Research Article Local Fractional Series Expansion Method for …downloads.hindawi.com/journals/aaa/2013/351057.pdf · 2019-07-31 · Fractional calculus theory [ ] has been applied

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 351057 5 pageshttpdxdoiorg1011552013351057

Research ArticleLocal Fractional Series Expansion Method for SolvingWave and Diffusion Equations on Cantor Sets

Ai-Min Yang12 Xiao-Jun Yang3 and Zheng-Biao Li4

1 College of Mechanical Engineering Yanshan University Qinhuangdao 066004 China2 College of Science Hebei United University Tangshan 063009 China3Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Campus XuzhouJiangsu 221008 China

4College of Mathematics and Information Science Qujing Normal University Qujing Yunnan 655011 China

Correspondence should be addressed to Ai-Min Yang aimin heut163com

Received 6 May 2013 Accepted 22 May 2013

Academic Editor Dumitru Baleanu

Copyright copy 2013 Ai-Min 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

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets Some examplesare given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equationswithin the local fractional derivatives

1 Introduction

Fractional calculus theory [1ndash3] has been applied to a wideclass of complex problems encompassing physics biologymechanics and interdisciplinary areas [4ndash9] Various meth-ods for example the Adomian decomposition method [10]the Rach-Adomian-Meyers modified decompositionmethod[11] the variational iteration method [12 13] the homotopyperturbation method [13 14] the fractal Laplace and Fouriertransforms [15] the homotopy analysismethod [16] the heat-balance integral method [17ndash19] the fractional variationaliterationmethod [20ndash22] the fractional subequationmethod[23 24] and the generalized Exp-functionmethod [25] havebeen utilized to solve fractional differential equations [3 15]

The characteristics of fractal materials have local andfractal behaviors well described by nondifferential functionsHowever the classic fractional calculus is not valid fordifferential equation onCantor sets due to its no-local natureIn contrast the local fractional calculus is one of the bestcandidates for dealing with such problems [26ndash44]The localfractional calculus theory has played crucial applications inseveral fields such as theoretical physics transport problemsin fractalmedia described by nondifferential functionsThereare some versions of the local fractional calculus where

different approaches in definition of the local fractionalderivative exist among them the local fractional derivativeof Kolwankar et al [32ndash38] the fractal derivative of Chen etal [39 40] the fractal derivative of Parvate et al [41 42] themodifiedRiemann-Liouville of Jumarie [43 44] and versionsdescribed in [45ndash52]

In order to deal with local fractional ordinary and partialdifferential equations there are somedeveloped technologiesfor example the local fractional variational iteration method[45 46] the local fractional Fourier series method [4748] the Cantor-type cylindrical-coordinate method [49]the Yang-Fourier transform [50 51] and the Yang-Laplacetransform [52]

The local fractional derivative is defined as follows [26ndash31 45ndash52]

119891(120572) (1199090) =119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr119909

0

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572 (1)

where Δ120572(119891(119909)minus119891(1199090)) cong Γ(1+120572)Δ(119891(119909)minus119891(1199090)) and 119891(119909)is satisfied with the condition [26 47]

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 le 120591120572

1003816100381610038161003816119909 minus 11990901003816100381610038161003816120572 (2)

2 Abstract and Applied Analysis

so that [26ndash31]1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816 lt 120576120572 (3)

with 119880 |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877The main idea of this paper is to present the local

fractional series expansion method for effective solutionsof wave and diffusion equations on Cantor sets involvinglocal fractional derivatives The paper has been organizedas follows Section 2 gives a local fractional series expansionmethod Some illustrative examples are shown in Section 3The conclusions are presented in Section 4

2 Analysis of the Method

Let us consider the local fractional differential equation

119906119899120572119905 = 119871120572119906 (4)

where 119871 is a linear local operator with respect to 119909 119899 isin 1 2In accordance with the results in [28 47] there are

multiterm separated functions of independent variables 119905 and119909 namely

119906 (119909 119905) =infin

sum119894=0

119879119894 (119905) 119883119894 (119909) (5)

where 119879119894(119905) and 119883119894(119909) are local fractional continuous func-tions

