Research Article Prandtl s Boundary Layer Equation for Two ...downloads.hindawi.com/journals/mpe/2013/724385.pdf · Exact Solutions via the Simplest Equation Method TahaAziz, 1 A.Fatima,

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 724385 5 pageshttpdxdoiorg1011552013724385

Research ArticlePrandtlrsquos Boundary Layer Equation for Two-Dimensional FlowExact Solutions via the Simplest Equation Method

Taha Aziz1 A Fatima1 C M Khalique2 and F M Mahomed1

1 Centre for Differential Equations Continuum Mechanics and Applications School of Computational and Applied MathematicsUniversity of the Witwatersrand Wits 2050 South Africa

2 Department of Mathematical Science International Institute of Symmetry Analysis and Mathematical ModelingNorth-West University Mafikeng Campus Private Bag X 2046 Mmabatho 2735 South Africa

Correspondence should be addressed to Taha Aziz tahaaziz77yahoocom

Received 12 March 2013 Accepted 18 March 2013

Academic Editor H Jafari

Copyright copy 2013 Taha Aziz et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The simplest equation method is employed to construct some new exact closed-form solutions of the general Prandtlrsquos boundarylayer equation for two-dimensional flow with vanishing or uniformmainstream velocity We obtain solutions for the case when thesimplest equation is the Bernoulli equation or the Riccati equation Prandtlrsquos boundary layer equation arises in the study of variousphysical models of fluid dynamics Thus finding the exact solutions of this equation is of great importance and interest

1 Introduction

Many scientific and engineering problems and phenomenaare modeled by nonlinear differential equations Thereforethe study of nonlinear differential equations has been anactive area of research from the past few decades Consider-able attention has been devoted to the construction of exactsolutions of nonlinear equations because of their importantrole in the study of nonlinear physical models For nonlineardifferential equations we donot have the freedom to computeexact (closed-form) solutions and for analytical work wehave to rely on some approximate analytical or numericaltechniques which may be helpful for us to understand thecomplex physical phenomena involvedThe exact solutions ofthe nonlinear differential equations are of great interest andphysically more importantThese exact solutions if reportedfacilitate the verification of complex numerical codes andare also helpful in a stability analysis for solving specialnonlinear problems In recent years much attention has beendevoted to the development of several powerful and usefulmethods for finding exact analytical solutions of nonlineardifferential equations These methods include the powerfulLie group method [1] the sine-cosine method [2] the tanhmethod [3 4] the extended tanh-function method [5]

the Backlund transformation method [6] the transformedrational function method [7] the (1198661015840119866)-expansion method[8] the exponential function rational expansion method [9]and the Adomianrsquos decomposition method [10]

Prandtl [11] introduced boundary layer theory in 1904 tounderstand the flow behavior of a viscous fluid near a solidboundary Prandtl gave the concept of a boundary layer inlarge Reynolds number flows and derived the boundary layerequations by simplifying the Navier-Stokes equations to yieldapproximate solutions Prandtlrsquos boundary layer equationsarise in various physical models of fluid mechanics Theequations of the boundary layer theory have been the subjectof considerable interest since they represent an importantsimplification of the original Navier-Stokes equations Theseequations arise in the study of steady flows produced bywall jets free jets and liquid jets the flow past a stretchingplatesurface flow induced due to a shrinking sheet andso on These boundary layer equations are usually solvedsubject to certain boundary conditions depending upon thespecific physical model considered Blasius [12] solved thePrandtlrsquos boundary layer equations for a flat moving plateproblem and gave a power series solution of the problemFalkner and Skan [13] generalized the Blasius boundary layerproblem by considering the boundary layer flow over a wedge

