620319330307200 cos (V) 1199042V2+ 6620319330307200 sin (V) 1199042Vminus 159532850484001199044minus 1719823266001199046+ 5555175075V10+ 181421311199048+ 1111035015V10 cos (2V)minus 181421311199048 cos (2V)+ 29105188245V8 cos (2V)+ 1719823266001199046 cos (2V)+ 158291884800V6 cos (2V)+ 159532850484001199044 cos (2V)minus 111672995338800V4 cos (2V)+ 17776560240V9 sin (2V)+ 1242534411000V7 sin (2V)+ 36350729704800V5 sin (2V)minus 63813140193600V3 sin (2V)minus 177765602401199042V7 sin (2V)minus 14380765736401199042V5 sin (2V)


minus 459817399944001199042V3 sin (2V)minus 638131401936001199042V sin (2V)minus 60473771199048V2 cos (2V)+ 3763923821199046V4 cos (2V)minus 14813800201199042V8 cos (2V)+ 584566193461199046V2 cos (2V)minus 1408733461801199042V6 cos (2V)+ 53177616828001199044V2 cos (2V)minus 97756117326001199042V4 cos (2V)minus 31906570096800V21199042 cos (2V)

1198799 = 9930478995460800V21199042minus 27798749545392001199044V2minus 5710160677081199046V2+ 604737701199048V2minus 37639238201199046V4+ 148138002001199042V8minus 1050119214858001199042V6+ 45690256736460001199042V4minus 204430442760 sin (V) V9+ 44441400600 cos (V) V8minus 15689049845400 sin (V) V7minus 3439646532000 cos (V) V6minus 483042711060000 sin (V) V5minus 319065700968000 cos (V) V4+ 63813140193600 sin (V) V3minus 2295045137952000V6+ 1655079832576800V4+ 70008688347210V8+ 204430442760 sin (V) 1199042V7minus 311089804200 cos (V) 1199042V6+ 15884592008040 sin (V) 1199042V5

minus 17198232660000 cos (V) 1199042V4+ 492673721349600 sin (V) 1199042V3minus 319065700968000 cos (V) 1199042V2+ 63813140193600 sin (V) 1199042V+ 16550798325768001199044+ 3439646532001199046minus 11110350150V10minus 362842621199048minus 2222070030V10 cos (2V)+ 362842621199048 cos (2V)+ 14214167564310V8 cos (2V)minus 3439646532001199046 cos (2V)minus 1183631345265600V6 cos (2V)minus 16550798325768001199044 cos (2V)+ 11585558828037600V4 cos (2V)+ 35553120480V9 sin (2V)minus 168783466467600V7 sin (2V)+ 5171871901492800V5 sin (2V)minus 6620319330307200V3 sin (2V)minus 355531204801199042V7 sin (2V)+ 1683923821423201199042V5 sin (2V)minus 39922541750448001199042V3 sin (2V)minus 66203193303072001199042V sin (2V)+ 120947541199048V2 cos (2V)minus 7527847641199046V4 cos (2V)+ 29627600401199042V8 cos (2V)minus 1169132386921199046V2 cos (2V)minus 211268202188401199042V6 cos (2V)minus 5302847106144001199044V2 cos (2V)+ 12087095663412001199042V4 cos (2V)+ 3310159665153600V21199042 cos (2V)


+ 356809448520001199044V4+ 34396465320001199042V4 cos (3V)+ 63813140193600V21199042 cos (3V)+ 63813140193600V3 sin (3V)+ 821253508200V7 sin (3V)+ 27462410488800V5 sin (3V)+ 8888280120V9 sin (3V)+ 63813140193600V4 cos (3V)minus 8888280120V8 cos (3V)+ 687929306400V6 cos (3V)minus 88882801201199042V7 sin (3V)+ 622179608401199042V6 cos (3V)+ 71361889704001199044V4 cos (2V)minus 6257113455601199042V5 sin (3V)minus 178314001992001199042V3 sin (3V)+ 638131401936001199042V sin (3V)

119879denom3= 1111035015V10 cos (2V)+ 17776560240 sin (V) V9+ 17776560240V9 sin (2V)+ 5555175075V10+ 17776560240 cos (V) V8+ 29105188245V8 cos (2V)minus 83991760628400 sin (V) V7+ 1242534411000V7 sin (2V)+ 641047557915V8+ 84258409032000 cos (V) V6+ 158291884800V6 cos (2V)+ 1448447428814400 sin (V) V5+ 36350729704800V5 sin (2V)+ 33124136824800V6

minus 6620319330307200 cos (V) V4minus 111672995338800V4 cos (2V)+ 6620319330307200 sin (V) V3minus 63813140193600V3 sin (2V)+ 239299275726000V4

(D1)

Additional Points

Theodore E Simos is highly cited researcher active memberof the European Academy of Sciences and Arts activemember of the European Academy of Sciences and corre-sponding member of European Academy of Arts Sciencesand Humanities

Competing Interests


References

[1] A C Allison ldquoThe numerical solution of coupled differentialequations arising from the Schrodinger equationrdquo Journal ofComputational Physics vol 6 no 3 pp 378ndash391 1970

[2] J D Lambert and I A Watson ldquoSymmetric multistep methodsfor periodic initial values problemsrdquo Journal of the Institute ofMathematics and Its Applications vol 18 no 2 pp 189ndash2021976

[3] T E Simos and P S Williams ldquoA finitendashdifference method forthe numerical solution of the Schrodinger equationrdquo Journal ofComputational and AppliedMathematics vol 79 no 2 pp 189ndash205 1997

[4] R MThomas ldquoPhase properties of high order almost P-stableformulaerdquo BIT Numerical Mathematics vol 24 no 2 pp 225ndash238 1984

[5] A D Raptis and T E Simos ldquoA four-step phase-fitted methodfor the numerical integration of second order initial valueproblemsrdquo BIT Numerical Mathematics vol 31 no 1 pp 160ndash168 1991

[6] Z A Anastassi and T E Simos ldquoA parametric symmetriclinear four-step method for the efficient integration of theSchrodinger equation and related oscillatory problemsrdquo Journalof Computational and Applied Mathematics vol 236 no 16 pp3880ndash3889 2012

[7] Z A Anastassi and T E Simos ldquoAn optimized Runge-Kuttamethod for the solution of orbital problemsrdquo Journal of Compu-tational and Applied Mathematics vol 175 no 1 pp 1ndash9 2005

[8] P Atkins and R Friedman Molecular Quantum MechanicsOxford University Press Oxford UK 2011

[9] M M Chawla and P S Rao ldquoA Noumerov-type methodwith minimal phase-lag for the integration of second orderperiodic initial-value problems II explicit methodrdquo Journal ofComputational and Applied Mathematics vol 15 no 3 pp 329ndash337 1986

[10] M M Chawla and P S Rao ldquoAn explicit sixthndashorder methodwith phasendashlag of order eight for 11991010158401015840 = 119891(119905 119910)rdquo Journal of


Computational and Applied Mathematics vol 17 no 3 pp 363ndash368 1987

[11] C J Cramer Essentials of Computational Chemistry JohnWileyamp Sons Chichester UK 2004

[12] J M Franco and M Palacios ldquoHigh-order P-stable multistepmethodsrdquo Journal of Computational and Applied Mathematicsvol 30 no 1 pp 1ndash10 1990

[13] L G Ixaru andM Rizea ldquoComparison of some four-stepmeth-ods for the numerical solution of the Schrodinger equationrdquoComputer Physics Communications vol 38 no 3 pp 329ndash3371985

[14] L G Ixaru and M Micu Topics in Theoretical Physics CentralInstitute of Physics Bucharest Romania 1978

[15] L G Ixaru and M Rizea ldquoA numerov-like scheme for thenumerical solution of the Schrodinger equation in the deepcontinuum spectrum of energiesrdquo Computer Physics Commu-nications vol 19 no 1 pp 23ndash27 1980

[16] F Jensen Introduction to Computational Chemistry JohnWileyamp Sons Chichester UK 2007

[17] Z Kalogiratou T Monovasilis and T E Simos ldquoNewmodifiedRunge-Kutta-Nystrom methods for the numerical integrationof the Schrodinger equationrdquo Computers amp Mathematics withApplications vol 60 no 6 pp 1639ndash1647 2010

[18] A A Kosti Z A Anastassi and T E Simos ldquoConstructionof an optimized explicit Runge-Kutta-Nystrom method forthe numerical solution of oscillatory initial value problemsrdquoComputersamp Mathematics with Applications vol 61 no 11 pp3381ndash3390 2011

[19] J D Lambert Numerical Methods for Ordinary DifferentialSystems The Initial Value Problem John Wiley amp Sons NewYork NY USA 1991

[20] A R Leach Molecular ModellingmdashPrinciples and ApplicationsPearson Essex UK 2001

[21] T Lyche ldquoChebyshevian multistep methods for ordinary differ-ential equationsrdquoNumerische Mathematik vol 19 no 1 pp 65ndash75 1972

[22] T Monovasilis Z Kalogiratou and T E Simos ldquoA familyof trigonometrically fitted partitioned Runge-Kutta symplecticmethodsrdquo Applied Mathematics and Computation vol 209 no1 pp 91ndash96 2009

[23] T Monovasilis Z Kalogiratou and T E Simos ldquoExponentiallyfitted symplectic RungendashKuttandashNystrom methodsrdquo AppliedMathematicsamp Information Sciences vol 7 no 1 pp 81ndash85 2013

[24] G A Panopoulos and T E Simos ldquoAn eight-step semi-embedded predictor-corrector method for orbital problemsand related IVPs with oscillatory solutions for which thefrequency is unknownrdquo Journal of Computational and AppliedMathematics vol 290 pp 1ndash15 2015

[25] G A Panopoulos and T E Simos ldquoAn optimized symmetric8-step semi-embedded predictor-corrector method for IVPswith oscillating solutionsrdquo Applied Mathematics amp InformationSciences vol 7 no 1 pp 73ndash80 2013

[26] G A Panopoulos and T E Simos ldquoA new optimized symmetricEmbedded Predictor-Corrector Method (EPCM) for initial-value problemswith oscillatory solutionsrdquoAppliedMathematicsand Information Sciences vol 8 no 2 pp 703ndash713 2014

[27] D F Papadopoulos and T E Simos ldquoA modified Runge-Kutta-Nystrom method by using phase lag properties for thenumerical solution of orbital problemsrdquo Applied Mathematicsamp Information Sciences vol 7 no 2 pp 433ndash437 2013

[28] D F Papadopoulos and T E Simos ldquoThe use of phase lagand amplification error derivatives for the construction of amodified Runge-Kutta-Nystrommethodrdquo Abstract and AppliedAnalysis vol 2013 Article ID 910624 11 pages 2013

[29] G D Quinlan and S Tremaine ldquoSymmetric multistep methodsfor the numerical integration of planetary orbitsrdquo AstronomicalJournal vol 100 no 5 pp 1694ndash1700 1990

[30] A D Raptis and A C Allison ldquoExponential-fitting methodsfor the numerical solution of the Schrodinger equationrdquo PhysicsCommunications vol 14 no 1-2 pp 1ndash5 1978

[31] D P Sakas and T E Simos ldquoMultiderivative methods ofeighth algebraic orderwithminimal phase-lag for the numericalsolution of the radial Schrodinger equationrdquo Journal of Com-putational and Applied Mathematics vol 175 no 1 pp 161ndash1722005

[32] T E Simos ldquoDissipative trigonometrically-fitted methods forlinear second-order IVP s with oscillating solutionrdquo AppliedMathematics Letters vol 17 no 5 pp 601ndash607 2004

[33] T E Simos ldquoClosed NewtonndashCotes trigonometrically-fittedformulae of high order for long-time integration of orbitalproblemsrdquoAppliedMathematics Letters vol 22 no 10 pp 1616ndash1621 2009

[34] T E Simos ldquoExponentially and trigonometrically fitted meth-ods for the solution of the Schrodinger equationrdquo Acta Appli-candae Mathematicae vol 110 no 3 pp 1331ndash1352 2010

[35] T E Simos ldquoNew stable closed Newton-Cotes trigonometri-cally fitted formulae for long-time integrationrdquo Abstract andApplied Analysis vol 2012 Article ID 182536 15 pages 2012

[36] T E Simos ldquoOptimizing a hybrid two-step method for thenumerical solution of the Schrodinger equation and relatedproblems with respect to phase-lagrdquo Journal of Applied Math-ematics vol 2012 Article ID 420387 17 pages 2012

[37] T E Simos ldquoHigh order closed NewtonndashCotestrigonometricallyndashfitted formulae for the numerical solutionof the Schrodinger equationrdquo Applied Mathematics andComputation vol 209 no 1 pp 137ndash151 2009

[38] H Ramos Z KalogiratouThMonovasilis and T E Simos ldquoAnoptimized two-step hybrid block method for solving generalsecond order initial-value problemsrdquo Numerical Algorithmsvol 72 no 4 pp 1089ndash1102 2016

[39] T Monovasilis Z Kalogiratou and T E Simos ldquoConstructionof exponentially fitted symplectic Runge-Kutta-Nystrommeth-ods from partitioned Runge-Kutta methodsrdquo MediterraneanJournal of Mathematics vol 13 no 4 pp 2271ndash2285 2016

[40] T E Simos ldquoOn the explicit four-step methods with vanishedphase-lag and its first derivativerdquo Applied Mathematics 120572 Infor-mation Sciences vol 8 no 2 pp 447ndash458 2014

[41] T E Simos ldquoExponentially fitted Runge-Kutta methods for thenumerical solution of the Schrodinger equation and relatedproblemsrdquo Computational Materials Science vol 18 no 3-4 pp315ndash332 2000

[42] S Stavroyiannis and T E Simos ldquoOptimization as a functionof the phasendashlag order of nonlinear explicit twondashstep P-stablemethod for linear periodic IVPsrdquoAppliedNumericalMathemat-ics vol 59 no 10 pp 2467ndash2474 2009

[43] E Stiefel and D G Bettis ldquoStabilization of Cowellrsquos methodrdquoNumerische Mathematik vol 13 pp 154ndash175 1969

[44] K Tselios andT E Simos ldquoRunge-Kuttamethodswithminimaldispersion and dissipation for problems arising from com-putational acousticsrdquo Journal of Computational and AppliedMathematics vol 175 no 1 pp 173ndash181 2005


[45] C Tsitouras I T Famelis and T E Simos ldquoOn modifiedRunge-Kutta trees and methodsrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 2101ndash2111 2011

[46] R Vujasin M Sencanski J Radic-Peric and M Peric ldquoAcomparison of various variational approaches for solving theone-dimensional vibrational Schrodinger equationrdquo MATCHCommunications in Mathematical and in Computer Chemistryvol 63 no 2 pp 363ndash378 2010

[47] A D Raptis and J R Cash ldquoA variable step method forthe numerical integration of the onendashdimensional Schrodingerequationrdquo Computer Physics Communications vol 36 no 2 pp113ndash119 1985

[48] J R Dormand M E A El-Mikkawy and P J Prince ldquoFamiliesof Runge-Kutta-Nystrom formulaerdquo IMA Journal of NumericalAnalysis vol 7 no 2 pp 235ndash250 1987

[49] J R Dormand and P J Prince ldquoA family of embeddedRunge-Kutta formulaerdquo Journal of Computational and AppliedMathematics vol 6 no 1 pp 19ndash26 1980

[50] K Mu and T E Simos ldquoA RungendashKutta type implicit highalgebraic order two-step method with vanished phase-lag andits first second third and fourth derivatives for the numericalsolution of coupled differential equations arising from theSchrodinger equationrdquo Journal of Mathematical Chemistry vol53 no 5 pp 1239ndash1256 2015

[51] H Ning and T E Simos ldquoA low computational cost eightalgebraic order hybrid method with vanished phase-lag and itsfirst second third and fourth derivatives for the approximatesolution of the Schrodinger equationrdquo Journal of MathematicalChemistry vol 53 no 6 pp 1295ndash1312 2015

[52] M Liang and T E Simos ldquoA new four stages symmetric two-step method with vanished phase-lag and its first derivative forthe numerical integration of the Schrodinger equationrdquo Journalof Mathematical Chemistry vol 54 no 5 pp 1187ndash1211 2016