Moreover there is a nondifferential series term

119879119894 (119905) = 119901119894119905119894120572

Γ (1 + 119894120572) (6)

where 119901119894 is a coefficientIn view of (6) we may present the solution in the form

119906 (119909 119905) =infin

sum119894=0

119901119894119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (7)

Then following (7) we have

119906 (119909 119905) =infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (8)

Hence

119906119899120572119905 =infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+1 (119909) =

infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+119899 (119909)

119871120572119906 = 119871120572 [infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909)] =

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572119883119894) (119909)

(9)

In view of (9) we have

infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+119899 (119909) =

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572119883119894) (119909) (10)

Hence from (10) we can obtain a recursion namely

119883119894+119899 (119909) = (119871120572119883119894) (119909) (11)

with 119899 = 1 we arrive at the following relation

119883119894+1 (119909) = (119871120572119883119894) (119909) (12)

with 119899 = 2 we may rewrite (11) as

119883119894+2 (119909) = (119871120572119883119894) (119909) (13)

By the recursion formulas we can obtain the solution of (4)as

119906 (119909 119905) =infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (14)

The convergent condition is

lim119899rarrinfin

[119905119894120572

Γ (1 + 119894120572)119883119894 (119909)] = 0 (15)

This approach is termed the local fractional series expansionmethod (LFSEM)

3 Applications to Wave and DiffusionEquations on Cantor Sets

In this section four examples for wave and diffusion equa-tions on Cantor sets will demonstrate the efficiency ofLFSEM

Example 1 Let us consider the diffusion equation on Cantorset

119906120572119905 (119909 119905) minus 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (16)

with the initial condition

119906 (119909 0) =119909120572

Γ (1 + 120572) (17)

Following (12) we have recursive formula

119883119894+1 (x) =1205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) =119909120572

Γ (1 + 120572)

(18)

Hence we get

1198830 (119909) =119909120572

Γ (1 + 120572)

1198831 (119909) = 0

1198832 (119909) = 0

(19)

and so onTherefore through (19) we get the solution

119906 (119909 119905) =119909120572

Γ (1 + 120572) (20)

Abstract and Applied Analysis 3


119906120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (21)


119906 (119909 0) =1199092120572

Γ (1 + 2120572) (22)

Following (12) we get

119883119894+1 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883i (119909)

1205971199092120572

1198830 (119909) =1199092120572

Γ (1 + 2120572)

(23)

By using the recursive formula (23) we get consequently

1198830 (119909) =1199092120572

Γ (1 + 2120572)

1198831 (119909) =1199092120572

Γ (1 + 2120572)

1198832 (119909) =1199092120572

Γ (1 + 2120572)

(24)

As a direct result of these recursive calculations we arrive at

119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)=

1199092120572

Γ (1 + 2120572)119864120572 (119905120572)

(25)

Example 3 Let us consider the following wave equation onCantor sets

1199062120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (26)


119906 (119909 0) =119909120572

Γ (1 + 120572) (27)

In view of (14) we obtain

119883i+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) =119909120572

Γ (1 + 120572)

119883119894+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

120597x2120572

1198831 (119909) = 119906(120572)119909 (119909 0) = 1

(28)

Hence using the relations (29) the recursive calculationsyield

1198830 (119909) =119909120572

Γ (1 + 120572)

1198831 (119909) = 1

(29)

1198832 (119909) = 0

1198833 (119909) = 0

1198834 (119909) = 0

(30)

and so onFinally we obtain

119906 (119909 119905) =119909120572

Γ (1 + 120572)+

1199052120572

Γ (1 + 2120572) (31)

Example 4 Let us consider the wave equation on Cantor sets[26 30]

1199062120572119905 (119909 119905) minus cu2120572119909 (119909 119905) = 0 0 lt 120572 le 1 (32)

where c is a constantThe initial condition is

119906 (119909 0) = 119864120572 (119909120572) (33)