2 Mathematical Problems in Engineering

inclined at a certain angle Sakiadis [14] initiated the studyof the boundary layer flow over a continuously moving rigidsurface with a uniform speed Crane [15] was the first onewho studied the boundary layer flow due to a stretchingsurface and developed the exact solutions of boundary layerequations with parameter 120574 = 0 P S Gupta and A SGupta [16] extended the Cranersquos work and for the first timeintroduced the concept of heat transfer with the stretchingsheet boundary layer flow The numerical solution for afree two-dimensional jet was obtained by Schlichting [17]and later an analytic study was made by Bickley [18] Riley[19] derived the solution for a radial liquid jet Recentlythe similarity solution of axisymmetric non-Newtonian walljet with swirl effects was investigated by Kolar [20] Nazet al [21] and Mason [22] have investigated the generalboundary layer equations for two-dimensional and radialflows by using the classical Lie group approach and veryrecently Naz et al [23] have provided the similarity solutionsof the Prandtlrsquos boundary layer equations by implementingthe nonclassicalconditional symmetry method

The simplest equation method is a powerful mathemat-ical tool for finding exact solutions of nonlinear ordinarydifferential equations It has been developed by Kudryashov[24 25] and used successfully bymany researchers for findingexact solutions of nonlinear ordinary differential equations[26ndash28] The purpose of the present work is to find the exactclosed-form solutions of Prandtlrsquos boundary layer equationfor two-dimensional flow with constant or uniform mainstream velocity by the use of simplest equation method

Prandtlrsquos boundary layer equation for the stream function120595(119909 119910) for an incompressible steady two-dimensional flowwith uniform or vanishing mainstream velocity is [29]

120597120595

120597119910

1205972120595

120597119909120597119910minus

120597120595

120597119909

1205972120595

1205971199102minus ]

1205973120595

1205971199103= 0 (1)

Here (119909 119910) denote the Cartesian coordinates parallel andperpendicular to the boundary 119910 = 0 and ] is the kinematicviscosityThe velocity components 119906 (119909 119910) and V (119909 119910) in the119909 and 119910 directions are related to stream function 120595 (119909 119910) as

119906 (119909 119910) =120597120595

120597119910 V (119909 119910) = minus

120597120595

120597119909 (2)

By the use of Lie group theoretic method of infinitesimaltransformations [1] the general formof similarity solution for(1) is

120595 (119909 119910) = 1199091minus120574

119865 (120578) 120578 =119910

119909120574 (3)

where 120574 is the constant determined from the further condi-tions and 120578 is the similarity variable By the substitution of(3) into (1) the third-order nonlinear ordinary differentialequation in 119865(120578) results namely in

]1198893119865

1198891205783+ (1 minus 120574) 119865

1198892119865

1198891205782+ (2120574 minus 1) (

119889119865

119889120578)

2

= 0 (4)

Equation (4) gives the general form of Prandtlrsquos boundarylayer equation for two-dimensional flow of a viscous incom-pressible fluid The boundary layer equation (4) is usually

solved subject to certain boundary conditions dependingupon the particular physical model considered Here wepresent the exact closed-form solutions of (4) using thesimplest equation method We organize the paper as fol-lows In Section 2 we describe briefly the simplest equationmethod In Section 3 we apply thismethod to solve nonlinearPrandtlrsquos boundary layer equation for two-dimensional flowFinally some closing remarks are presented in Section 4

2 A Description of the SimplestEquation Method

Here we present a brief description of the simplest equationmethod for solving nonlinear ordinary differential equations

Step 1 We first consider a general form of a nonlinear ordi-nary differential equation

119864[119865119889119865

1198891205781198892119865

11988912057821198893119865

1198891205783 ] = 0 (5)

where 119865 is the dependent variable and 120578 is the independentvariable

Step 2 The basic idea of the simplest equation methodconsists in expanding the solutions of the previous ordinarydifferential equation in a finite series

119865 (120578) =

119872

sum119894=0

119860119894(119866 (120578))

119894

(6)

where 119866(120578) is a solution of some ordinary differential equa-tions These ordinary differential equations are called thesimplest equations The main property of the simplest equa-tion is that we know the general solution of the simplestequation or we at least know the exact analytical solutions ofthe simplest equation The parameters 119860

0 1198601 119860

119872are to

be determined from the further conditions

In this paper we use the Bernoulli and Riccati equationsas the simplest equations These equations are well-knownnonlinear ODEs whose solutions can be expressed in termsof elementary functions