[53] X Xi and T E Simos ldquoA new high algebraic order fourstages symmetric two-step method with vanished phase-lagand its first and second derivatives for the numerical solutionof the Schrodinger equation and related problemsrdquo Journal ofMathematical Chemistry vol 54 no 7 pp 1417ndash1439 2016

[54] Z Zhou and T E Simos ldquoA new two stage symmetric two-step method with vanished phase-lag and its first second thirdand fourth derivatives for the numerical solution of the radialSchrodinger equationrdquo Journal of Mathematical Chemistry vol54 no 2 pp 442ndash465 2016

[55] T E Simos ldquoMultistage symmetric two-step P-stable methodwith vanished phase-lag and its first second and third deriva-tivesrdquo Applied and Computational Mathematics vol 14 no 3pp 296ndash315 2015

[56] R B Bernstein A Dalgarno H Massey and I C Perci-val ldquoThermal scattering of atoms by homonuclear diatomicmoleculesrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 274 no 1359 pp 427ndash442 1963

[57] R B Bernstein ldquoQuantum mechanical (phase shift) analysis ofdifferential elastic scattering of molecular beamsrdquo The Journalof Chemical Physics vol 33 no 3 pp 795ndash804 1960

Research ArticleStability of the Cauchy Additive Functional Equation onTangle Space and Applications

Soo Hwan Kim

Department of Mathematics Dong-eui University Busan 614-714 Republic of Korea

Correspondence should be addressed to Soo Hwan Kim sh-kimdeuackr

Received 18 July 2016 Accepted 9 October 2016


Copyright copy 2016 Soo Hwan Kim This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce real tangle and its operations as a generalization of rational tangle and its operations to enumerating tangles byusing the calculus of continued fraction and moreover we study the analytical structure of tangles knots and links by using newoperations between real tangles which need not have the topological structure As applications of the analytical structure we provethe generalized Hyers-Ulam stability of the Cauchy additive functional equation 119891(119909 oplus 119910) = 119891(119909) oplus 119891(119910) in tangle space which is aset of real tangles with analytic structure and describe the DNA recombination as the action of some enzymes on tangle space

1 Introduction

In 1970 Conway introduced rational tangles and algebraictangles for enumerating knots and links by using Conwaynotation The rational tangles are defined as the familyof tangles that can be transformed into the trivial tangleby sequence of twisting of the endpoints Given a tangletwo operations called the numerator and denominator byconnecting the endpoints of the tangle produce knots or 2-component links To enumerating and classifying knots thetheory of general tangles has been introduced in [1]

Moreover the rational tangles are classified by theirfractions by means of the fact that two rational tangles areisotopic if and only if they have the same fraction [1] Thisimplies the known result that the rational tangles correspondto the rational numbers one to one It is clear that everyrational number can be written as continued fractions withall numerators equal to 1 and that every real number 119903corresponds to a unique continued fraction which is finiteif 119903 is rational and infinite if 119903 is irrational Thus the con-tinued fractions give the relationship between the analyticalstructure and topological structure under a certain restrictedoperator See [2] for example There are some operationsthat can be performed on tangles as the sum multiplicationrotation mirror image and inverted image

Topologically the sum and multiplication on tangles aredefined as connecting two endpoints of one tangle to twoendpoints of another However they are not commutative anddo not preserve the class of rational tangles Furthermore thesum and multiplication of two rational tangles are a rationaltangle if and only if one of two is an integer tangle [3]Thus theset of rational tangles is not a group because it was discoveredthat not all rational tangles form a closed set under the sumand multiplication Considering a braid of rational tangles aseries of strands that are always descending the set of braidsis a group under braid multiplication

In 1940 Ulam introduced the stability problem of func-tional equations during talk before a Mathematical Collo-quium at the University of Wisconsin [4]

Given a group 1198661 a metric group (1198662 119889) and a positivenumber 120598 does there exist a number 120575 gt 0 such thatif a function 119891 1198661 rarr 1198662 satisfies the inequality119889(119891(119909119910) 119891(119909)119891(119910)) lt 120575 for all 119909 119910 isin 1198661 there exists ahomomorphism 119879 1198661 rarr 1198662 such that 119889(119891(119909) 119879(119909)) lt 120598for all 119909 isin 1198661

Analytically the stability problem of functional equationsoriginated from a question of Ulam concerning the stabilityof group homomorphisms The functional equation

119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW

Figure 1 Rational trivial prime locally knotted tangles

is called the Cauchy additive functional equation In particu-lar every solution of the Cauchy additive functional equationis said to be an additive mapping In [5] Hyers gave the firstaffirmative partial answer to the question of Ulam for Banachspaces In [6] Hyersrsquo theorem was generalized by Aoki foradditive mappings and by Rassias for linear mappings byconsidering an unbounded Cauchy difference in [7] In [8]a generalization of the Rassias theorem was obtained byGavruta by replacing the unbounded Cauchy difference bya general control function in Rassiasrsquo approach There aremany interesting stability problems of several functionalequations that have been extensively investigated by a numberof authors See [9ndash14]

In recent years new applications of tangles to the field ofmolecular biology have been developed In particular knottheory gives a nice way to model DNA recombination Therelationship between topology and DNA began in the 1950swith the discovery of the helical Crick-Watson structure ofduplex DNA The mathematical model is the tangle modelfor site-specific recombination which was first introduced bySumners [15] This model uses knot theory to study enzymemechanisms Therefore rational tangles are of fundamentalimportance for the classification of knots and the study ofDNA recombination In this paper we introduce new tanglescalled real tangles to apply the stability problem and DNArecombination on tangles

In Section 2 we introduce real tangles and operationsbetween tangles which can be performed to make up tanglespace and having analytical structure Moreover we showthat the operations together with two real tangles will alwaysgenerate a real tangle In Section 3 we prove the Hyers-Ulamstability of the Cauchy additive functional equation in tanglespace and study the DNA recombination on real tangles asapplications of knots or links

2 Continued Fractions and Tangle Space

A rational tangle is a proper embedding of two unorientedarcs (strings) 1199051 and 1199052 in 3-ball 1198613 so that the endpoints ofthe arcs go to a specific set of 4 points on the equator of 1198613usually labeled NWNE SW SE This is equivalent to sayingthat rational tangles are defined as the family of tangles thatcan be transformed into the trivial tangle by a sequence oftwisting of the endpoints Note that there are tangles thatcannot be obtained in this fashion they are the prime tanglesand locally knotted tangles For example see Figure 1

Geometrically we have the following operations betweenrational tangles the integer (the horizontal) tangles denoted

by 119877(119899) consist in 119899 horizontal twists 119899 isin 119885 the mirrorimage of 119877(119899) denoted by minus119877(119899) or 119877(minus119899) is obtainedfrom 119877(119899) by switching all the crossing and the rotationof 119877(119899) denoted by 119877119877(119899) is obtained by rotation 119877(119899)counterclockwise by 90∘ Moreover the inverse (the vertical)tangle of 119877(119899) denoted by 119877(1119899) is defined by minus119877119877(119899) or119877119877(minus119899) as the composition of the rotation andmirror of119877(119899)For example 119877(13) = minus119877119877(3) and 119877(1minus3) = minus119877119877(minus3) Forthe trivial tangle 119877(0) we define 119877(infin) = 119877119877(0) or 119877(10)

Generally every rational tangle can be represented bythe continued fractions119862(1198861 1198862 119886119899) as following Conwaynotation

119862 (1198861 1198862 119886119899) = 1198861+ 11198862 + sdot sdot sdot + (1 (119886(119899minus1) + (1119886119899)))

(2)

for 1198861 isin 119885 1198862 119886119899 isin 119885 minus 0 and 119899 even or odd and wedenote it by 119877(1198861 1198862 119886119899) See Figure 2

By Conway [1] rational tangles are classified by fractionsby fact of the following two rational tangles are isotopicif and only if they have the same fraction For example119862(2 minus2 3) and 119862(1 2 2) represent the same tangles upto isotopy because they have a fraction 75 Thereforethe rational tangles 119877(1198861 1198862 119886119899) with the exception of119877(0) 119877(plusmn1) 119877(infin) are said to be in canonical form if |119886119899| gt1 119886119894 = 0 for 2 le 119894 le 119899 Note that all nonzero entrieshave the same sign and every rational tangle has a uniquecanonical form The canonical form of the example above is119877(1 2 2) and the following corollary which is a direct resultof Conwayrsquos theorem [1] will give us a means of classifyingrational tangles by way of fractions

Corollary 1 There is a one-to-one correspondence betweencanonical rational tangles and rational numbers 120573120572 isin 119876 cupinfin where120572 isin 119873cup0120573 isin 119885 gcd(120572 120573) = 1119877(infin) = 119877(10)

Now we define infinite tangles by infinite continuedfractions of irrational numbers that the chain of fractionsnever ends as the following

119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2

Figure 2 Rational tangles 119877(1198861 1198862 119886119899) for 119899 odd or even

where 1198861 is allowed to be 0 but all subsequent terms 119886119894 mustbe positive that is 1198861 1198862 isin 119885 119886119894 gt 0 for 119894 ge 2 Note thatlet 119862119894 = 119862(1198861 1198862 119886119894) for 119894 ge 1 and then the limit

119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)

is a unique irrational number and that let 119877119894 =119877(1198861 1198862 119886119894) for 119894 ge 1 be canonical rational tanglesand then the infinite tangles denoted by 119877(1198861 1198862 1198863 ) aredefined by the limit of canonical rational tangles 119877119894 as

119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)

Example 2 Let 1198621 = 119862(1) 1198622 = (1 1) 119862119894 = 119862(1 11 1) and then the limit has an irrational number

lim119894rarrinfin119862119894 = 119862 (1 1 1 ) = 1 + radic22 (6)

Corollary 3 There is a one-to-one correspondence betweeninfinite tangles and irrational numbers

Note that 120572 isin 119877minus119876 is quadratic irrational if and only if itis of the form

120572 = 119886 + radic119887119888 (7)

where 119886 119887 119888 isin 119885 119887 gt 0 119888 = 0 and 119887 is not the squareof a rational number Thus an irrational number is calledquadratic irrational if it is a solution of a quadratic equation1198601199092 +119861119909+119863 = 0 where119860 119861119863 isin 119885 and119860 = 0 Moreover 120572is eventually periodic of the form

119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)

where the bar indicates the periodic part with 119901 terms Thusan infinite tangle is said to be periodic if it has eventuallyperiodic of the form

119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)

See Figure 5 for119873 = 1 and 119901 = 2

Figure 3 Canonical rational tangle 119877(2 1 1 3 4)

Corollary 4 There is a one-to-one correspondence betweeninfinite periodic tangles and quadratic irrational numbers

Finally tangles are said to be real if it is rational tangles orinfinite tangles and so the real tangles are finite if it is rationaltangles and infinite if it is infinite tangles Thus continuedfractions of finite real tangles are rational and continuedfractions of infinite real tangles are irrational Moreoverinfinite real tangle is periodic if it is infinite periodic tanglesand so continued fractions of infinite periodic real tanglesare quadratic irrational numbers Let 119903 be a real number thatcorresponds to finite or infinite continued fractionsThen bycorollaries the fact that 119877(119903) has a unique real tangle can beproved The following is examples of the corollaries above

Example 5 (1) Let 7730 isin 119876 be a rational number Then119877(7730) = 119877(2 1 1 3 4) is a canonical rational tangle of7730 See Figure 3(2) Let 119890 = 2718281 sdot sdot sdot isin 119877 minus 119876 be an irrational numberused as the base of the natural logarithm function Then


middot middot middot⋱

Figure 4 Infinite tangle 119877(2 1 2 1 1 4 )

⋱

Figure 5 Infinite periodic tangle 119877(3 2 1)

119877(119890) = 119877(2 1 2 1 1 4 1 1 ) is an infinite tangle of 119890 SeeFigure 4(3)Considering a quadratic irrational number (5+radic3)2119877((5 + radic3)2) = 119877(3 2 1 2 1 ) = 119877(3 2 1) is an infiniteperiodic tangle of quadratic irrational number (5+radic3)2 SeeFigure 5 In Figure 5 the boxesmean periodic parts as119877(2 1)

Now we introduce the operations on real tangles withanalytical structure which need not have the topologicalstructure However on rational tangles our operations areapplicable to geometrical results obtained from topologicalstructure Our operations need to discuss the generalizedHyers-Ulam stability of the Cauchy additive functional equa-tion119891(119909+119910) = 119891(119909)+119891(119910) andDNA recombinations in nextsection

Let functions 119901 119877 times 119877 rarr 119877 defined by 119901(1199031 1199032) = 1199031 +1199032 and 119898 119877 times 119877 rarr 119877 defined by 119898(1199031 1199032) = 1199031 times 1199032be two binary operators on 119877 and 120601 a map from the set ofreal tangles 119879 to the set of real number 119877 in which tangles119877(1198861 1198862 119886119899) or 119877(1198861 1198862 1198863 ) are corresponding to the

rational numbers or irrational numbers respectively one tooneThen for 120601 119879 rarr 119877 and each tangles 1199051 1199052 isin 119879 we definea map 120601lowast 119879 times 119879 rarr 119877 times 119877 by 120601lowast(1199051 1199052) = (120601(1199051) 120601(1199052)) andtwo binary operators oplus and otimes on a nonempty set 119879 by

oplus 119879 times 119879 997888rarr 119879otimes 119879 times 119879 997888rarr 119879 (10)

where

oplus (1199051 1199052) = 120601minus1119901120601lowast (1199051 1199052) otimes (1199051 1199052) = 120601minus1119898120601lowast (1199051 1199052)

(11)

For convenience we write 1199051 oplus 1199052 and 1199051 otimes 1199052 by oplus(1199051 1199052)and otimes(1199051 1199052) respectivelyLemma 6 Let 120601 119879 rarr 119877 be a map from the set of real tanglesto the set of real numbers at which real tangles 119877(1198861 1198862 119886119899)or 119877(1198861 1198862 ) are corresponding to the real number 119903 where 119903has continued fraction 119862(1198861 1198862 119886119899) or 119862(1198861 1198862 )Then120601 satisfies the following properties for all 119905 1199051 1199052 isin 119879

(1) 120601 (minus119905) = minus120601 (119905)(2) 120601 (minus119905119877) = 1

120601 (119905)(3) 119905 oplus (minus119905) = 119877 (0)(4) 119905 otimes (minus119905119877) = 119877 (1)(5) 120601 (1199051 oplus 1199052) = 120601 (1199051) + 120601 (1199052)(6) 120601 (1199051 otimes 1199052) = 120601 (1199051) times 120601 (1199052)

(12)

Proof Let 119905 be a real tangle in 119879 and 120601(119905) = 119903 isin 119877 continuedfraction corresponding to 119905(1) Since minus119905 is obtained from 119905 by switching all thecrossing 120601(minus119905) = minus119903 and so 120601(minus119905) = minus120601(119905)(2) Since 119905119877 is obtained by rotation 119905 counterclockwise by90∘ 120601(119905119877) = minus1119903 and so 120601(minus119905119877) = 1119903 by (1)Thus 120601(minus119905119877) =1120601(119905)(3) 119905 oplus (minus119905) = 120601minus1119901120601lowast(119905 minus119905) = 120601minus1119901(120601(119905) 120601(minus119905)) =120601minus1(120601(119905) + 120601(minus119905)) = 120601minus1(0) = 119877(0) by (1)(4) 119905 otimes (minus119905119877) = 120601minus1119898120601lowast(119905 minus119905119877) = 120601minus1119898(120601(119905) 120601(minus119905119877)) =120601minus1(120601(119905) times 120601(minus119905119877)) = 120601minus1(1) = 119877(1) by (2)(5) 120601(1199051 oplus 1199052) = 120601(120601minus1119901120601lowast(1199051 1199052)) = 119901(120601(1199051) 120601(1199052)) =120601(1199051) + 120601(1199052)(6) 120601(1199051 otimes 1199052) = 120601(120601minus1119898120601lowast(1199051 1199052)) = 119898(120601(1199051) 120601(1199052)) =120601(1199051) times 120601(1199052)