By using (14) we have

119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) = 119864120572 (119909120572)

119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906(120572)119909 (119909 0) = 119864120572 (119909120572)

(34)

Then through the iterative relations (35) we have

1198830 (119909) = 119864120572 (119909120572)

1198831 (119909) = 119864120572 (119909120572)

(35)

1198832 (119909) = 119888119864120572 (119909120572)

1198833 (119909) = 119888119864120572 (119909120572)

1198834 (119909) = 1198882119864120572 (119909120572)

(36)


Therefore we obtain

119906 (119909 119905) = 119864120572 (119909120572)infin

sum119894=0

1198881198941199052119894120572

Γ (1 + 2119894120572)

+ 119864120572 (119909120572)infin

sum119894=0

119888119894119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

= 119864120572 (119909120572) [cosh120572 (119888119905

120572) + sin120572 (119888119905120572)]

(37)

where

cosh120572 (119905120572) =infin

sum119894=0

1199052119894120572

Γ (1 + 2119894120572)

sinh120572 (119905120572) =infin

sum119894=0

119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(38)

For more details concerning (38) we refer to [26ndash28]

4 Conclusions

In this work the local fractional series expansion methodis demonstrated as an effective method for solutions of awide class of problems Analytical solutions of the wave anddiffusion equations on Cantor sets involving local fractionalderivatives are successfully developed by recurrence relationsresulting in convergent series solutions In this context thesuggested method is a potential tool for development ofapproximate solutions of local fractional differential equa-tions with fractal initial value conditions which of coursedraws new problems beyond the scope of the present work

Acknowledgments

The first author was supported by the National Scientificand Technological Support Projects (no 2012BAE09B00)the National Natural Science Foundation of China (no11126213 and no 61170317) and the National Natural ScienceFoundation of Hebei Province (no E2013209123) The thirdauthor is supported in part by NSF11061028 of China andYunnan Province NSF Grant no 2011FB090

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[4] R L Magin Fractional Calculus in Bioengineering BegerllHouse West Redding Conn USA 2006

[5] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2011

[6] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[7] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[8] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Fields and Media SpringerBerlin Germany 2011

[9] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[10] J S Duan T Chaolu R Rach and L Lu ldquoTheAdomian decom-position method with convergence acceleration techniquesfor nonlinear fractional differential equationsrdquo Computers ampMathematics With Applications 2013

[11] J S Duan T Chaolu and R Rach ldquoSolutions of the initial valueproblem for nonlinear fractional ordinary differential equa-tions by the Rach-Adomian-Meyers modified decompositionmethodrdquo Applied Mathematics and Computation vol 218 no17 pp 8370ndash8392 2012

[12] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[13] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[14] S Momani and Z Odibat ldquoHomotopy perturbation methodfor nonlinear partial differential equations of fractional orderrdquoPhysics Letters A vol 365 no 5-6 pp 345ndash350 2007

[15] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Boston Mass USA2012

[16] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

[17] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[18] J Hristov ldquoIntegral-balance solution to the stokesrsquo first problemof a viscoelastic generalized second grade fluidrdquo ThermalScience vol 16 no 2 pp 395ndash410 2012

[19] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

[20] G C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A vol 374 no 25pp 2506ndash2509 2010

[21] Y Khan N Faraz A Yildirim and Q Wu ldquoFractionalvariational iteration method for fractional initial-boundaryvalue problems arising in the application of nonlinear sciencerdquoComputers amp Mathematics with Applications vol 62 no 5 pp2273ndash2278 2011

[22] G C Wu ldquoA fractional variational iteration method for solvingfractional nonlinear differential equationsrdquoComputers ampMath-ematics with Applications vol 61 no 8 pp 2186ndash2190 2011

[23] 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


[24] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[25] S Zhang Q A Zong D Liu and Q Gao ldquoA generalized exp-function method for fractional Riccati differential equationsrdquoCommunications in Fractional Calculus vol 1 pp 48ndash52 2010