For the Bernoulli equation

119889119866

119889120578= 119889119866 (120578) + 119890119866

2(120578) (7)

where 119889 and 119890 are constants independent of 120578The solution of(7) is

119866 (120578) = 119889 [cosh [119889 (120578 + 119862)] + sinh [119889 (120578 + 119862)]

1 minus 119890 cosh [119889 (120578 + 119862)] minus 119890 sinh [119889 (120578 + 119862)]]

(8)

where 119862 is a constant of integrationFor the Riccati equation

119889119866

119889120578= 119889119866 (120578) + 119890119866

2(120578) + 119891 (9)

Mathematical Problems in Engineering 3

where 119889 119890 and 119891 are constants we will use the solutions

119866 (120578) = minus119889

2119890minus

120579

2119890tanh [

1

2120579 (120578 + 119862)]

119866 (120578) = minus119889

2119890minus

120579

2119890tanh(

1

2120579120578)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)

(10)

where

1205792= 1198892minus 4119890119891 gt 0 (11)

and 119862 is a constant of integration

Step 3 One of the main steps in using the simplest equationmethod is to determine the positive number 119872 in (6) Thepositive number 119872 can be determined by considering thehomogeneous balance between the highest order derivativesand nonlinear terms appearing in (5)

Step 4 By the substitution of (6) into (5) and with (7) or(9) the left hand side of (5) is converted into a polynomialin 119866(120578) Equating each coefficient of the polynomial to zeroyields a set of algebraic equations for 119860

119894 119889 119890 119891

Step 5 By solving the algebraic equations obtained in Step 4and substituting the results into (6) we obtain the exactsolutions of ODE (5)

3 Application of the SimplestEquation Method

In this section we employ the simplest equation methodand obtain exact closed-form solutions of Prandtlrsquos boundarylayer equation (4)

31 Solutions of Boundary Layer Equation Using the Equationof Bernoulli as the Simplest Equation The balancing proce-dure yields119872 = 1 Thus we search for a solution of (4) of theform

119865 (120578) = 1198600+ 1198601119866 (120578) (12)

where 119866(120578) satisfies the Bernoulli equation and 1198600and 119860

1

are the parameters to be determinedBy the substitution of (12) into (4) and making use of

the Bernoulli equation (7) and then equating all coefficientsof the functions 119866

119894 to zero we obtain an algebraic systemof equations in terms of 119860

0and 119860

1 Solving this system of

algebraic equations we obtain the values of the constants 1198600

and 1198601 Therefore the solution of Prandtlrsquos boundary layer

equation (4) with 120574 = 23 is given by

119865 (120578) = minus3]119889

minus 6]119890119889 [cosh [119889 (120578+119862)] + sinh [119889 (120578+119862)]

1minus119890 cosh [119889 (120578 + 119862)]minus119890 sinh [119889 (120578+119862)]]

(13)

and hence the corresponding stream function becomes

120595 (119909 119910)

= minus3]11988911990913

minus 6]11989011988911990913

times [ (cosh [119889 (119909minus23

119910 + 119862)] + sinh [119889 (119909minus23

119910 + 119862)])

times (1minus119890 cosh [119889 (119909minus23

119910+119862)]

minus119890 sinh [119889 (119909minus23

119910+119862)])minus1

]

(14)

Special Cases By taking 119889 = minus1 and 119890 = 1 in the previoussolution we obtain a special solution given by

120595 (119909 119910) = 3]11990913 coth [

1

2(119909minus23

119910 + 119862)] (15)

Likewise if we take 119889 = minus1 and 119890 = minus1 we deduce

120595 (119909 119910) = 3]11990913 tanh [

1

2(119909minus23

119910 + 119862)] (16)

32 Solutions of Boundary Layer Equation Using the Equationof Riccati as the Simplest Equation The balancing procedureyields119872 = 1 Thus the solution of (4) is written in the form

119865 (120578) = 1198600+ 1198601119866 (120578) (17)