In the following we show that operators oplus and otimes togetherwith two real tangles will always generate a real tangle

Theorem 7 Let 119879 be the set of real tangles and oplus the binaryoperation on 119879 Then (119879 oplus) is a groupProof To show associative of oplus let 1199051 1199052 1199053 isin 119879


Then

(1199051 oplus 1199052) oplus 1199053 = 120601minus1119901120601lowast (120601minus1119901120601lowast (1199051 1199052) 1199053)= 120601minus1119901120601lowast (120601minus1 (120601 (1199051) + 120601 (1199052)) 1199053)= 120601minus1 ((120601 (1199051) + 120601 (1199052)) + 120601 (1199053))= 120601minus1 (120601 (1199051) + (120601 (1199052) + 120601 (1199053)))= 120601minus1119901120601lowast (1199051 120601minus1 (120601 (1199052) + 120601 (1199053)))= 120601minus1119901120601lowast (1199051 120601minus1119901120601lowast (1199052 1199053))= 1199051 oplus (1199052 oplus 1199053)

(13)

For the remainder the identity element is the trivial tangle119877(0) isin 119879 and the inverse of 119905 isin 119879 is minus119905 See Lemma 6 Thusthe set 119879 forms a group with respect to oplus

In particular for 1199051 1199052 isin 119879 we write 1199051 ⊖ 1199052 for 1199051 oplus (minus1199052)Theorem 8 Let 119879 be the set of real tangles and otimes the binaryoperation on 119879 Then (119879 otimes) is a groupProof To show associative of otimes let 1199051 1199052 1199053 isin 119879 Then

(1199051 otimes 1199052) otimes 1199053 = 120601minus1119898120601lowast (120601minus1119898120601lowast (1199051 1199052) 1199053)= 120601minus1119898120601lowast (120601minus1 (120601 (1199051) times 120601 (1199052)) 1199053)= 120601minus1 ((120601 (1199051) times 120601 (1199052)) times 120601 (1199053))= 120601minus1 (120601 (1199051) times (120601 (1199052) times 120601 (1199053)))= 120601minus1119898120601lowast (1199051 120601minus1 (120601 (1199052) times 120601 (1199053)))= 120601minus1119898120601lowast (1199051 120601minus1119898120601lowast (1199052 1199053))= 1199051 otimes (1199052 otimes 1199053)

(14)

For the remainder the identity element is the integer tangle119877(1) isin 119879 and the inverse of 119905 isin 119879 is minus119905119877 and denoted as119905minus1 where 119905119877 means rotation counterclockwise by 90∘ SeeLemma 6Thus the set119879 forms a group with respect to otimesCorollary 9 (119879 oplus) and (119879 otimes) are abelian groups

For the symbolization we write 119899119905 for 119905 oplus 119905 oplus sdot sdot sdot oplus 119905(119899summands) and write 119905119899 for 119905 otimes 119905 otimes sdot sdot sdot otimes 119905(119899 products) Notethat by distributive law between two operations we have

(1199051 oplus 1199052) otimes (1199053 oplus 1199054) lArrrArr(1199051 otimes 1199053) oplus (1199051 otimes 1199054) oplus (1199052 otimes 1199053) oplus (1199052 otimes 1199054) (15)

For 1199051 1199052 119905 isin 119879 and 119903 isin 119877 let oplus 119879 times 119879 rarr 119879and sdot 119877 times 119879 rarr 119879 be addition and scalar multiplicationoperators respectively defined by oplus(1199051 1199052) = 1199051 oplus 1199052 andsdot(119903 119905) = 120601minus1119898(119903 120601(119905)) isin 119879 denoted by 119903 sdot 119905 where (sdot sdot) 119877 times 119879 rarr 119877 times 119877 is a map Then the set 119879 of the real tangleswith addition and scalar multiplication operators satisfy thefollowing result

Theorem 10 (119879 oplus sdot) is a vector spaceProof ByTheorem 7 (119879 oplus) is a groupMoreover we have thefollowing properties

(1) 119903 sdot (1199051 oplus 1199052) = 120601minus1119898(119903 120601 (1199051 oplus 1199052))= 120601minus1119898(119903 120601120601minus1119901120601lowast (1199051 1199052))= 120601minus1119898(119903 120601 (1199051) + 120601 (1199052))= 120601minus1 (119903 times (120601 (1199051) + 120601 (1199052)))= 120601minus1 (119903 times 120601 (1199051) + 119903 times 120601 (1199052))= 120601minus1119901 (120601120601minus1119898(119903 120601 (1199051)) 120601120601minus1119898(119903 120601 (1199052)))= 120601minus1119901 (120601 (119903 sdot 1199051) 120601 (119903 sdot 1199052))= 120601minus1119901120601lowast (119903 sdot 1199051 119903 sdot 1199052)= (119903 sdot 1199051) oplus (119903 sdot 1199052)

(16)

Similarly

(2) (1199031 + 1199032) sdot 119905 = (1199031 sdot 119905) oplus (1199032 sdot 119905) (3) 1199031 sdot (1199032 sdot 119905) = (11990311199032) sdot 119905(4) 1 sdot 119905 = 119905

(17)

hold Thus (119879 oplus sdot) is a vector space over 119877In particular we write 119903(1199051 oplus 1199052) for 119903 sdot (1199051 oplus 1199052) that is the

operator sdot is often omitted

Remark 11 Wehave the following relations which are provedfrom the above facts

(1) minus (1199051 oplus 1199052) = minus1199051 oplus (minus1199052) (2) minus (1199051 ⊖ 1199052) = (minus1199051 oplus 1199052) (3) 1199051 ⊖ 1199052 = minus1199052 oplus 1199051

(18)

If we define a function 119863 119879 times 119879 rarr 119877+ as 119863(1199051 1199052) =|120601(1199051) minus120601(1199052)| for each 1199051 1199051 isin 119879 then (119879119863) is a metric spacefrom the following theorem

Theorem 12 (119879119863) is a metric space

Proof Let 1199051 1199052 1199053 isin 119879 Then

(1) 119863 (1199051 1199052) = 0 lArrrArr1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 0 lArrrArr120601 (1199051) = 120601 (1199052) lArrrArr1199051 = 1199052

(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2

Figure 6 Addition and multiplication of tangles

(3) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816= 1003816100381610038161003816120601 (1199051) minus 120601 (1199053) + 120601 (1199053) minus 120601 (1199052)1003816100381610038161003816le 1003816100381610038161003816120601 (1199051) minus 120601 (1199053)1003816100381610038161003816 + 1003816100381610038161003816120601 (1199053) minus 120601 (1199052)1003816100381610038161003816le 119863 (1199051 1199053) + 119863 (1199053 1199052)

(19)

Thus (119879119863) is a metric space

Define the norm of 119905 by 119905 = 119863(119877(0) 119905) Then we havethat 119905 = |120601(119905)| and 119863(1199051 1199052) = |120601(1199051) minus 120601(1199052)| = |120601(1199051) +120601(minus1199052)| = |120601(1199051 oplus (minus1199052))| = 1199051 oplus (minus1199052) = 1199051 ⊖ 1199052Remark 13 We have the following relations

(1) 10038171003817100381710038171199051 otimes 11990521003817100381710038171003817 = 119863 (119877 (0) 1199051 otimes 1199052) = 10038161003816100381610038160 minus 120601 (1199051 otimes 1199052)1003816100381610038161003816= 1003816100381610038161003816120601 (1199051) times 120601 (1199052)1003816100381610038161003816= 1003816100381610038161003816120601 (1199051)1003816100381610038161003816 times 1003816100381610038161003816120601 (1199052)1003816100381610038161003816= 100381710038171003817100381711990511003817100381710038171003817 times 100381710038171003817100381711990521003817100381710038171003817

(2) 10038171003817100381710038171199051 oplus 11990521003817100381710038171003817 = 119863 (119877 (0) 1199051 oplus 1199052) = 10038161003816100381610038160 minus 120601 (1199051 oplus 1199052)1003816100381610038161003816= 1003816100381610038161003816120601 (1199051) + 120601 (1199052)1003816100381610038161003816le 1003816100381610038161003816120601 (1199051)1003816100381610038161003816 + 1003816100381610038161003816120601 (1199052)1003816100381610038161003816= 100381710038171003817100381711990511003817100381710038171003817 + 100381710038171003817100381711990521003817100381710038171003817

(20)

Define an inequality ≺ (resp ≼) of 1199051 1199052 on 119879 as 1199051 ≺1199052 (resp 1199051 ≼ 1199052) hArr 120601(1199051) lt 120601(1199052) (resp 120601(1199051) le 120601(1199052))Remark 14 (1) For each 120598 gt 0 119905 lt 120598means that minus119905120598 ≺ 119905 ≺119905120598 for some 120601(119905120598) = 120598

(2) For each 120598 gt 0 we have that10038171003817100381710038171199051 ⊖ 11990521003817100381710038171003817 lt 120598 lArrrArr

minus119905120598 ≺ 1199051 ⊖ 1199052 ≺ 119905120598 lArrrArrminus119905120598 oplus 1199052 ≺ 1199051 ≺ 119905120598 oplus 1199052 lArrrArr1199052 ⊖ 119905120598 ≺ 1199051 ≺ 1199052 oplus 119905120598

(21)

for some 120601(119905120598) = 120598In order to determine a group (generally a vector space)

from the set of rational tangles (generally real tangles)two binary operators oplus and otimes are necessary For otheroperators restricted on rational tangles addition (denoteby ) and multiplication (denote by lowast) of horizontal andvertical rational tangles are considered in [1] In detail themultiplication of two rational tangles is defined as connectingthe top two ends of one tangle to the bottom two endpointsof another and the addition of two rational tangles is definedas connecting the two leftmost endpoints of one tangle withthe two rightmost points of the other as shown in Figure 6

However the addition of two rational tangles is notnecessarily rational but it can be algebraic tangle [1] Forexample it can be easily seen that the sum of 119877(12) and119877(12) is not a rational tangle As the results in [3] themultiplication (resp addition) of two rational tangles will berational tangle if one of two is a vertical (resp horizontal)tangle Note that as a special case of rational tangles the setof braids is a group under the multiplication Therefore twooperators oplus and otimes on rational tangles are a generalizationof operators and lowast introduced in [1] For two operators oplusand otimes on real tangles we do not know yet whether it has atopological or geometrical structure

In Section 3 we will study some applications for twooperators oplus and otimes on real tangles In this paper the set (119879119863)of the real tangles with a metric119863 is called the tangle space


3 Some Applications on Tangle Space

31 Tangle Space and Stability Let 119879 be the tangle space and119891 119879 rarr 119879 a mapping Then we prove the generalized Hyers-Ulam stability of the Cauchy additive functional equation asfollows

Theorem 15 Let 119891 119879 rarr 119879 be a mapping such that

1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)

for all 119909 119910 isin 119879 and for some 120598 gt 0 Then there exists a uniqueadditive mapping 119876 119879 rarr 119879 such that 119891(119909) ⊖ 119876(119909) lt 120598 forall 119909 isin 119879Proof Suppose that 119891 119879 rarr 119879 is a mapping such that

1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)

for all 119909 119910 isin 119879 and for some 120598 gt 0 Then we have

1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 997904rArr119891 (119909) oplus 119891 (119910) ⊖ 119886120598 ≺ 119891 (119909 oplus 119910) ≺ 119891 (119909) oplus 119891 (119910) oplus 119886120598 (I)

for some 120601(119886120598) = 120598(1) Putting 119909 = 119910 in (I)

2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984 (∵ by (III)) 997904rArr

119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)

(3) Putting 119909 = 4119909 in (II)2119891 (4119909) ⊖ 119886120598 ≺ 119891 (8119909) ≺ 2119891 (4119909) oplus 119886120598 997904rArr119891 (4119909)4 ⊖ 1198861205988 ≺

119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)

Putting recursively 119909 = (2119899minus1)119909 in (II) we have100381710038171003817100381710038171003817100381710038171003817119891 (2119899119909)2119899 ⊖ 119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

2119899 minus 12119899 120598 lt 120598 (V)

where 119899 ge 1Define 119876 119879 rarr 119879 by

119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)

for all 119909 isin 119879 that is lim119899rarrinfin119891(2119899119909)2119899 ⊖ 119876(119909) = 0Now putting 119909 = 2119899119909 and 119910 = 2119899119910 in 119891(119909 oplus 119910) ⊖ 119891(119909) ⊖119891(119910) lt 120598 we have1003817100381710038171003817119891 (2119899119909 oplus 2119899119910) ⊖ 119891 (2119899119909) ⊖ 119891 (2119899119910)1003817100381710038171003817 lt 120598 997904rArr100381710038171003817100381710038171003817100381710038171003817119891 (2119899 (119909 oplus 119910))

2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr

100381710038171003817100381710038171003817100381710038171003817 lim119899rarrinfin119891 (2119899 (119909 oplus 119910))

2119899 ⊖ lim119899rarrinfin

119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)

Thus 119876(119909 oplus 119910) ⊖ 119876(119909) ⊖ 119876(119910) = 0 that is 119876(119909 oplus 119910) =119876(119909) oplus 119876(119910) Moreover from (V)100381710038171003817100381710038171003817100381710038171003817 lim119899rarrinfin

119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)

Thus we have 119876(119909) ⊖119891(119909) lt 120598Therefore this means that119876is an additive mapping such that 119876(119909) ⊖ 119891(119909) lt 120598

To prove the uniqueness of the additive mapping 119876assume that there is another additive mapping 1198761015840 119879 rarr 119879such that

1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)


middot middot middot

R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7

From the fact 1198761015840(2119899119909) = 21198991198761015840(119909) we have 1198761015840(119909) =1198761015840(2119899119909)2119899 Since 119891(119909) ⊖ 1198761015840(119909) lt 120598 we have that1198761015840 (119909) ⊖ 119886120598 ≺ 119891 (119909) ≺ 1198761015840 (119909) oplus 119886120598 997904rArr1198761015840 (2119899119909) ⊖ 119886120598 ≺ 119891 (2119899119909) ≺ 1198761015840 (2119899119909) oplus 119886120598 997904rArr1198761015840 (2119899119909)2119899 ⊖ 1198861205982119899 ≺

119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr

100381710038171003817100381710038171003817100381710038171003817 lim119899rarrinfin119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)

where 120598 gt 0 and 120601(119886120598) = 120598 Thus we have 119876(119909) ⊖ 1198761015840(119909) = 0that is 119876(119909) = 1198761015840(119909) This completes the proof

For example let 119891 119879 rarr 119879 be a mapping defined by119891(119909) = 1199031 sdot 119909 oplus 1199032 where 1199031 1199032 isin 119877 119909 isin 119879 In fact 1199032 means1199032 sdot119877(1)Then119891 is not additive mapping However amapping119891 119879 rarr 119879 defined by119891(119909) = 119903sdot119909 119903 isin 119877 and119909 isin 119879 is additiveIn tangle space 119879 let 119909 = 119877(1 1 2) 119891(119909) = 119909 oplus 3 and 119899 = 4then 119891(16119909) = 119877(29 1 2) and 119891(16119909)16 = 119877(1 1 5 1 6)where 16119909 = 119877(26 1 2) See Figure 7

Generally for each 119899 the real tangles are as the following119891 (2119899119909)2119899 = 119877(14 + sum119899119896=1 52119896minus121198993 ) (30)

and so the additive mapping 119876(119909) is real tangle as thefollowing

119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)

32 Tangle Space and Rational Knot or Link Suppose that1198791015840 sub 119879 is the set of rational tangles Given 119905 isin 1198791015840 thenumerator closure 119873(119905) is formed by connecting the NWand NE endpoints and the SW and SE endpoints and thedenominator closure 119863(119905) is formed by connecting the NWand SW endpoints and connecting the NE and SE endpointsWe note that two operations119873(119905) and119863(119905) by connecting theendpoints of 119905 produce knots or 2-component links calledrational knot or link if 119905 is a rational tangle and that every2-bridge knot is a rational knot because it can be obtained asthe numerator or denominator closure of a rational tangleSee Figure 8