[26] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[28] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

[29] W P Zhong X J Yang and F Gao ldquoA Cauchy problem forsome local fractional abstract differential equation with fractalconditionsrdquo Journal of Applied Functional Analysis vol 8 no 1pp 92ndash99 2013

[30] 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 no 1 pp 97ndash107 2013

[31] M S Hu D Baleanu and X J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

[32] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[33] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Journal of AppliedMathematics and Mechanics vol 90 no 3 pp 203ndash210 2010

[34] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons and Fractals vol 13 no 1 pp 85ndash94 2002

[35] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[36] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] Y Chen Y Yan and K Zhang ldquoOn the local fractional deriv-ativerdquo Journal of Mathematical Analysis and Applications vol362 no 1 pp 17ndash33 2010

[39] W Chen ldquoTime-space fabric underlying anomalous diffusionrdquoChaos Solitons and Fractals vol 28 no 4 pp 923ndash925 2006

[40] W Chen H Sun X Zhang and D Korosak ldquoAnomalous diffu-sion modeling by fractal and fractional derivativesrdquo Computersamp Mathematics with Applications vol 59 no 5 pp 1754ndash17582010

[41] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[42] A Parvate and A D Gangal ldquoFractal differential equations andfractal-time dynamical systemsrdquo Pramana vol 64 no 3 pp389ndash409 2005

[43] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation and

information of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[44] G Jumarie ldquoLaplacersquos transform of fractional order via theMittag-Leffler function andmodified Riemann-Liouville deriv-ativerdquoAppliedMathematics Letters vol 22 no 11 pp 1659ndash16642009

[45] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[46] W H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 1 pp 89ndash102 2013

[47] M SHu R P Agarwal andX-J Yang ldquoLocal fractional Fourierseries with application to wave equation in fractal vibratingstringrdquo Abstract and Applied Analysis vol 2012 Article ID567401 15 pages 2012

[48] G A Anastassiou and O Duman Advances in Applied Math-ematics and Approximation Theory Springer New York NYUSA 2013

[49] X J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[50] X Yang M Liao and J Chen ldquoA novel approach to pro-cessing fractal signals using the Yang-Fourier transformsrdquo inProceedings of the International Workshop on Information andElectronics Engineering (IWIEE rsquo12) vol 29 pp 2950ndash2954March 2012

[51] W Zhong F Gao and X Shen ldquoApplications of Yang-Fouriertransform to local fractional equations with local fractionalderivative and local fractional integralrdquo Advanced MaterialsResearch vol 461 pp 306ndash310 2012

[52] W P Zhong and F Gao ldquoApplication of the Yang-Laplace trans-forms to solution to nonlinear fractional wave equation withfractional derivativerdquo in Proceedings of the 3rd InternationalConference on Computer Technology and Development pp 209ndash213 ASME 2011

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Stochastic AnalysisInternational Journal of


so that [26ndash31]1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816 lt 120576120572 (3)

with 119880 |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877The main idea of this paper is to present the local

fractional series expansion method for effective solutionsof wave and diffusion equations on Cantor sets involvinglocal fractional derivatives The paper has been organizedas follows Section 2 gives a local fractional series expansionmethod Some illustrative examples are shown in Section 3The conclusions are presented in Section 4

2 Analysis of the Method

Let us consider the local fractional differential equation

119906119899120572119905 = 119871120572119906 (4)

where 119871 is a linear local operator with respect to 119909 119899 isin 1 2In accordance with the results in [28 47] there are

multiterm separated functions of independent variables 119905 and119909 namely

119906 (119909 119905) =infin

sum119894=0

119879119894 (119905) 119883119894 (119909) (5)

where 119879119894(119905) and 119883119894(119909) are local fractional continuous func-tions

Moreover there is a nondifferential series term

119879119894 (119905) = 119901119894119905119894120572

Γ (1 + 119894120572) (6)

where 119901119894 is a coefficientIn view of (6) we may present the solution in the form