By the insertion of (17) into (4) andmaking use of the Riccatiequation (9) and proceeding as above we obtain algebraicsystem of equations in terms of 119860

0and 119860

1 Solving this

system we obtain the solutions of Prandtlrsquos boundary layerequation (4) for 120574 = 23 as

119865 (120578) = minus3]119889 minus 6]119890 [minus119889

2119890minus

120579

2119890tanh(

1

2120579 (120578 + 119862))]

119865 (120578) = minus3]119889minus6]119890 [minus119889

2119890minus

120579

2119890tanh(

120578120579

2)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)]

(18)

and the solutions for corresponding stream functions are

120595 (119909 119910) = minus 3]11988911990913

minus 6]11989011990913

[minus119889

2119890minus

120579

2119890tanh(

1

2120579 (119909minus23

119910+119862))]

(19)


120595 (119909 119910) = minus3]11988911990913

minus 6]11989011990913

[ minus119889

2119890minus

120579

2119890tanh(

120579119909minus23119910

2)

+ (sech(120579119909minus23119910

2))

times (119862 cosh(120579119909minus23119910

2) minus

2119890

120579

times sinh(120579119909minus23119910

2))

minus1

]

(20)

where 1205792 = 1198892 minus 4119890119891 and 119862 is a constant of integrationBy taking 119889 = 3 119890 = 1 and 119891 = 1 in (19) we deduce a

special solution of stream function 120595 given by

120595 (119909 119910) = minus 9]11990913

+ 3]11990913

[3 + radic5 tanh(radic5

2(119909minus23

119910 + 119862))]

(21)

4 Concluding Remarks

In this study we have utilized the method of simplest equa-tion for obtaining exact closed-form solutions of the well-known Prandtlrsquos boundary layer equation for two-dim-ensional flow with uniform mainstream velocity As thesimplest equations we have used the Bernoulli and Riccatiequations Prandtlrsquos boundary layer equations arise in var-ious physical models of fluid dynamics and thus the exactsolutions obtained may be very useful and significant for theexplanation of some practical physical models dealing withPrandtlrsquos boundary layer theoryWe have also verified that thesolutions obtained here are indeed the solutions of Prandtlrsquosboundary layer equation

Acknowledgments

T Aziz and A Fatima would like to thank the School ofComputational and Applied Mathematics and the FinancialAid and Scholarship Office University of the Witwatersrandfor financial support and research grant

References

[1] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[2] A-M Wazwaz ldquoThe sine-cosine method for obtaining solu-tions with compact and noncompact structuresrdquoAppliedMath-ematics and Computation vol 159 no 2 pp 559ndash576 2004

[3] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[4] A-M Wazwaz ldquoThe tanh method solitons and periodic solu-tions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equationsrdquo Chaos Solitons amp Fractals vol 25no 1 pp 55ndash63 2005

[5] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

[6] M R Miura Backlund Transformation Springer Berlin Ger-many 1978

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

[8] R Abazari ldquoApplication of (119866119866)-expansion method to trav-elling wave solutions of three nonlinear evolution equationrdquoComputers amp Fluids vol 39 no 10 pp 1957ndash1963 2010

[9] H Xin ldquoThe exponential function rational expansion methodand exact solutions to nonlinear lattice equations systemrdquoApplied Mathematics and Computation vol 217 no 4 pp 1561ndash1565 2010

[10] J-L Li ldquoAdomianrsquos decomposition method and homotopyperturbation method in solving nonlinear equationsrdquo Journalof Computational and Applied Mathematics vol 228 no 1 pp168ndash173 2009

[11] L Prandtl ldquoUber Flussigkeitsbewegungen bei sehr kleiner Rei-bungrdquo in Verhandlungen des III Internationalen MathematikerKongresses pp 484ndash491 Heidelberg Germany 1904

[12] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zeitschrift fur Mathematik und Physik vol 56 pp 1ndash371908