Let 1198791015840 sub 119879 be the set of rational tangles and 119870 theset of rational knots or links Then for given 119905 isin 1198791015840 itallows defining a function 119873 1198791015840 rarr 119870 in order that 119873(119905)is the numerator closure The following theorem discussesequivalence of rational knots or links obtained by taking thenumerator closure of rational tangles We call this theoremthe tangle classification theorem

Theorem 16 (see [16]) Let 119877(119901119902) and 119877(11990110158401199021015840) be the ratio-nal tangles with reduced fractions 119901119902 and 11990110158401199021015840 respectivelyThen119873(119877(119901119902)) and119873(119877(11990110158401199021015840)) are topologically equivalentif and only if 119901 = 1199011015840and 119902plusmn equiv 1199021015840mod119901

For example 119873(119877(0 3 2)) = 119873(119877(27)) = 119873(119877(21)) =119873(119877(2)) because 7 equiv 1 mod 2


NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)

Figure 8 The numerator closure and denominator closure of tangle

Corollary 17 If two rational tangles are isotopic then theireach numeratorrsquos closures are topological equivalent

Proof Let 119877(119901119902) and 119877(11990110158401199021015840) be the rational tangles withreduced fractions 119901119902 and 11990110158401199021015840 respectively Then 119901 = 1199011015840and 119902 = 1199021015840 because 119877(119901119902) and 119877(11990110158401199021015840) are isotopic

Thus by Theorem 16 119873(119877(119901119902)) and 119873(119877(11990110158401199021015840)) aretopological equivalent

However there is a counterexample for the converse ofCorollary 17 as follows

Example 18 Let 119877(1 2 2) and 119877(2 3) be two rational tangleswith fractions 75 and 73 respectively By Theorem 16119873(119877(75)) and119873(119877(73)) are topological equivalent but twotangles 119877(75) and 119877(73) are not isotopic

Define the numerator closure of the sum of two rationaltangles as the following

119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)

where gcd(1199091 1199101) = gcd(1199092 1199102) = 1 Note that the rationalknot or link

119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)

is denoted by 119887(11990921199101+11990911199102 11990911199092) called the 2-bridge knot orlink and that 119887(11990921199101 + 11990911199102 11990911199092) is to be the 2-bridge knotif 11990921199101 + 11990911199102 is odd number and the 2-bridge link if not

A tangle equation is an equation of the form119873(119860 oplus 119861) =119870 where 119860 119861 isin 1198791015840 and 119870 isin 119870 Solving equations of thistype will be useful in the tangle model and gaining a betterunderstanding of certain enzyme mechanisms [15]

Example 19 Considering rational tangles 119877(2) and 119877(2317)then119877(2)oplus119877(2317) is the rational tangle119877(3 2 1 5) becauseof 119877(2317) = 119877(1 2 1 5) Thus a tangle equation 119873(119877(2) oplus119877(2317)) = 119873(119877(3 2 1 5)) is representing the 2-bridgeknot 119887(57 17) from the computation of the numerator closureabove

If one of the tangles in the equation is unknown and theother tangle and the knot 119870 are known then there is onetangle as the solution of equation but it is not unique In factlet 119860 be known rational tangle and 119870 rational knot or linkThen there are two different rational tangles as the solutionof the equation 119873(119883 oplus 119860) = 119870 which is the topologicalequivalent under numerator operation inTheorem 16

Example 20 Let 119860 = 119877(13) 2-bridge knot 119870 = 119887(5 2)known and 119883 = 119877(119909) unknown Then 119883 = 119877(136) is asolution of the equation119873(119883 oplus 119860) = 119870 However 119887(5 2) and119887(5 3) are topological equivalent fromTheorem 16Thus119883 =119877(43) is the other solution of the equation119873(119883 oplus 119860) = 119870 if119870 = 119887(5 3)

FromTheorem 16 and Corollary 17 we obtain the follow-ing corollary by the method as in Example 20

Corollary 21 Let119860 and119870 be known rational tangle in1198791015840 andrational knot or link in 119870 respectively Then there exist twosolutions119883 isin 1198791015840 of the equation119873(119883 oplus 119860) = 11987033 Tangle Space and DNA Suppose that tangles 119878 119879and 119877 below are rational As discussed in the introductionof Section 1 DNA must be topologically manipulated byenzymes in order for vital life processes to occur Theactions of some enzymes can be described as site-specific


Specific site

(a) (b)

Recombination

Recombinaseenzyme

Figure 9 Example of a site-specific recombination

recombination Site-specific recombination is a process bywhich a piece of DNA is moved to another position on themolecule or to import a foreign piece of a DNA moleculeinto it Recombination is used for gene rearrangement generegulation copy number control and gene therapy Thisprocess is mediated by an enzyme called a recombinase Asmall segment of the genetic sequence of the DNA that isrecognized by the recombinase is called a recombination siteor a specific site See Figure 9 Note that the tangle in Figure 9is where the enzyme acts

The DNA molecule and the enzyme itself are calledthe synaptic complexes Before recombination the DNAmolecule is called the substract that is it is unchanged bythe enzyme After recombination the DNAmolecule is calledthe product In Figure 9 (a) is the substract and (b) is theproductThis is the result which replaces a tangle (or enzyme)with a new tangle called the recombination tangle Thus thefollowing tangle equations hold

119873(119878 oplus 119879) = the substract119873 (119878 oplus 119877) = the product (34)

where the product is a result that the enzyme replaces atangle 119879 with a tangle 119877 Generally it will repeat the tanglereplacement a number of times If it is possible to observethe substract and the product then the ideal situation wouldbe to determinate tangles 119878 119879 and 119877 from the tangleequations However it is a hard question in general to solvethe tangle equations because there are only two equationsbut three unknowns As above the tangle model has beenused to mathematically show the enzyme mechanism ofrecombination See [17] for similar examples

Example 22 Let the knot types of the substrate and theproduct yielding equations in the recombination variables 119878119879 and 119877 be as follows

119873(119878 oplus 119879) = the unknot 119887 (1 1) 119873 (119878 oplus 119877) = the trefoil knot 119887 (3 1) (35)

Then solutions of the equations are either (119878 119879 119877) =(119877(minus12) 119877(0) 119877(2)) or (119878 119879 119877) = (119877(12) 119877(0) 119877(minus2))

In our study of tangle space with operator oplus it is stillunknown how to construct a link or knot associated with agiven real tangle and analyze DNAmolecules by real tangles

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J H Conway ldquoAn enumeration of knots and links and some oftheir algebraic propertiesrdquo in Proceedings of the Conference onComputational Problems in Abstract Algebra J Leech Ed pp329ndash358 Pergamon Press Oxford UK 1970

[2] A Ya Khinchin Continued Fractions Dover 1997 (republica-tion of the 1964 edition of Chicago Univ Press)

[3] L H Kauffman and S Lambropoulou ldquoOn the classification ofrational tanglesrdquo Advances in Applied Mathematics vol 33 no2 pp 199ndash237 2004

[4] SMUlamProblems inModernMathematicsWiley NewYorkNY USA 1960

[5] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[6] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[7] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[8] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[9] J Aczel and J Dhombres Functional Equations in SeveralVariables Cambridge University Press Cambridge UK 1950

[10] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[11] K Cieplinski ldquoApplications of fixed point theorems to theHyers-Ulam stability of functional equationsmdasha surveyrdquoAnnalsof Functional Analysis vol 3 no 1 pp 151ndash164 2012

[12] S Czerwik Functional Equations and Inequalities in SeveralVariables World Scientific Publishing River Edge NJ USA2002

[13] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[14] T M Rassias ldquoOn the stability of functional equations inBanach spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 251 no 1 pp 264ndash284 2000

[15] D W Sumners ldquoUntangling DNArdquo The Mathematical Intelli-gencer vol 12 no 3 pp 71ndash80 1990

[16] L H Kauffman and S Lambropoulou ldquoFrom tangle fractions toDNArdquo in Topology inMolecular Biology Biological andMedicalPhysics Biomedical Engineering pp 69ndash110 Springer BerlinHeidelberg Berlin Germany 2007

[17] M Vazquez and D W Sumners ldquoTangle analysis of Ginsite-specific recombinationrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 136 no 3 pp 565ndash5822004

Research ArticleNew Periodic Solutions for a Class of Zakharov Equations

Cong Sun12 and Shuguan Ji1

1College of Mathematics Jilin University Changchun 130012 China2The City College of Jilin Jianzhu University Changchun 130111 China

Correspondence should be addressed to Shuguan Ji jisgjlueducn

Received 10 May 2016 Revised 15 August 2016 Accepted 1 September 2016

Academic Editor Mariano Torrisi

Copyright copy 2016 C Sun and S Ji This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Through applying the Jacobian elliptic function method we obtain the periodic solution for a series of nonlinear Zakharovequations which contain Klein-Gordon Zakharov equations Zakharov equations and Zakharov-Rubenchik equations

1 Introduction

For most of nonlinear evolution equations we have manymethods to obtain their exactly solutions such as hyperbolicfunction expansion method [1] the transformation method[2] the trial function method [3] the automated method [4]and the extended tanh-function method [5] But these meth-ods can only obtain solitary wave solution and cannot be usedto deduce periodic solutions The Jacobian function methodprovides a way to find periodic solutions for some nonlinearevolution equations In particular in the research of plasmaphysics theory quantum mechanics fluid mechanics andoptical fiber communication we frequently meet kinds ofZakharov equations

Hence in this paper inspired by Angulo Pavarsquos work [6]we are concerned to obtain exact periodic solutions of a seriesof Zakharov equations

119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 119906119905119905 minus 119906119909119909 minus |V|119909119909 = 0119894V119905 + 120572V119909119909 minus 119906V = 0(1)

and Zakharov-Rubenchik equations

119894119861119905 + 120596119861119909119909 minus 119896 (119906 minus 12120582120588 + 119902 |119861|2 119861) = 0120579120588119905 + (119906 minus 120582120588)119909 = minus119896 |119861|2119909 120579119906119905 + (120573120588 minus 120582119906)119909 = 12119896120582 |119861|2119909 (2)

2 The Periodic Solution for Klein-GordonZakharov Equations

The Klein-Gordon Zakharov equations are used to showthe interaction between langmuir wave and ion wave in theplasma which has the following form

119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)

where 119906(119909 119905) denotes the biggest moment scale componentproduced by electron in electric field 119899(119909 119905) denotes thespeed of deviations between the ions at any position and thatat equilibrium position

Now we suppose that it possesses solitary wave solutionsof the following form

119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)

where 119888 is a traveling wave speed and 1198882 lt 1 is a constantBy substituting (4) into (3) we can obtain

119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)

where 1198881119898 are integration constantsIn what follows we are concerned with the periodic

solution of (3) thus we need to require 1198881 = 0Therefore (7) implies

119899 = 1205932 + 1198981198882 minus 1 (8)

Moreover through (8) and (5) we have that

(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)

Multiplying (9) by 1205931015840 and integrating once we obtain

1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)

where ℎ is a nonzero integration constant Furthermore ℎ119898and 119888 satisfy the following condition

(1 minus 1198882 minus 119898)2 minus 4ℎ (1 minus 1198882) ge 0(1 minus 1198882 minus 119898) minus radic(1 minus 1198882 minus 119898)2 minus 4 (1 minus 1198882) ℎ ge 00 lt ℎ lt (1 minus 1198882 minus 119898)24 (1 minus 1198882)

(11)

so that

(1205931015840)2 = 12 1(1198882 minus 1)2 (12057821 minus 1205932) (1205932 minus 12057822) (12)

where minus1205781 1205781 minus1205782 and 1205782 are the zeros of the polynomial119865(120582) = minus((1198882minus1+119898)(1198882minus1)2)1205822minus(12)(1205824(1198882minus1)2)+2ℎ(1198882minus1) Without losing generality we assume that 1205781 gt 1205782 gt 0Therefore we can deduce that 1205781 le 120593 le 1205782 1205781 and 1205782 satisfy

12057821 + 12057822 = 2 (1 minus 1198882) minus 21198981205782112057822 = 4ℎ (1 minus 1198882) (13)

Let 120601 = 1205931205781 1198962 = (12057821minus12057822)12057821 Hence (12) can be writtenas

(1206011015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1206012) (1206012 minus 1 + 1198962) (14)

Moreover we define a new variable120595 which satisfies1205951015840 ge0120595(0) = 0 by the relation1206012 = 1minus1198962sin2120595 through a tediouscomputation we obtain that

(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0

119889120591radic1 minus 1198962sin2120591 = radic22 11 minus 1198882 1205781120585 (16)

According to the definition of the Jacobian elliptic func-tion 119910 = sn(119906 119896) we can obtain that sin120595 = sn(1205781radic120573120585 119896)Here 120573 = radic22(1 minus 1198882) So

120601 (120585) = radic1 minus 1198962sn2 (1205781radic22 11 minus 1198882 120585 119896)= dn(1205781radic22 11 minus 1198882 120585 119896)

(17)

By returning to initial variable we obtain that

120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)

is a dnoidal solution of (10)Furthermore dn has fundamental period 2119870(119896) that is

dn(119906 + 2119896 119896) = dn(119906 119896) and 119870(119896) is the complete ellipticintegral of first kind So the dnoidal wave solution 120593 hasfundamental period 119879 given by

119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)

So by applying the method of the Jacobian ellipticfunction and inspired by Angulo Pavarsquos ideas we obtain that(3) has the periodic traveling solution of the following form

119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

= (1205781dn (1205781 (radic22) (1 (1 minus 1198882)) 120585 119896))2 + 1198981198882 minus 1 (20)

Moreover 119879 and 1198962 can be also rewritten as the followingform

119879 (1205782) = 2radic2119870 (119896 (1205782)) (1 minus 1198882)radic2 (1 minus 1198882) minus 2119898 minus 12057822

1198962 = 2 (1 minus 1198882) minus 2119898 minus 2120578222 (1 minus 1198882) minus 2119898 minus 12057822 (21)

Furthermore if 1205782 rarr 0 then 119896(1205782) rarr 1 Hence119870(119896(1205782)) rarr +infin so that 119879(1205782) rarr +infin If 1205782 rarr radic1 minus 1198882 minus 119898then 119896(1205782) rarr 0 Hence 119870(119896(1205782)) rarr 1205872 so that 119879(1205782) rarrradic2120587(1 minus 1198882)radic1 minus 1198882 minus 119898

Next we will show that for an arbitrary but

fixed 119871 radic1 minus (1198712 + 119871radic1198712 minus 8120587119898)4120587 lt |1198880| ltradic1 + (1198712 + 119871radic1198712 minus 8120587119898)4120587 We can obtain that there


is a unique 12057820 = 12057820(1198880) isin (0 radic1 minus 11988820 minus 119898) such that119879(1205782) = 119879(1205782(119888)) = 119871 is a fundamental period for the dnoidalwave solution (18)

Theorem 1 Let 119871 gt 0 be arbitrary but fixedConsider radic1 minus (1198712 + 119871radic1198712 minus 8120587119898)4120587 lt |1198880| ltradic1 + (1198712 + 119871radic1198712 minus 8120587119898)4120587 and unique 12057820 = 12057820(1198880) isin(0 radic1 minus 11988820 minus 119898) such that 119879(1205782) = 119871 Then there existan interval 119880(1198880) an interval 119868(12057820) and a unique smoothfunction 119865 119880(1198880) rarr 119868(12057820) such that 119865(119888) = 1205782 and

119871 = 2radic2119870 (119896 (1205782)) (1 minus 1198882)radic2 (1 minus 1198882) minus 2119898 minus 12057822 (22)

where 119888 isin 119880(1198880) 1205782 isin 119880(12057820)Proof Based on the ideas establish in Angulo Pavarsquos work [6]we will give a brief proof Now we consider the open set

Ω = (1205782 119888) 1205782 isin (0radic1 minus 11988820 minus 119898) 119888

isin (minusradic1 + 1198712 + 119871radic1198712 minus 81205871198984120587 radic1 + 1198712 + 119871radic1198712 minus 81205871198984120587 ) cup(minusinfin

minus radic1 minus 1198712 + 119871radic1198712 minus 81205871198984120587 )

cup(radic1 minus 1198712 + 119871radic1198712 minus 81205871198984120587 +infin) isin 1198772