119906 (119909 119905) =infin

sum119894=0

119901119894119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (7)

Then following (7) we have

119906 (119909 119905) =infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (8)

Hence

119906119899120572119905 =infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+1 (119909) =

infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+119899 (119909)

119871120572119906 = 119871120572 [infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909)] =

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572119883119894) (119909)

(9)

In view of (9) we have

infin

sum119894=0

1

Γ (1 + 119894120572)119905119894120572119883119894+119899 (119909) =

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572119883119894) (119909) (10)

Hence from (10) we can obtain a recursion namely

119883119894+119899 (119909) = (119871120572119883119894) (119909) (11)

with 119899 = 1 we arrive at the following relation

119883119894+1 (119909) = (119871120572119883119894) (119909) (12)

with 119899 = 2 we may rewrite (11) as

119883119894+2 (119909) = (119871120572119883119894) (119909) (13)

By the recursion formulas we can obtain the solution of (4)as

119906 (119909 119905) =infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)119883119894 (119909) (14)

The convergent condition is

lim119899rarrinfin

[119905119894120572

Γ (1 + 119894120572)119883119894 (119909)] = 0 (15)

This approach is termed the local fractional series expansionmethod (LFSEM)

3 Applications to Wave and DiffusionEquations on Cantor Sets

In this section four examples for wave and diffusion equa-tions on Cantor sets will demonstrate the efficiency ofLFSEM


119906120572119905 (119909 119905) minus 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (16)


119906 (119909 0) =119909120572

Γ (1 + 120572) (17)

Following (12) we have recursive formula

119883119894+1 (x) =1205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) =119909120572

Γ (1 + 120572)

(18)

Hence we get

1198830 (119909) =119909120572

Γ (1 + 120572)

1198831 (119909) = 0

1198832 (119909) = 0

(19)

and so onTherefore through (19) we get the solution

119906 (119909 119905) =119909120572

Γ (1 + 120572) (20)



119906120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (21)


119906 (119909 0) =1199092120572

Γ (1 + 2120572) (22)


119883119894+1 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883i (119909)

1205971199092120572

1198830 (119909) =1199092120572

Γ (1 + 2120572)

(23)


1198830 (119909) =1199092120572

Γ (1 + 2120572)

1198831 (119909) =1199092120572

Γ (1 + 2120572)

1198832 (119909) =1199092120572

Γ (1 + 2120572)

(24)


119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)=

1199092120572

Γ (1 + 2120572)119864120572 (119905120572)

(25)


1199062120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (26)


119906 (119909 0) =119909120572

Γ (1 + 120572) (27)


119883i+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) =119909120572

Γ (1 + 120572)

119883119894+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

120597x2120572

1198831 (119909) = 119906(120572)119909 (119909 0) = 1

(28)


1198830 (119909) =119909120572

Γ (1 + 120572)

1198831 (119909) = 1

(29)

1198832 (119909) = 0

1198833 (119909) = 0

1198834 (119909) = 0

(30)


119906 (119909 119905) =119909120572

Γ (1 + 120572)+

1199052120572

Γ (1 + 2120572) (31)


1199062120572119905 (119909 119905) minus cu2120572119909 (119909 119905) = 0 0 lt 120572 le 1 (32)


119906 (119909 0) = 119864120572 (119909120572) (33)


119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) = 119864120572 (119909120572)

119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906(120572)119909 (119909 0) = 119864120572 (119909120572)

(34)


1198830 (119909) = 119864120572 (119909120572)

1198831 (119909) = 119864120572 (119909120572)

(35)

1198832 (119909) = 119888119864120572 (119909120572)

1198833 (119909) = 119888119864120572 (119909120572)

1198834 (119909) = 1198882119864120572 (119909120572)

(36)


Therefore we obtain

119906 (119909 119905) = 119864120572 (119909120572)infin

sum119894=0

1198881198941199052119894120572

Γ (1 + 2119894120572)