[13] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12pp 865ndash896 1931

[14] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurface I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[15] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[16] P S Gupta and A S Gupta ldquoHeat and mass transfer ona stretching sheet with suction and blowingrdquo The CanadianJournal of Chemical Engineering vol 55 pp 744ndash746 1977

[17] H Schlichting ldquoLaminare Strahlausbreitungrdquo Zeitschrift furAngewandte Mathematik und Mechanik vol 13 pp 260ndash2631933

[18] W G Bickley ldquoThe plane jetrdquo Philosophical Magazine vol 23pp 727ndash731 1937

[19] N Riley ldquoAsymptotic expansions in radial jetsrdquo Journal ofMathematical Physics vol 41 pp 132ndash146 1962

[20] V Kolar ldquoSimilarity solution of axisymmetric non-Newtonianwall jets with swirlrdquo Nonlinear Analysis Real World Applica-tions vol 12 no 6 pp 3413ndash3420 2011

[21] R Naz F M Mahomed and D P Mason ldquoSymmetry solutionsof a third-order ordinary differential equation which arisesfrom Prandtl boundary layer equationsrdquo Journal of NonlinearMathematical Physics vol 15 no 1 pp 179ndash191 2008

[22] D P Mason ldquoGroup invariant solution and conservation lawfor a free laminar two-dimensional jetrdquo Journal of NonlinearMathematical Physics vol 9 no 2 pp 92ndash101 2002

[23] R Naz M D Khan and I Naeem ldquoNonclassical symmetryanalysis of boundary layer equationsrdquo Journal of Applied Math-ematics vol 2012 Article ID 938604 7 pages 2012


[24] N A Kudryashov ldquoSimplest equation method to look for exactsolutions of nonlinear differential equationsrdquo Chaos Solitons ampFractals vol 24 no 5 pp 1217ndash1231 2005

[25] N A Kudryashov ldquoExact solitary waves of the Fisher equationrdquoPhysics Letters A vol 342 no 1-2 pp 99ndash106 2005

[26] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[27] N Taghizadeh and M Mirzazadeh ldquoThe simplest equationmethod to study perturbed nonlinear Schrodingerrsquos equationwith Kerr law nonlinearityrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 4 pp 1493ndash14992012

[28] N K Vitanov andZ I Dimitrova ldquoApplication of themethod ofsimplest equation for obtaining exact traveling-wave solutionsfor two classes of model PDEs from ecology and populationdynamicsrdquo Communications in Nonlinear Science and Numer-ical Simulation vol 15 no 10 pp 2836ndash2845 2010

[29] L Rosenhead Laminar Boundary Layers pp 254ndash256 Claren-don Press Oxford UK 1963

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Stochastic AnalysisInternational Journal of


inclined at a certain angle Sakiadis [14] initiated the studyof the boundary layer flow over a continuously moving rigidsurface with a uniform speed Crane [15] was the first onewho studied the boundary layer flow due to a stretchingsurface and developed the exact solutions of boundary layerequations with parameter 120574 = 0 P S Gupta and A SGupta [16] extended the Cranersquos work and for the first timeintroduced the concept of heat transfer with the stretchingsheet boundary layer flow The numerical solution for afree two-dimensional jet was obtained by Schlichting [17]and later an analytic study was made by Bickley [18] Riley[19] derived the solution for a radial liquid jet Recentlythe similarity solution of axisymmetric non-Newtonian walljet with swirl effects was investigated by Kolar [20] Nazet al [21] and Mason [22] have investigated the generalboundary layer equations for two-dimensional and radialflows by using the classical Lie group approach and veryrecently Naz et al [23] have provided the similarity solutionsof the Prandtlrsquos boundary layer equations by implementingthe nonclassicalconditional symmetry method

The simplest equation method is a powerful mathemat-ical tool for finding exact solutions of nonlinear ordinarydifferential equations It has been developed by Kudryashov[24 25] and used successfully bymany researchers for findingexact solutions of nonlinear ordinary differential equations[26ndash28] The purpose of the present work is to find the exactclosed-form solutions of Prandtlrsquos boundary layer equationfor two-dimensional flow with constant or uniform mainstream velocity by the use of simplest equation method