(23)

We define Φ Ω rarr 119877 by

Φ(1205782 119888) = 2radic2119870 (119896 (1205782)) (1 minus 1198882)radic2 (1 minus 1198882) minus 2119898 minus 12057822 (24)

Here 1198962 = (2(1 minus 1198882) minus 2119898 minus 212057822)(2(1 minus 1198882) minus 2119898 minus 12057822)Hypothesise Φ(12057820 1198880) = 119871 In what follows we will provethat 120597Φ1205971205782 lt 0

From (24) we can obtain that

120597Φ1205971205782 =2radic21205782119870(119896 (1205782)) (1 minus 1198882)(2 (1 minus 1198882) minus 2119898 minus 12057822)32+ 2radic2 (1 minus 1198882)radic2 (1 minus 1198882) minus 2119898 minus 12057822

119889119870119889119896 1205971198961205971205782 (25)

From 1198962 we can deduce 119896(1205782 119888) is a strictly decreasingfunction of 1205782

According to Jacobian elliptic function theory [7] wehave

119889119870119889119896 = 119864 minus (1 minus 1198962)119870119896 (1 minus 1198962) (26)

119889119864119889119896 = 119864 minus 119870119896 (27)

Here 119864 is the complete elliptic integral of second kindNext we adopt the reduction to absurdity to prove120597Φ1205971205782 lt 0 Now we assume that 120597Φ1205971205782 ge 0 So we have

the following inequality

119896 (2 (1 minus 1198882 minus 119898) minus 12057822)119870 (119896 (1205782))ge 2 (1 minus 1198882 minus 119898) 119889119870119889119896

(28)

Indeed substituting (26) into (25) and using the methodof enlarging and reducing we obtain that

(1 minus 1198962) [4 (1 minus 1198882 minus 119898) minus 212057822]119870 (119896)ge 2 (1 minus 1198882 minus 119898)119864 (119896) (29)

From 1198962 we can easily deduce that

1 minus 1198962 = 120578222 (1 minus 1198882) minus 2119898 minus 12057822 2 minus 1198962 = 2 (1 minus 1198882) minus 21198982 (1 minus 1198882) minus 2119898 minus 12057822

(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)

Let 1205742(1205782 119888) = 1 minus 1198962 1205741015840 = minus119896(1205971198961205971205782)(1120574) gt 0 So 120574is an increasing function of 1205782 isin (0 radic1 minus 119888 2 minus 119898) Moreover120574(0) = 1 120574(radic1 minus 1198882 minus 119898) = 1

Define

119891 (120574) = (1 + 1205742) 119864 (radic1 minus 1205742) minus 21205742119870(radic1 minus 1205742) (32)

Due to 120597Φ1205971205782 ge 0 119891(120574) le 0However by (27) and a simple computation

1198911015840 (120574) = 3120574 (119864(radic1 minus 1205742) minus 119870(radic1 minus 1205742)) lt 0 (33)

so 119891(120574) is a decreasing function Furthermore 119891(1) = 0 sofor 120574 isin (0 1) 119891(120574) gt 0

It is in conflict with our assumption So we obtain ouraffirmation that 120597Φ1205971205782 lt 0 Hence by the implicit functiontheorem there exists a unique smooth function 119865 definedin a neighborhood 119880(1198880) of 1198880 so that Φ(1205782(119888) 119888) = 119871 for119888 isin 119880(1198880) Hence we obtain (22)


3 The Periodic Solution forZakharov Equations

Now we consider the following Zakharov equations

119906119905119905 minus 119906119909119909 minus (|V|)2119909119909 = 0119894V119905 minus 120572V119909119909 minus 119906V = 0 (34)

which describe the high frequency moment of plasmawhere 119906(119909 119905) denotes the ion number density variationV(119909 119905) denotes the electric field intensity of slowly varyingamplitude and 120572 isin 119877

We seek the solitary solutions of (34) in the form

119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)

where 119906 and 120593 are real functions 119888 is a traveling speed and1198882 = 1Substituting (35) into (34) we can obtain that

(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)

minus12057212059310158401015840 minus (119888 + 2120572) 1198941205931015840 + (119888 + 120572) 120593 minus 119906120593 = 0 (37)

By (36) we can deduce

(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)

Here 1198882 119903 are integration constantsNext we consider periodic solutions of (34) So we need

to require 1198882 = 0 Therefore by (38)

119906 = 1205932 + 1199031198882 minus 1 (39)

Furthermore by (37) we can obtain that

119888 = minus2120572 (40)

So by (39) and (40) (37) can be rewritten

12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)

Multiplying (41) by 1205931015840 and integrating once we obtain12 (1205931015840)2 + 121205932 (1 + 2119903119888 minus 1198883 ) + 121205934 = ℎ (42)

where ℎ is a nonzero integration constantHence

(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)

Here minus119886119894 119886119894 minus119887 and 119887 are polynomial roots of 119865(119905) = minus1199054 minus1199052(1198862 minus 1198872) + 2ℎ 119887 gt 0 and1198862 minus 1198872 = 1 + 2119903119888 minus 1198883 11988621198872 = 2ℎ (44)

Let 120594 = 120593119887 (suppose 120594(0) = 0) and 1198962 = 1198872(1198862 + 1198872)Hence (43) becomes

(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)

Now we define 1205942 = 1 minus sin2120595 so we get that(1205951015840)2 = (1198862 + 1198872) (1 minus 1198962sin2120595) (46)

Let 120591 = radic1198862 + 1198872 According to the definition of theJacobian elliptic function sn we can obtain sin120595(120585) =sn(120591120585 119896) So that 1205942 = 1 minus sn2(120591120585 119896)

So we obtain the solution of (41)120593 (120585) = 119887 cn (120591120585 119896) (47)

According to Jacobian elliptic functions theory cn hasperiod 4119870(119896) and we can obtain that the cnoidal wavesolution has period 119879 which is given by

119879 = 4119870 (119896)120591 (48)

So we can obtain the periodic solutions of (34)

119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)

Furthermore it follows that for 1198962 = 1198872(1198862+1198872) we have1198962 isin (0 12)If 120591 rarr 0 then 119896 rarr 0 and 119870(119896) rarr 1205872 119879(120591) rarr +infin

Furthermore if 120591 rarr +infin 119879(120591) rarr 0Next we will prove that for an arbitrary but fixed 119871 gt 0

there exists a unique 120572 = 120572(119888) isin (0 +infin) such that 119879(120572(119888)) =119871 is a fundamental period of the cnoidal wave solution (47)So we have the following theorem

Theorem 2 Let 119871 gt 0 but fixed Consider 1198880 gt 0 and theunique 1205910 = 1205910(1198880) isin (0 +infin) such that 119879(1205910) = 119871 Then thereexists an internal119860(1198880) around 1198880 an internal 119861(1198870) around 1198870and a unique smooth function Υ such that Υ(1198880) = 1205720 and

119871 = 4119870 (119896)120591 (50)

where 119888 isin 119860(1198880) 120591 = Υ(119888)Proof The proof is similar to that of Theorem 1 For detailssee Theorem 1 (see also Angulo Pava [6 8 9])

4 The Periodic Solution forZakharov-Rubenchik Equations

Now we consider the Zakharov-Rubenchik equations

119894119861119905 + 120596119861119909119909 minus ℓ (119906 minus 12120582120588 + 119902 |119861|2 119861) = 0120579120588119905 + (119906 minus 120582120588)119909 = minusℓ |119861|2119909

120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)


which describe the dynamics of small amplitude Alfvenwaves propagating in a plasma Here 119861(119909 119905) denotes themagnetic field 119906(119909 119905) the fluid speed and 120588(119909 119905) the densityof mass Moreover 120596 ℓ 120582 120579 120573 119902 are real constants

Motivated by Oliveira [10] we look for the solitary wavesof (51) as follows

119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)

where 119886 119887 and 119860 are real functions and 119888 denotes travelingspeed

Putting (52) into (51) from the second and third equa-tions of (51) we can deduce

119886 = 119886 (119888) = ℓ [(1205822) (120582 + 119888120579) minus 120573]120573 minus (119888120579 + 120582)2 119887 = 119887 (119888) = ℓ (minus119888120579 minus 1205822)120573 minus (119888120579 + 120582)2

(53)

Moreover from the first equation of (51) we can obtainthat

(119888 minus 120596)119860 + 119894 (2120596 minus 119888) 1198601015840 + 12059611986010158401015840minus ℓ (119886 minus 12120582119887 + 119902)1198603 = 0

(54)

So by (54) it implies

119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)

Let 1198871 = minusℓ(119886 minus (1205822)119887 + 119902)119888 gt 0 and multiplying (56) by1198601015840 and integrating once we obtain

[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)

where minus1198861119894 1198861119894 minus1198872 1198872 are the polynomial roots of 119865(119903) =minus11988711199034 minus 1199032 + ℎ Moreover

11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)

Let 120594 = 1198601198872 (suppose 120594(0) = 0) and 1198962 = 11988722 (11988621 + 11988722 )Hence (57) becomes

(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)

Now we define 1205942 = 1 minus sin2120595 so we get that(1205951015840)2 = 1198871 (11988621 + 11988722 ) (1 minus 1198962sin2120595) (60)

Let 120572 = radic1198871(11988621 + 11988722 ) According to the definition ofthe Jacobin elliptic function sn we can obtain sin120595(120585) =sn(120572120585 119896) So 1205942 = 1 minus sn2(120572120585 119896)

So we obtain the solution of (56)

119860 (120585) = 1198872cn (120572120585 119896) (61)

Since cn has fundamental period 4119870(119896) we can obtainthat solution (61) has fundamental period 119879 which is

119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)

Hence (51) have the periodic solutions of the followingform

119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)

Moreover from (58) and the definition of 1198962 it followsthat

1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)

Moreover if 1198861 rarr 0 1198962 rarr 1 119879(1198861) rarr +infin If 1198861 rarr +infin1198962 rarr 12 119879(1198861) rarr 0Next we will show that for an arbitrary but fixed 119871 gt 0

there exists 1198861 = 1198861(119888) isin (0 +infin) such that 119879(1198861(119888)) = 119871is a fundamental period of the cnoidal wave solution (61)Motivated by Angulo Pavarsquos result [6] we have the followingtheorem

Theorem3 For 119871 gt 0 arbitrary and fixed consider 1198880 gt 0 andthe unique 1198861 = 1198861(1198880) isin (0 +infin) Then there exist an interval119862(1198880) around 1198880 an interval 119863(1198861) around 1198861 and a uniquesmooth function119867 119862(1198880) rarr 119863(1198861) such that119867(1198880) = 1198861 and

119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)

where 119888 isin 119862(1198880) 1198861 = 119867(119888)Proof The idea andmethod are similar to those ofTheorem 1For details please see Theorem 1 (see also Angulo Pava [6 89])


5 Conclusion

Inspired by Angulo Pavarsquos idea by applying Jacobian ellipticfunctionmethod we have obtained newperiodwave solutionfor Klein-Gordon Zakharov equations Zakharov equationsand Zakharov-Rubenchik equations In particular the solu-tions of (18) (47) and (58) were not found in the previousworkThemethod can help to look for periodic solution for aclass of nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This work was supported by Scientific Research Foundationof the City College of Jilin Jianzhu University (no 2016002)NSFC Grant (nos 11322105 J1310022 and 11671071) and 973Program (nos 2012CB821200 and 2013CB834102)

References

[1] M L Wang ldquoSolitary wave solutions for variant Boussinesqequationsrdquo Physics Letters A vol 199 no 3-4 pp 169ndash172 1995

[2] C T Yan ldquoA simple transformation for nonlinear wavesrdquoPhysics Letters A vol 224 no 1-2 pp 77ndash84 1996

[3] S K Liu Z T Fu S D Liu and Q Zhao ldquoA simple fast methodin finding particular solutions of some nonlinear PDErdquo AppliedMathematics and Mechanics vol 22 no 3 pp 326ndash331 2001

[4] E J Parkes and B R Duffy ldquoTravelling solitary wave solutionsto a compound KdV-Burgers equationrdquo Physics Letters A vol229 no 4 pp 217ndash220 1997

[5] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

[6] J Angulo Pava ldquoNonlinear stability of periodic traveling wavesolutions to the Schrodinger and the modified Korteweg-deVries equationsrdquo Journal of Differential Equations vol 235 no1 pp 1ndash30 2007

[7] P F Byrd and M D Friedman Handbook of Elliptic Integralsfor Engineers and Scientists Springer New York NY USA 2ndedition 1971

[8] J Angulo Pava ldquoStability of cnoidal waves to Hirota-Satsumasystemsrdquo Matematica Contemporanea vol 27 pp 189ndash2232004

[9] J Angulo Pava Nonlinear Dispersive Equations Existence andStability of Solitary and Periodic Travelling Wave Solutionsvol 156 of Mathematical Surveys and Monographs AmericanMathematical Society Providence RI USA 2009

[10] F Oliveira ldquoStability of the solitons for the one-dimensionalZakharov-Rubenchik equationrdquo Physica D vol 175 no 3-4 pp220ndash240 2003







Editorial Board




Contents




























Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References
















Editorial Board




Contents




























Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References













Editorial Board




Contents




























Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References











Editorial Board




Contents




























Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References











Contents




























Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References































Acknowledgments













1 Introduction



















2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References

























































































1 Introduction












(1)



119884119909119910 (119892) = 119892119909119911 + 1198921199091198921199111198841199092119911 (119892) = 1198921199092119911 + 1198921199091198922119911 + 2119892119909119911119892119911 + 1198921199091198922119911 sdot sdot sdot

(2)



(3)


120591119909 (120592) = 1205921199091205912119909 (120592 120596) = 1205922119909 + 120596119909119909

120591119909119910 = 120596119909119910 + 1205921199091205921199101205913119909 (120592 120596) = 1205923119909 + 31205921199091205962119909 + 12059231199091205912119909119910 (120592 120596) = 1205922119909119910 + 2120592119909120596119909119910 + 1205922119909120592119910 + 1205921199101205962119909 sdot sdot sdot

(4)



(5)



(6)



=

0 119903sum119894=1119899119894 is odd

11987511989911199091119899119903119909119903 (119902) 119903sum119894=1119899119894 is even

(7)


1198752119909 (119902) = 11990221199091198754119909 (119902) = 1199024119909 + 311990222119909

11987521199092119910 (119902) = 11990221199092119910 + 11990221199091199022119910 + 21199022119909119910119875119909119910 (119902) = 1199021199091199101198753119909119910 (119902) = 1199023119909119910 + 31199021199091199101199022119909 sdot sdot sdot

(8)




119906119909119909119909119910 + 3 (119906119906119910)119909 + 3119906119909119909120597minus1119909 119906119910 + 3119906119909119906119910 + 2119906119910119905 minus 3119906119909119911= 0 (9)




119902 = 2 ln119865 lArrrArr 119906 = 119902119909119909 = 2 (ln119865)119909119909 (12)





119863119898119901119909119863119899119901119905119865 sdot 119865 = ( 120597120597119909 + 1205721199011205971205971199091015840)119898 ( 120597120597119905 + 120572119901

1205971205971199051015840)119899

sdot 119865 (119909 119905) 119865 (1199091015840 1199051015840)100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905 =119898sum119894=0119899sum119895=0(119898119894 )(

119899119895)

sdot 120572119894119901120572119895119901 120597119898minus119894120597119909119898minus119894

1205971198941205971199091015840119894

120597119899minus119895120597119905119899minus119895

1205971198951205971199051015840119895119865 (119909 119905)

sdot 119865 (1199091015840 1199051015840)1003816100381610038161003816100381610038161003816100381610038161199091015840=1199091199051015840=119905= 119898sum119894=0119899sum119895=0(119898119894 )(


120597119894+119895119865 (119909 119905)120597119909119894120597119905119895 119898 119899 ge 0

(14)


120572119904119901 = (minus1)119903119901(119904) 119904 = 119903119901 (119904) mod 119901


+ 61198651199091199101198652119909

(15)



(16)



119865 = 3sum119894=02sum119904=02sum119897=0

2sum119895=0119888119894119904119897119895119909119894119910119904119911119897119905119895 (17)





(18)




(19)