+ 119864120572 (119909120572)infin

sum119894=0

119888119894119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

= 119864120572 (119909120572) [cosh120572 (119888119905

120572) + sin120572 (119888119905120572)]

(37)

where

cosh120572 (119905120572) =infin

sum119894=0

1199052119894120572

Γ (1 + 2119894120572)

sinh120572 (119905120572) =infin

sum119894=0

119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(38)


4 Conclusions


Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (21)


119906 (119909 0) =1199092120572

Γ (1 + 2120572) (22)


119883119894+1 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883i (119909)

1205971199092120572

1198830 (119909) =1199092120572

Γ (1 + 2120572)

(23)


1198830 (119909) =1199092120572

Γ (1 + 2120572)

1198831 (119909) =1199092120572

Γ (1 + 2120572)

1198832 (119909) =1199092120572

Γ (1 + 2120572)

(24)


119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum119894=0

119905119894120572

Γ (1 + 119894120572)=

1199092120572

Γ (1 + 2120572)119864120572 (119905120572)

(25)


1199062120572119905 (119909 119905) minus1199092120572

Γ (1 + 2120572)sdot 1199062120572119909 (119909 119905) = 0 0 lt 120572 le 1 (26)


119906 (119909 0) =119909120572

Γ (1 + 120572) (27)


119883i+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) =119909120572

Γ (1 + 120572)

119883119894+2 (119909) =1199092120572

Γ (1 + 2120572)

1205972120572119883119894 (119909)

120597x2120572

1198831 (119909) = 119906(120572)119909 (119909 0) = 1

(28)


1198830 (119909) =119909120572

Γ (1 + 120572)

1198831 (119909) = 1

(29)

1198832 (119909) = 0

1198833 (119909) = 0

1198834 (119909) = 0

(30)


119906 (119909 119905) =119909120572

Γ (1 + 120572)+

1199052120572

Γ (1 + 2120572) (31)


1199062120572119905 (119909 119905) minus cu2120572119909 (119909 119905) = 0 0 lt 120572 le 1 (32)


119906 (119909 0) = 119864120572 (119909120572) (33)


119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906 (119909 0) = 119864120572 (119909120572)

119883119894+2 (119909) = 1198881205972120572119883119894 (119909)

1205971199092120572

1198830 (119909) = 119906(120572)119909 (119909 0) = 119864120572 (119909120572)

(34)


1198830 (119909) = 119864120572 (119909120572)

1198831 (119909) = 119864120572 (119909120572)

(35)

1198832 (119909) = 119888119864120572 (119909120572)

1198833 (119909) = 119888119864120572 (119909120572)

1198834 (119909) = 1198882119864120572 (119909120572)

(36)


Therefore we obtain

119906 (119909 119905) = 119864120572 (119909120572)infin

sum119894=0

1198881198941199052119894120572

Γ (1 + 2119894120572)

+ 119864120572 (119909120572)infin

sum119894=0

119888119894119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

= 119864120572 (119909120572) [cosh120572 (119888119905

120572) + sin120572 (119888119905120572)]

(37)

where

cosh120572 (119905120572) =infin

sum119894=0

1199052119894120572

Γ (1 + 2119894120572)

sinh120572 (119905120572) =infin

sum119894=0

119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(38)


4 Conclusions


Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Therefore we obtain

119906 (119909 119905) = 119864120572 (119909120572)infin

sum119894=0

1198881198941199052119894120572

Γ (1 + 2119894120572)

+ 119864120572 (119909120572)infin

sum119894=0

119888119894119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

= 119864120572 (119909120572) [cosh120572 (119888119905

120572) + sin120572 (119888119905120572)]

(37)

where

cosh120572 (119905120572) =infin

sum119894=0

1199052119894120572

Γ (1 + 2119894120572)

sinh120572 (119905120572) =infin

sum119894=0

119905(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(38)


4 Conclusions


Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Local Fractional Series Expansion Method for …downloads.hindawi.com/journals/aaa/2013/351057.pdf · 2019-07-31 · Fractional calculus theory [ ] has been applied