(6119909119905 + 5119909 + 4)2 (20)



(119909 + 119910)2 (21)




minus40minus20

020

40

minus40minus20020

40

minus12

minus10

minus8

minus6

minus4

minus2

x

t

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

t

(b)


minus40minus2002040


minus50

minus40

minus30

minus20

minus10

0

x

y

u

(a)


minus30

minus20

minus10

0

10

20

30

40

x

y

(b)







1198902120587119894119899120585+1205871198941198992120591 (22)



1198902120587119894119899120585+1205871198941198992120591


1198902120587119894119898120585+1205871198941198982120591


minus10

0

10


minus200

minus150

minus100

minus50

0

x

t

u

(a)

minus5 0 5

minus8

minus6

minus4

minus2

0

2

4

6

8

x

t

(b)




119869 (119863119909 119863119910 119863119911 119863119905) 1198902120587119894119899120585+1205871198941198992120591

sdot 1198902120587119894119898120585+1205871198941198982120591



119869 [2120587119894 (119899 minus 119898) (119896 119897 119886 119908)]sdot 1198902120587119894(119899+119898)120585+120587119894(1198992+1198982)120591



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)]


119869 (119902) 1198902120587119894119902120585(23)



119869 [2120587119894 (2119899 minus 119902) (119896 119897 119886 119908)] 119890120587119894(1198992+(119902minus119899)2)120591


119869 (2120587119894119861119896 2120587119894119861119897 2120587119894119861119886 2120587119894119861119908)


(24)


recursion formula

119869 (119902) = 119869 (0) 119890120587119894119899119902120591 119902 = 2119899119869 (1) 119890120587119894(21198992+2119899)120591 119902 = 2119899 + 1 (25)



(256120587411989941198963119897 minus 3212058721198992119897119908 + 4812058721198992119886119896

minus 120582) 11989021205871198941198992120591 = 0119869 (1) = 119899=infinsum



(26)



321205872119899211989711989021205871198941198992120591


(16 (2119899 minus 1)4 12058741198963119897 + 121205872 (2119899 minus 1)2 119886119896)

sdot 119890120587119894(21198992minus2119899+1)12059111990221 =


119890120587119894(21198992minus2119899+1)120591


81205872 (2119899 minus 1)2 119897119890120587119894(21198992minus2119899+1)120591


11989021205871198941198992120591


(256120587411989941198963119897 + 4812058721198992119886119896) 11989021205871198941198992120591

(27)


minus10minus5

05

10

minus10minus5

05

10minus4

minus3

minus2

minus1

0

x

y

u

(a)


minus5

0

5

10

x

y

(b)



11990211119908 + 11990213 minus 12058211990231 = 011990222119908 + 11990212 minus 12058211990221 = 0 (28)



(29)


119906 = 2 (ln119865)119909119909 (30)




119865 = 1 + 1205761198911 + 12057621198912 + 12057631198913 + sdot sdot sdot (31)


119865 = sum120583119894 120583119895=01

exp119873sum119894gt119895119860 119894119895120583119894120583119895 +

119873sum119895=1120583119895120578119895

(32)

where

120578119895 = 119896119895119909 + 119897119895119910 + 119886119895119911 + 119908119895119905


2 (119897119895 + 119897119894) (119908119894 + 119908119895) minus 3 (119886119894 + 119886119895) (119896119894 + 119896119895) + (119896119894 + 119896119895)3 (119897119895 + 119897119894) (33)



119865 = 1 + 11989111198911 = 119890120578

120578 = 119896119909 + 119897119910 + 119886119911 + 119908119905(34)


119906 = 2 (ln119865)119909119909 = 11989622 sech2

1205852 (35)


minus20

minus10

0

10

20 minus20

minus10

0

10

20minus002

0002

(a)

minus20minus15

minus10minus5

05

1015

20 minus20minus15

minus10minus5

05

1015

20

(b)



119865 = 1 + 1198911 + 11989121198911 = 1198901205781 + 1198901205782

1198912 = 1198901205781+1205782+11986012 (36)

where


3 (1198972 minus 1198971)2 (1198972 + 1198971) (1199081 + 1199082) minus 3 (1198861 + 1198862) (1198961 + 1198962) + (1198961 + 1198962)3 (1198972 + 1198971)

(37)


119906 = 2 (ln119865)119909119909 = 2 [ln (1 + 1198911 + 1198912)]119909119909 (38)




7 Conclusion



minus20minus10

010

20

minus20minus10

010

200

01

02

03

04

05

(a)minus60

minus40minus20

020

minus60minus40

minus200

20minus005

0

005

01

015

02

(b)


minus50

0

50 minus50

0

50

minus01

0

01

02

minus50

0

50

minus50

0

50

minus01

0

01

02

minus50

0

50 minus50

0

50

minus005

0

005

01

015

02



Appendix


1198652 = 119888101011988821111199111199051198883110+ (11988811101198882111

2 + 119888111121198883110 + 6119888211111988831102) 11991011991111990511988821111198883110

+ 11990511990921199101199111198882111 + 11988802111199102119911119905 minus 119888101011988811111199111198882111 + 1198881010119909119911+ 1199091199101199111198881110minus 1198881111 (11988811101198882111

2 + 119888111121198883110 + 9119888211111988831102) 11991011991111988821113

+ 11990931199101199111198883110 minus 1198880211119888111111988831101199102119911

11988821112 + 1198880211119888311011990911991021199111198882111+ 1199051199091199101199111198881111

1198653 = 11990521199101199111198880112minus (1198880112

21198882111 minus 119888011211988811111198881112 minus 3119888111211988821112) 11991011991111990511988811122

+ 11990511990921199101199111198882111 + 119888011211988812111199102119911119905

1198881112 + 11988812111199091199102119911119905

+ 211988801121198881211119910119911211990531198882111 + 1199051199091199101199111198881111 + 2119888111211988812111199091199101199112119905

31198882111+ 11990521199091199101199111198881112

1198654 = 11990521199101199111198880112 + 1198881112119888121111991021199111199052


21198882111 minus 119888011211988811111198881112 + 3119888111211988821112) 11991011991111990511988811122

+ 1199051199091199101199111198881111 minus 1198881211 (11988801121198882111 minus 11988811111198881112) 1199102119911119905

11988811121198882111+ 11988812111199091199102119911119905 + 11988810111198881112119911119905

21198882111 + 11990521199091199101199111198881112 + 1198881011119909119911119905

+ 11990511990921199101199111198882111

1198655 = minus3119888011111988830111199091199111199051198882111 + 11988830111199093119911119905 + 1199051199101199111198880111+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 611988830111199111199052 + 1198880011119911119905

1198656 = 1198880011119911119905 + 1199051199101199111198880111 + 11990511990921199101199111198882111 + 11988802111199102119911119905+ 11988821112119909119910211991111990531198883011 + 11988830111199093119911119905 minus 211988821111199112119905 minus 311988830111199111199052

1198657 = 611988831111199101199111199052 + 13119888101111988821111199111199051198883111 + 1198881011119909119911119905 + 1199051199101199111198880111

+ 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111 + 11988802111199102119911119905+ 31198880211119888311111990911991021199111199051198882111

1198658 = 611988831111199101199111199052 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111 + 11990511990931199101199111198883111+ 11988812111198882111119910211991111990531198883111 + 11988812111199091199102119911119905 + 1199051199101199111198880111

1198659 = 11990511990931199101199111198883111 + 11990511990921199101199111198882111 minus 311988831111199101199111199052 + 1199051199091199101199111198881111+ 11988802111199102119911119905 + 1199051199101199111198880111 + 1198880011119911119905

11986510 = 611988831111199101199111199052 + 13(311988801111198883011 + 11988810111198882111) 1199111199051198883111

+ 1198881011119909119911119905 + 11988830111199093119911119905 + 1199051199091199101199111198881111 + 11990511990921199101199111198882111+ 11990511990931199101199111198883111 + 611988830111199111199052 + 1199051199101199111198880111

11986511 = 11990521199091199101199111198881112 + 11988810021199091199052 + 11988811021199091199101199052 + 119888120211990911991021199052

+ 11988810121199091199111199052 + 119888002111988810021199051198881022 + 119888102211990911991121199052

+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022+ 119888102111988811121199091199101199111199051198881022 + 1198881021119888121211990911991021199111199051198881022+ 1198880021119888112211991011991121199051198881022 + 119888112211990911991011991121199052 + 1198881222119909119910211991121199052

+ 11988800211198881222119910211991121199051198881022 + 1198880021119888121211991021199111199051198881022 + 119888102111988811021199091199101199051198881022+ 1198881021119888120211990911991021199051198881022 + 119888002111988811121199101199111199051198881022 + 1198880021119888120211991021199051198881022+ 119888002111988811021199101199051198881022 + 119888002111988810121199111199051198881022 + 119888100211988810211199091199051198881022+ 119888101211988810211199091199111199051198881022 + 11988810211199091199112119905 + 11988800211199112119905 + 119888121211990911991021199111199052


11986512 = 119888102111988811221199091199101199112119905

1198881022 + 119888102211990911991121199052 + 119888012211991011991121199052

+ 119888112211990911991011991121199052 + 11988810211199091199112119905 + 11988811021199091199101199052 + 11988810121199091199111199052

+ 1198880122119888112211988810211199102119911211990511988810222 + 11988801221198881012119888102111991011991111990511988810222

+ 119888012211988810211198881112119910211991111990511988810222 + 119888102111988811121199091199101199111199051198881022+ 119888102111988811021199091199101199051198881022 + 119888101211988810211199091199111199051198881022 + 119888012211988811221199102119911211990521198881022+ 119888012211988810211198881102119910211990511988810222 + 11988801221198881002119888102111991011990511988810222

+ 1198880122119888100211991011990521198881022 + 11988801221198881102119910211990521198881022 + 119888100211988810211199091199051198881022+ 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 11988801221198881112119910211991111990521198881022 + 11990521199091199101199111198881112 + 11988810021199091199052

11986513 = 1198881011119909119911119905 + 1198881000119909 + 11988810211199091199112119905 + 11988810021199091199052

+ 11988810121199091199111199052 + 119888102211990911991121199052 + 1198880222119888100211991021199052

1198881022+ 1198880222119888100111991021199051198881022 + 119888100011988811221199091199101198881022 + 1198881000119888122211990911991021198881022+ 1198880222119888101011991021199111198881022 + 1198881222119909119910211991121199052 + 11988810101198881222119909119910

21199111198881022

+ 119888012211988810101199101199111198881022 + 119888012211988810111199101199111199051198881022 + 119888101011988811221199091199101199111198881022+ 119888012211991011991121199052 + 119888100111988811221199091199101199051198881022 + 1198880222119910211991121199052 + 1198881001119909119905

+ 119888012211988810001199101198881022 + 1198880222119888100011991021198881022 + 1198881010119909119911

+ 1198880122119888100211991011990521198881022 + 1198880122119888101211991011991111990521198881022 + 1198880122119888102111991011991121199051198881022+ 119888012211988810011199101199051198881022 + 119888112211990911991011991121199052 + 11988810121198881222119909119910

211991111990521198881022

+ 119888101111988811221199091199101199111199051198881022 + 1198881011119888122211990911991021199111199051198881022+ 1198881021119888112211990911991011991121199051198881022 + 11988810211198881222119909119910211991121199051198881022

+ 1198881001119888122211990911991021199051198881022 + 11988802221198881021119910211991121199051198881022+ 1198881012119888112211990911991011991111990521198881022 + 1198881002119888112211990911991011990521198881022+ 11988810021198881222119909119910211990521198881022 + 11988802221198881012119910211991111990521198881022+ 1198880222119888101111991021199111199051198881022

(A1)






1199063 = minus11988811122 (911990521198881112411988821112+ 121199051199111198881112411988812111198882111 + 181199051199091198881112311988821113+ 1811990511991011988811123119888121111988821112+ 1811990911991011988811122119888121111988821113+ 1211990911991111988811123119888121111988821112 + 1811990921198881112211988821114+ 91199102119888111221198881211211988821112














11990611 = minus (1199051198881022 + 1198881021)2 (1199051199091198881022 + 1199091198881021+ 1198880021)minus2


(B1)

Competing Interests


Acknowledgments


References















































1 Introduction


119863119905119906 (119909 119905) + 1198634119909119906 (119909 119905) + 1198632119909119906 (119909 119905) + 119906 (119909 119905)119863119909119906 (119909 119905)= 0 (1)








119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (2)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (3)

with initial value

119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (4)


119906 (119909 119905) = 119909 minus 119905 (5)


119906 (119909 119905) = 1199091 + 119905 (6)



(7)



(8)








1199050119868120572119905119891 (119905)

fl

1Γ (120572) int119905

1199050


(9)


(10)


1199050119863120572119905119891 (119905) fl



(Pro1) 1199050119863120572119905 1199050119863120573119905 119891 (119905) = 1199050119863120572+120573119905 119891 (119905)= 1199050119863120573119905 1199050119863120572119905 119891 (119905)


(12)


1199050119863120572119905119906 (119909 119905)

fl1199050119868119899minus120572119905


(13)



1199090119863120573119909119906 (119909 119905)

fl1199090119868119898minus120573119909


(14)



(15)



119891 (119905) = infinsum119899=0

119888119899 (119905 minus 1199050)119899120572 0 ⩽ 119898 minus 1 lt 120572 ⩽ 119898 1199050 ⩽ 119905 lt 1199050 + 119877

(16)


119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) (17)




0120572

119905119891 (119905)

Γ (0120572 + 1) (18)


1198881 = 1199050119863120572

119905119891 (119905)

Γ (120572 + 1) (19)


1198882 = 11990501198632120572

119905119891 (119905)

Γ (2120572 + 1) (20)



119888119899 = 1199050119863119899120572

119905119891 (119905)

Γ (119899120572 + 1) 119899 = 0 1 2 (21)




119906 (119909 119905) = infinsum119899=0


(22)



119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) = 0119906 (119909 0) = 1198860 (119909) 119863120572119905 119906 (119909 119905)1003816100381610038161003816119905=0= 1198861 (119909) 119863(119898minus1)120572119905 119906 (119909 119905)|119905=0 = 119886119898minus1 (119909)

(23)




119906 (119909 119905) = infinsum119899=0

119906119899 (119909 119905) (25)


119906119899 (119909 119905) = 119862119899 (119909) 119905119899120572119909 isin R |119905| lt 119877 119899 = 0 1 2 (26)


119886119894 (119909) = 119863119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0 = 119862119894 (119909) Γ (119894120572 + 1) 997904rArr119906119894 (119909 119905) = 119862119894 (119909) 119905119894120572 = 119863

119894120572119905 119906 (119909 119905)10038161003816100381610038161003816119905=0Γ (119894120572 + 1) 119905119894120572

= 119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 119894 = 0 1 119898 minus 1(27)



(28)


119906119896 (119909 119905) fl 119906initial (119909 119905) + 119896sum119894=119898

119862119894 (119909) 119905119894120572119896 = 119898119898 + 1119898 + 2

(29)


Res (119909 119905) fl 119863119898120572119905 119906 (119909 119905) + 119866 (119909 119905) (30)


Res119896 (119909 119905) fl 119863119898120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905) (31)

where

119866119896 (119909 119905) = 119865 (119906119896 119863120572119905 119906119896 1198632120572119905 119906119896 119863(119898minus1)120572119905 119906119896 1198631205731119909 1199061198961198631205732119909 119906119896 1198631205731119899119909 119906119896 1198631205731198971199091119906119896)

(32)



119906119896 (119909 119905) = 119906 (119909 119905) (2) Res (119909 119905) = 0(3) Resinfin (119909 119905) = lim


119909 isin R |119905| lt 119877(33)




119862119896 (119909) = minus 119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0Γ (119896120572 + 1)

119896 = 119898119898 + 1119898 + 2

(34)


119862119896 (119909) = 119891119896 (119909)Γ (119896120572 + 1) 119891119896 (119909) = minus119863(119896minus119898)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0

119896 = 119898119898 + 1119898 + 2 (35)



119862119894 (119909) 119905119894120572

= 119898minus1sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 +infinsum119894=119898

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572(36)





119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905) + 120575119863120578119909119906 (119909 119905)= 0 (37)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (38)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (39)


119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(40)



(41)

which gives

119891119896 (119909) = minus119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (42)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909) + 1205751198631205781199091198860 (119909))

≜ 1198861 (119909) 1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909) + 1205751198631205781199091198861 (119909))

≜ 1198862 (119909) 1198913 (119909) = minus (1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909) + 1205751198631205781199091198862 (119909))

≜ 1198863 (119909)

(43)



(44)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(45)



1198861 (119909) = minus1119886119896 (119909) = 0 119896 = 2 3 (46)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 = 119909 minus 119905 (47)


119906 (119909 119905) = lim119896rarrinfin

119906119896 (119909 119905) = 119909 minus 119905 (48)


119863120572119905 119906 (119909 119905) + 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905) 119863120578119909119906 (119909 119905) = 0 (49)


119906 (119909 119905) = infinsum119899=0

119891119899 (119909)Γ (119899120572 + 1) 119905119899120572 (50)


119906 (119909 0) = 1198860 (119909) isin 119862infin (R) (51)



119866 (119909 119905) = 1205731198634120591119909 119906 (119909 119905) + 1205741198632120590119909 119906 (119909 119905)+ 120575119906 (119909 119905)119863120578119909119906 (119909 119905)

Res (119909 119905) = 119863120572119905 119906 (119909 119905) + 119866 (119909 119905) 119906119896 (119909 119905) = 119896sum

119895=0

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572

= 1198860 (119909) + 119896sum119895=1

119891119895 (119909)Γ (119895120572 + 1) 119905119895120572119866119896 (119909 119905) = 1205731198634120591119909 119906119896 (119909 119905) + 1205741198632120590119909 119906119896 (119909 119905)

+ 120575119906119896 (119909 119905) 119863120578119909119906119896 (119909 119905) Res119896 (119909 119905) = 119863120572119905 119906119896 (119909 119905) + 119866119896 (119909 119905)

(52)




(53)

which gives

119891119896 (119909) = minus 119863(119896minus1)120572119905 119866119896 (119909 119905)10038161003816100381610038161003816119905=0 (119896 = 1 2 3 ) (54)


1198910 (119909) = 1198860 (119909) 1198911 (119909) = minus (1205731198634120591119909 1198860 (119909) + 1205741198632120590119909 1198860 (119909)+ 1205751198860 (119909)1198631205781199091198860 (119909)) ≜ 1198861 (119909)

1198912 (119909) = minus (1205731198634120591119909 1198861 (119909) + 1205741198632120590119909 1198861 (119909)+ 1205751198860 (119909)1198631205781199091198861 (119909) + 1205751198861 (119909)1198631205781199091198860 (119909)) ≜ 1198862 (119909)

1198913 (119909) = minus(1205731198634120591119909 1198862 (119909) + 1205741198632120590119909 1198862 (119909)+ 1205751198860 (119909)1198631205781199091198862 (119909) + 120575 Γ (2120572 + 1)(Γ (120572 + 1))2 1198861 (119909)1198631205781199091198861 (119909)+ 1205751198862 (119909)1198631205781199091198860 (119909)) ≜ 1198863 (119909)

(55)



+ 120575119896minus1sum119895=0


sdot 119863120578119909119886119896minus1minus119895 (119909)) ≜ 119886119896 (119909)

(56)


119906119896 (119909 119905) = 119896sum119894=0

119891119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572119896 = 1 2 3

(57)


1198861 (119909) = minus1199091198862 (119909) = 21199091198863 (119909) = minus3119909






(58)


119906119896 (119909 119905) = 119896sum119894=0

119886119894 (119909)Γ (119894120572 + 1) 119905119894120572 =119896sum119894=0

(minus1)119894 119894119909119894 119905119894

= 119909 119896sum119894=0



119896rarrinfin119906119896 (119909 119905) = lim




Competing Interests


Acknowledgments


References




























































1 Introduction


[ 11988921198891199092 + 1198962119894 minus 119897119894 (119897119894 + 1)1199092 minus 119881119894119894]119910119894119895 = 119873sum

119898=1

119881119894119898119910119898119895 (1)


119910119894119895 = 0 at 119909 = 0 (2)

119910119894119895 sim 119896119894119909119895119897119894 (119896119894119909) 120575119894119895 + ( 119896119894119896119895)12119870119894119895119896119894119909119899119897119894 (119896119894119909) (3)



1198701015840119894119895 = ( 119896119894119896119895)12119870119894119895

119872119894119895 = 119896119894119909119895119897119894 (119896119894119909) 120575119894119895119873119894119895 = 119896119894119909119899119897119894 (119896119894119909) 120575119894119895

(4)


y sim M + NK1015840 (5)





[ 11988921198891199092 + 11989621198951015840119895 minus 1198971015840 (1198971015840 + 1)1199092 ]119910119869119895119897

11989510158401198971015840(119909)

= 2120583ℏ2sum11989510158401015840

sum11989710158401015840

⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ 1199101198691198951198971198951015840101584011989710158401015840

(119909) (6)

where

1198961198951015840119895 = 2120583ℏ2 [119864 + ℏ22119868 119895 (119895 + 1) minus 1198951015840 (1198951015840 + 1)] (7)




119898sum119894=minus119898

119888119894119910119899+119894 = ℎ2 119898sum119894=minus119898

119887119894119891 (119909119899+119894 119910119899+119894) (8)




11991010158401015840 = minus1206012119910 (9)









1205821 = 119890119894120579(V)1205822 = 119890minus119894120579(V)10038161003816100381610038161205821198941003816100381610038161003816 le 1 119894 = 3 (1) 2119898

(12)



119905 = V minus 120579 (V) (13)







(15)






(16)





(17)




(18)





1198871= 51911383292 + 268231049V101572361991867354688


1198872= minus1997095823 minus 268231049V10393090497966838672


1198864= minus2 + 134317V1615752663892725760

+ 1170023347V184217775757277322240000 + sdot sdot sdot

(19)



40

20

0

minus20

minus40

minus60

10 20 3010 20 30

10 20 30

2

1

0

minus1



b2



b1

10 20 30

05

0

minus1

minus05

minus15

06

04

02

0

minus02

minus04

minus06

minus08



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(20)



11991010158401015840 (119909) = (119881 (119909) minus 119881119888 + 119866) 119910 (119909) (21)




LTECL = 134317551343236245401600ℎ16119910(16)119899 + 119874 (ℎ18) (22)



LTENM2S5S3DV = 134317551343236245401600ℎ16 (119910(16)119899+ 5612060110119910(6)119899 + 14012060112119910(4)119899 + 12012060114119910(2)119899 + 3512060116119910119899)+ 119874 (ℎ18)

(23)


LTE = ℎ119901 119898sum119894=0

119886119894119866119894 (24)








(25)


LTENM2S5S3DV = 1343179845414932953600sdot ℎ16 [(20119892 (119909) 119910 (119909) 11988921198891199092119892 (119909)+ 15 ( 119889119889119909119892 (119909))

2 119910 (119909)+ 10( 11988931198891199093119892 (119909)) 119889119889119909119910 (119909)+ 31( 11988941198891199094119892 (119909))119910 (119909))1198665 + sdot sdot sdot] + 119874 (ℎ18)

(26)






11991010158401015840 = minus1205962119910 (27)



1198601 (119904 V) (119910119899+1 + 119910119899minus1) + 1198600 (119904 V) 119910119899 = 0 (28)

where

1198601 (119904 V) = 1 + 18 (81198871 + 41198872) 1199042 + 181198872 (minus81198863 + 1) 1199044+ 411988631198872 (V211988601198862 + 121198861) 1199046minus 411988601198861119886311988721199048

1198600 (119904 V) = 1198864 + (1198870 + 1198872) 1199042 + 411988631198872 (V21198862 + 12) 1199044


(met

hod)

s (test problem)

30

30

25

25

20

20

15

15

10

10

5

500


minus 811988631198872 (V211988601198862 + 121198861) 1199046+ 811988601198861119886311988721199048

(29)



1198601 (119904 V) = 1198798119879denom3 1198600 (119904 V) = 1198799119879denom3

(30)











4 Numerical Results





LTE = 10038161003816100381610038161003816119910119867119899+1 minus 119910119871119899+110038161003816100381610038161003816 (31)











119881(119909 k1198951015840119895k119895119895) = 1198810 (119909) 1198750 (k1198951015840119895k119895119895)+ 1198812 (119909) 1198752 (k1198951015840119895k119895119895) (32)


⟨11989510158401198971015840 119869 |119881| 1198951015840101584011989710158401015840 119869⟩ = 1205751198951015840119895101584010158401205751198971015840119897101584010158401198810 (119909)+ 1198912 (11989510158401198971015840 1198951015840101584011989710158401015840 119869) 1198812 (119909) (33)


11991011986911989511989711989510158401198971015840

(119909) = 0 at 119909 = 0 (34)

11991011986911989511989711989510158401198971015840


12 119878119869 (119895119897 11989510158401198971015840) exp [119894 (1198961198951015840119895119909 minus 12 1198971015840120587)] (35)

where




2120583ℏ2 = 10000120583119868 = 2351119864 = 11

1198810 (119909) = 111990912 minus 2 11199096 1198812 (119909) = 022831198810 (119909)

(37)






Method III4 0007 4515 90 times 100916









11991011986911989511989711989510158401198971015840

(1199090) = 0 (38)














5 Conclusions










Appendix







minus8195406183360001198872V4188196191 (1198870 + 83791198864259082 + 134392821198871129541 + 68491821198872129541 minus 513135129541)


minus1639081236672000V21198872188196191 (1198864 + 13439282129541 )

+ (1667689651584000001198864188196191 minus 333537930316800000188196191 ) 1198872


minus1653746424225668236800V1811988723 minus 1153619606787815059200011988722

sdot (1198870 + 21198871 + 22793073722811988724994087615 ) V16 minus 229595394226772318976000011988722


minus19473400902538405380096000011988722 (1198870 + 67307306011198864290822990464 + 379717697333119887118176436904+127130652893979671119887215971271578806720 minus 356864310411145411495232) V12


sdot (minus17739700676691198872273624597200 + (904054678811198870147249194400 + 5839353922711988646184466164800


+ 4546911988701005480) V8 minus 460959572442991450610073600000001198872 (342459111988722129541





+ (93801026033659087351971840000000001198864 minus 18760205206731817470394368000000000) 1198872+1876020520673181747039436800000000011988641198871 minus 187602052067318174703943680000000001198870

1198793 = 4274266643640461521(minus26571283200045469 + V81198872 + 70383601198872V645469 + 6528866401198872V445469





minus1167382756108800000011988722V16188196191 (2598184155864406610230311988722997636444251855529000+(14436736503248119887020643288160585 + 672095448881119886483900671932000 + 6137210358128978411988711135380848832175 minus 619438310185941950335966000) 1198872

+1198871 (1198870 + 21198871) ) minus 8451216098008388812800000011988722V148557092608579 (minus2002533238829945811198872215064365645390600




+((96906582357119886483904353405 minus 1455338241287683904353405 ) 1198871 + 1198870 + 5839353922711988641761991421505 + 439542930952251713060215) 1198872

+(29449838880119886416780870681 minus 360705169440016780870681 ) 11988712 + (minus 39952700921198864352398284301 + 51938511196352398284301) 1198871 + 99881752301198870117466094767)

+2314491183509758444697482690560000000001198872V4389082443819478551 ((minus141198864 + 12) 11988722

+((minus1198864 + 1) 1198871 + 121198870 + 1266896819911988642731802061120 + 4306381525791365901030560) 1198872



+((29436101984019845312296648704000000000001198864389082443819478551 minus 5887220396803969062459329740800000000000389082443819478551 ) 11988723

+((176616611904119071873779892224000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 1198871

+661502702451465533356077219840000000009489815702914111 minus 58872203968039690624593297408000000000001198870389082443819478551+ 1735868387632318833523112017920000000001198864389082443819478551 ) 11988722

+((353233223808238143747559784448000000000001198864389082443819478551 minus 23548881587215876249837318963200000000000389082443819478551 ) 11988712


minus 2354888158721587624983731896320000000000011988712 (minus11988641198871 + 1198870)389082443819478551 ) V2

+(11774440793607938124918659481600000000000389082443819478551 minus 58872203968039690624593297408000000000001198864389082443819478551 ) 11988722

+((23548881587215876249837318963200000000000389082443819478551 minus 235488815872158762498373189632000000000001198864389082443819478551 ) 1198871












minus 1718677320V445469 + 29638785120V245469+ 13546796256045469 ) sin (V) + 2V7 + 12521841V545469

+ 674562000V345469 + 7181753040V45469 ) 1198797 = (minus727504V5 + 56306880V3 + 5223093120V)



(B1)




LTECL

= LTENM2S5S3DV = 1198860

= 134317 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (11988931198891199093) 119892 (119909)3390294398400 + 14371919119892 (119909) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909))2 (119889119889119909) 119892 (119909)

153834608327400

+222563269 (119892 (119909))2 119910 (119909) ((11988971198891199097) 119892 (119909)) (119889119889119909) 119892 (119909)22972634843558400 + 3238785821 (119892 (119909))2 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)

137835809061350400

+42041221 (119892 (119909))4 119910 (119909) ((11988921198891199092) 119892 (119909))2

13127219910604800 + 134317 ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988981198891199098) 119892 (119909)36063790963200

+166687397119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)1378358090613504 + 757413563 (119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)5301377271590400

+5917066801119892 (119909) 119910 (119909) ((11988941198891199094) 119892 (119909)) ((11988931198891199093) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 5506997 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) (119889119889119909) 119892 (119909)615338433309600

+447409927 (119892 (119909))2 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)11486317421779200 + 11954213 (119892 (119909))2 119910 (119909) (1198891011988911990910) 119892 (119909)

45945269687116800

+6312899119892 (119909) ((119889119889119909) 119910 (119909)) (1198891111988911990911) 119892 (119909)68917904530675200 + 134317 (119892 (119909))6 119910 (119909) (11988921198891199092) 119892 (119909)2187869985100800

+7118801 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)

492270746647680 + 189252653 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988931198891199093) 119892 (119909))22812975695129600

+2552023 (119892 (119909))5 119910 (119909) ((119889119889119909) 119892 (119909))29845414932953600 + 5506997 (119892 (119909))2 119910 (119909) ((119889119889119909) 119892 (119909))4984541493295360

+68367353 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

6265264048243200 + 6312899 ((119889119889119909) 119892 (119909))2 119910 (119909) (11988981198891199098) 119892 (119909)1790075442355200

+134317 (119892 (119909))3 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909))370324392378240 + 99260263 ((119889119889119909) 119892 (119909))4 119910 (119909) (11988921198891199092) 119892 (119909)4922707466476800

+4969729 (119892 (119909))4 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

6563609955302400 + 134317 (119892 (119909))5 119910 (119909) (11988941198891199094) 119892 (119909)307669216654800

+3089291 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988941198891199094) 119892 (119909))2

108191372889600 + 2011665709119892 (119909) 119910 (119909) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))334458952265337600

+2283389119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909)) ((11988931198891199093) 119892 (119909))2

18031895481600 + 686494187119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119892 (119909))26265264048243200

+8999239 ((11988921198891199092) 119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988951198891199095) 119892 (119909)

168297691161600 + 134317 (119892 (119909))4 ((119889119889119909) 119910 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)70324392378240

+13834651 (119892 (119909))4 119910 (119909) ((119889119889119909) 119892 (119909)) (11988931198891199093) 119892 (119909)2812975695129600 + 75620471 ((11988921198891199092) 119892 (119909)) 119910 (119909) ((11988931198891199093) 119892 (119909)) (11988951198891199095) 119892 (119909)

1514679220454400

+59502431 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988931198891199093) 119892 (119909)703243923782400 + 6312899 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)162734131123200

+429411449 ((119889119889119909) 119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988921198891199092) 119892 (119909)2461353733238400 + 1112010443 ((119889119889119909) 119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)

9845414932953600

+134317 (119892 (119909))6 ((119889119889119909) 119910 (119909)) (119889119889119909) 119892 (119909)9845414932953600 + 29952691119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988941198891199094) 119892 (119909)1566316012060800


+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) (1198891011988911990910) 119892 (119909)255725063193600 + 134317 ((11988931198891199093) 119892 (119909)) 119910 (119909) (11988991198891199099) 119892 (119909)233027572377600 + 134317 (119892 (119909))8 119910 (119909)

551343236245401600

+134317 ((119889119889119909) 119892 (119909))5 (119889119889119909) 119910 (119909)44751886058880 + 134317 ((1198891411988911990914) 119892 (119909)) 119910 (119909)551343236245401600 + 134317 ((1198891311988911990913) 119892 (119909)) (119889119889119909) 119910 (119909)

39381659731814400

+134317 ((11988931198891199093) 119892 (119909))3 (119889119889119909) 119910 (119909)

4507973870400 + 134317 ((11988921198891199092) 119892 (119909))4 119910 (119909)8975876861952 + 134317 ((11988961198891199096) 119892 (119909))2 119910 (119909)

183597481267200

+155673403119892 (119909) 119910 (119909) ((11988961198891199096) 119892 (119909)) ((119889119889119909) 119892 (119909))2

5301377271590400 + 671585 ((11988921198891199092) 119892 (119909))2 119910 (119909) (11988961198891199096) 119892 (119909)40391445878784 + 134317 ((119889119889119909) 119892 (119909)) 119910 (119909) (1198891111988911990911) 119892 (119909)

1458579990067200

+671585 ((119889119889119909) 119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988941198891199094) 119892 (119909)

17900754423552 + 20013233 ((119889119889119909) 119892 (119909))3 119910 (119909) (11988951198891199095) 119892 (119909)895037721177600 + 3089291119892 (119909) 119910 (119909) (1198891211988911990912) 119892 (119909)

137835809061350400

+123437323 (119892 (119909))3 119910 (119909) ((11988931198891199093) 119892 (119909))2

8614738066334400 + 1581851309 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909))268917904530675200 + 134317 (119892 (119909))5 ((119889119889119909) 119910 (119909)) (11988931198891199093) 119892 (119909)

703243923782400

+134317 ((11988921198891199092) 119892 (119909)) 119910 (119909) (1198891011988911990910) 119892 (119909)504893073484800 + 2552023 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)1514679220454400

+134317 ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988961198891199096) 119892 (119909)17212263868800 + 136869023119892 (119909) 119910 (119909) ((11988951198891199095) 119892 (119909))212530528096486400

+134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909))2

2712235518720 + 59502431 (119892 (119909))3 119910 (119909) (11988981198891199098) 119892 (119909)68917904530675200 + 2283389 (119892 (119909))3 ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)

2153684516583600+2552023 ((11988951198891199095) 119892 (119909)) 119910 (119909) (11988971198891199097) 119892 (119909)1927773553305600 + 53861117119892 (119909) ((119889119889119909) 119910 (119909)) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909)) (11988921198891199092) 119892 (119909)

351621961891200

+279782311 (119892 (119909))4 119910 (119909) (11988961198891199096) 119892 (119909)

275671618122700800 + 134317 ((11988941198891199094) 119892 (119909)) 119910 (119909) (11988981198891199098) 119892 (119909)137698110950400

+4163827 (119892 (119909))2 ((119889119889119909) 119910 (119909)) (11988991198891199099) 119892 (119909)

7657544947852800 + 165344227119892 (119909) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988941198891199094) 119892 (119909)1566316012060800

+426187841 (119892 (119909))3 119910 (119909) ((11988951198891199095) 119892 (119909)) (119889119889119909) 119892 (119909)34458952265337600 + 134317 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))3

1823224987584

+4160200441 (119892 (119909))2 119910 (119909) ((11988921198891199092) 119892 (119909))3

137835809061350400 + 134317 ((11988931198891199093) 119892 (119909))2 119910 (119909) (11988941198891199094) 119892 (119909)4161206649600

+134317 ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119910 (119909)) (11988971198891199097) 119892 (119909)21906517651200 + 159702913 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) ((119889119889119909) 119892 (119909))24922707466476800

+1578090433 (119892 (119909))2 119910 (119909) ((11988941198891199094) 119892 (119909)) ((119889119889119909) 119892 (119909))234458952265337600 + 13566017 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (119889119889119909) 119892 (119909)

1093934992550400

+89589439 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988921198891199092) 119892 (119909)

3281804977651200 + 432635057119892 (119909) 119910 (119909) ((11988981198891199098) 119892 (119909)) (11988921198891199092) 119892 (119909)68917904530675200

+876955693119892 (119909) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988931198891199093) 119892 (119909)68917904530675200 + 84754027 (119892 (119909))3 119910 (119909) ((119889119889119909) 119892 (119909))2 (11988921198891199092) 119892 (119909)4922707466476800

+6312899119892 (119909) 119910 (119909) ((11988991198891199099) 119892 (119909)) (119889119889119909) 119892 (119909)2871579355444800 + 589248679119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)4922707466476800

+60308333119892 (119909) ((119889119889119909) 119910 (119909)) ((11988981198891199098) 119892 (119909)) (119889119889119909) 119892 (119909)11486317421779200 + 248352133119892 (119909) ((119889119889119909) 119910 (119909)) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)17229476132668800

+66218281119892 (119909) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2461353733238400 + 50906143119892 (119909) ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119892 (119909))31230676866619200


+37743077 (119892 (119909))2 ((119889119889119909) 119910 (119909)) ((11988921198891199092) 119892 (119909))2 (119889119889119909) 119892 (119909)

895037721177600 + 22430939119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) ((119889119889119909) 119892 (119909))2546967496275200

+41235319 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988921198891199092) 119892 (119909))2 (11988931198891199093) 119892 (119909)

262544398212096 + 170716907 ((119889119889119909) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988961198891199096) 119892 (119909)) (11988921198891199092) 119892 (119909)3281804977651200

+30758593 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988971198891199097) 119892 (119909)) (11988921198891199092) 119892 (119909)1837810787484672 + 64337843 ((119889119889119909) 119892 (119909)) 119910 (119909) ((11988961198891199096) 119892 (119909)) (11988931198891199093) 119892 (119909)2187869985100800

+28340887 ((11988921198891199092) 119892 (119909)) ((119889119889119909) 119910 (119909)) ((11988931198891199093) 119892 (119909)) (11988941198891199094) 119892 (119909)189334902556800

+80455883 (119892 (119909))3 119910 (119909) ((11988941198891199094) 119892 (119909)) (11988921198891199092) 119892 (119909)3445895226533760 + 1477487119892 (119909) ((119889119889119909) 119910 (119909)) ((11988951198891199095) 119892 (119909)) (11988941198891199094) 119892 (119909)

40683532780800 (C1)












(D1)

Additional Points


Competing Interests


References






























































Soo Hwan Kim







1 Introduction







119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)



NENW

SESW











(2)





119862 (1198861 1198862 1198863 ) = 1198861+ 11198862 + (1 (1198863 + (1 (1198864 + sdot sdot sdot ))))

(3)


NENE

NWNW

SESE

SWSW

a1a1

a2a2

an

an

an

minus1

anminus1

anminus2anminus3

anminus2



119862 (1198861 1198862 1198863 ) = lim119894rarrinfin119862119894 (4)


119877 (1198861 1198862 1198863 ) = lim119894rarrinfin119877119894 (5)





120572 = 119886 + radic119887119888 (7)


119862 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (8)


119877 (1198861 1198862 119886119873 119886119873+1 119886119873+119901) (9)









⋱







where


(11)




(12)





Then


(13)




(14)







(16)

Similarly


(17)





(18)





(2) 119863 (1199051 1199052) = 1003816100381610038161003816120601 (1199051) minus 120601 (1199052)1003816100381610038161003816 = 1003816100381610038161003816120601 (1199052) minus 120601 (1199051)1003816100381610038161003816= 119863 (1199052 1199051)


NENW

SESW

NENW

SESW

NENW

SESW

NENW

SESW

T1 T2

T1

T2



(19)





(20)




(21)









1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (22)


1003817100381710038171003817119891 (119909 oplus 119910) ⊖ (119891 (119909) oplus 119891 (119910))1003817100381710038171003817 lt 120598 (23)




2119891 (119909) ⊖ 119886120598 ≺ 119891 (2119909) ≺ 2119891 (119909) oplus 119886120598 997904rArr (II)119891 (119909) ⊖ 1198861205982 ≺

119891 (2119909)2 ≺ 119891 (119909) oplus 1198861205982 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (2119909)2 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

1205982

(III)

(2) Putting 119909 = 2119909 in (II)

2119891 (2119909) ⊖ 119886120598 ≺ 119891 (4119909) ≺ 2119891 (2119909) oplus 119886120598 997904rArr119891 (2119909)2 ⊖ 1198861205984 ≺

119891 (4119909)4 ≺ 119891 (2119909)2 oplus 1198861205984 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ≺

119891 (4119909)4


119891 (119909) ⊖ 34119886120598 ≺119891 (4119909)4 ≺ 119891 (119909) oplus 34119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (4119909)4 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

34120598

(IV)


119891 (8119909)8 ≺ 119891 (4119909)4 oplus 1198861205988 997904rArr

119891 (119909) ⊖ 1198861205982 ⊖1198861205984 ⊖

1198861205988 ≺119891 (8119909)8

≺ 119891 (119909) oplus 1198861205982 oplus1198861205984

oplus 1198861205988 (∵ by (IV)) 997904rArr119891 (119909) ⊖ 78119886120598 ≺

119891 (8119909)8 ≺ 119891 (119909) oplus 78119886120598 997904rArr10038171003817100381710038171003817100381710038171003817

119891 (8119909)8 ⊖ 119891 (119909)10038171003817100381710038171003817100381710038171003817 lt

78120598

(24)


2119899 minus 12119899 120598 lt 120598 (V)


119876 (119909) = lim119899rarrinfin

119891 (2119899119909)2119899 (25)


2119899 ⊖ 119891 (2119899119909)2119899 ⊖ 119891 (2119899119910)2119899100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr



119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin

119891 (2119899119910)2119899

100381710038171003817100381710038171003817100381710038171003817lt lim119899rarrinfin

1205982119899 = 0

(26)


119891 (2119899119909)2119899 ⊖ lim

119899rarrinfin119891 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin(2119899 minus 12119899 ) 120598 = 120598 (27)



1198761015840 (119909 oplus 119910) = 1198761015840 (119909) oplus 1198761015840 (119910) 10038171003817100381710038171003817119891 (119909) ⊖ 1198761015840 (119909)10038171003817100381710038171003817 lt 120598

(28)



R(1 1 2)

f(16x)

f(16x)

16

R(1 1 5 1 6)

R(29 1 2)

29 crossings

Figure 7


119891 (2119899119909)2119899 ≺ 1198761015840 (2119899119909)2119899 oplus 1198861205982119899 997904rArr

1198761015840 (119909) ⊖ 1198861205982119899 ≺119891 (2119899119909)2119899 ≺ 1198761015840 (119909) oplus 1198861205982119899 997904rArr100381710038171003817100381710038171003817100381710038171003817

119891 (2119899119909)2119899 ⊖ 1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt

1205982119899 997904rArr


119899rarrinfin1198761015840 (119909)100381710038171003817100381710038171003817100381710038171003817 lt lim

119899rarrinfin

1205982119899 = 0

(29)





119876 (119909) = lim119899rarrinfin

119877(14 + sum119899119896=1 52119896minus121198993 ) (31)






NENW

SESW

NENW

SESW

NENW

SESW

T

D(T)

N(T)








119873(119877(11991011199091) oplus 119877(11991021199092)) = 119873(119877(

11990921199101 + 1199091119910211990911199092 )) (32)


119873(119877(11990921199101 + 1199091119910211990911199092 )) (33)









Specific site

(a) (b)

Recombination

Recombinaseenzyme










Competing Interests


References


























1 Introduction








119906119905119905 minus 119906119909119909 + 119906 + 119899119906 = 0119899119905119905 minus 119899119909119909 = (|119906|2)119909119909 (3)



119906 (119909 119905) = 119890119894120579120593 (120585) 119899 (119909 119905) = 119899 (120585)

120585 = 119909 minus 119888119905(4)


119888212059310158401015840 minus 12059310158401015840 + 120593 + 120593119899 = 0 (5)

(1198882 minus 1) 11989910158401015840 = (1205932)10158401015840 (6)



By (6) we have

(1198882 minus 1) 119899 minus 1205932 = 1198881120585 + 119898 (7)



119899 = 1205932 + 1198981198882 minus 1 (8)


(1198882 minus 1) 12059310158401015840 + 120593 + 1205933 + 1205931198981198882 minus 1 = 0 (9)


1198882 minus 12 (1205931015840)2 + 12059322 + 1205934 + 211989812059321198882 minus 1 14 = ℎ (10)



(11)

so that







(1205951015840)2 = 12 1(1198882 minus 1)2 12057821 (1 minus 1198962sin2120595) (15)

Then we obtain

int120595(120585)0




(17)


120593 (120585) = 1205781dn(1205781radic22 11 minus 1198882 120585 119896) (18)



119879 = 2radic2119870 (119896) (1 minus 1198882)1205781 (19)


119906 (119909 119905) = 119890119894120579120593 (120585) = 1198901198941205791205781dn(1205781radic22 11 minus 1198882 120585 119896) 119899 (119909 119905)

















(23)






119889119870119889119896 1205971198961205971205782 (25)




119889119864119889119896 = 119864 minus 119870119896 (27)




(28)





(30)

Hence

2 (1 minus 1198962)119870 (119896) ge (2 minus 1198962) 119864 (119896) (31)


Define












119906 = 119906 (120585) V = 120593 (120585) 119890119894120585120585 = 119909 minus 119888119905

(35)


(1198882 minus 1) 11990610158401015840 minus (1205932)10158401015840 = 0 (36)



(1198882 minus 1) 119906 minus 1205932 = 1198882120585 + 119903 (38)



119906 = 1205932 + 1199031198882 minus 1 (39)


119888 = minus2120572 (40)


12059310158401015840 + 120593(1 + 2119903119888 minus 1198883 ) + 21205933119888 minus 1198883 = 0 (41)



(1205931015840)2 = (1198862 + 1205932) (1198872 minus 1205932) (43)



(1205941015840)2 = 1198872 (1 minus 1205942)(11988621198872 + 1205942) (45)





119879 = 4119870 (119896)120591 (48)


119906 (119909 119905) = 119906 (120585) = (119887 cn (120591120585 119896))2 + 1199031198882 minus 1 V (119909 119905) = V (120585) = 119887119890119894120585cn (120591120585 119896) (49)





119871 = 4119870 (119896)120591 (50)





120579119906119905 + (120573120588 minus 120582119906)119909 = 12ℓ120582 |119861|2119909 (51)




119861 (119909 119905) = 119890119894120585119860 (120585) 119906 (119909 119905) = 1198861198602 (120585) 120588 (119909 119905) = 1198871198602 (120585)

120585 = 119909 minus 119888119905(52)




(53)



(54)


119888 = 2120596 (55)

11986010158401015840 + 119860 minus 2ℓ (119886 minus (1205822) 119887 + 119902)119888 1198603 = 0 (56)


[1198601015840]2 = 1198871 (1198602 + 11988621) (11988722 minus 1198602) (57)


11988722 minus 11988621 = minus 11198871 1198872211988621 = ℎ1198871

(58)


(1205941015840)2 = 119887111988722 (1 minus 1205942)(1198862111988722 + 1205942) (59)




119860 (120585) = 1198872cn (120572120585 119896) (61)


119879 = 4119870 (119896)radic1198871 (11988621 + 11988722 ) (62)


119861 (120585) = 1198901198941205851198872cn (120572120585 119896) 119906 (120585) = 11988611988722 cn2 (120572120585 119896) 120588 (120585) = 11988711988722 cn2 (120572120585 119896)

(63)


1198962 = 11988621 minus 11198871211988621 minus 11198871 119879 (1198861) = 4119870 (119896)

radic2119886211198871 minus 1 (64)




119871 = 4119870 (119896)radic2119886211198871 minus 1 (65)



5 Conclusion


Competing Interests


Acknowledgments


References











Documents

Recent Advances in Symmetry Analysis and Exact Solutions