Prandtlrsquos boundary layer equation for the stream function120595(119909 119910) for an incompressible steady two-dimensional flowwith uniform or vanishing mainstream velocity is [29]

120597120595

120597119910

1205972120595

120597119909120597119910minus

120597120595

120597119909

1205972120595

1205971199102minus ]

1205973120595

1205971199103= 0 (1)

Here (119909 119910) denote the Cartesian coordinates parallel andperpendicular to the boundary 119910 = 0 and ] is the kinematicviscosityThe velocity components 119906 (119909 119910) and V (119909 119910) in the119909 and 119910 directions are related to stream function 120595 (119909 119910) as

119906 (119909 119910) =120597120595

120597119910 V (119909 119910) = minus

120597120595

120597119909 (2)

By the use of Lie group theoretic method of infinitesimaltransformations [1] the general formof similarity solution for(1) is

120595 (119909 119910) = 1199091minus120574

119865 (120578) 120578 =119910

119909120574 (3)

where 120574 is the constant determined from the further condi-tions and 120578 is the similarity variable By the substitution of(3) into (1) the third-order nonlinear ordinary differentialequation in 119865(120578) results namely in

]1198893119865

1198891205783+ (1 minus 120574) 119865

1198892119865

1198891205782+ (2120574 minus 1) (

119889119865

119889120578)

2

= 0 (4)

Equation (4) gives the general form of Prandtlrsquos boundarylayer equation for two-dimensional flow of a viscous incom-pressible fluid The boundary layer equation (4) is usually

solved subject to certain boundary conditions dependingupon the particular physical model considered Here wepresent the exact closed-form solutions of (4) using thesimplest equation method We organize the paper as fol-lows In Section 2 we describe briefly the simplest equationmethod In Section 3 we apply thismethod to solve nonlinearPrandtlrsquos boundary layer equation for two-dimensional flowFinally some closing remarks are presented in Section 4

2 A Description of the SimplestEquation Method

Here we present a brief description of the simplest equationmethod for solving nonlinear ordinary differential equations

Step 1 We first consider a general form of a nonlinear ordi-nary differential equation

119864[119865119889119865

1198891205781198892119865

11988912057821198893119865

1198891205783 ] = 0 (5)

where 119865 is the dependent variable and 120578 is the independentvariable

Step 2 The basic idea of the simplest equation methodconsists in expanding the solutions of the previous ordinarydifferential equation in a finite series

119865 (120578) =

119872

sum119894=0

119860119894(119866 (120578))

119894

(6)

where 119866(120578) is a solution of some ordinary differential equa-tions These ordinary differential equations are called thesimplest equations The main property of the simplest equa-tion is that we know the general solution of the simplestequation or we at least know the exact analytical solutions ofthe simplest equation The parameters 119860

0 1198601 119860

119872are to

be determined from the further conditions

In this paper we use the Bernoulli and Riccati equationsas the simplest equations These equations are well-knownnonlinear ODEs whose solutions can be expressed in termsof elementary functions

For the Bernoulli equation

119889119866

119889120578= 119889119866 (120578) + 119890119866

2(120578) (7)

where 119889 and 119890 are constants independent of 120578The solution of(7) is

119866 (120578) = 119889 [cosh [119889 (120578 + 119862)] + sinh [119889 (120578 + 119862)]

1 minus 119890 cosh [119889 (120578 + 119862)] minus 119890 sinh [119889 (120578 + 119862)]]

(8)

where 119862 is a constant of integrationFor the Riccati equation

119889119866

119889120578= 119889119866 (120578) + 119890119866

2(120578) + 119891 (9)



119866 (120578) = minus119889

2119890minus

120579

2119890tanh [

1

2120579 (120578 + 119862)]

119866 (120578) = minus119889

2119890minus

120579

2119890tanh(

1

2120579120578)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)

(10)

where

1205792= 1198892minus 4119890119891 gt 0 (11)




119894 119889 119890 119891





119865 (120578) = 1198600+ 1198601119866 (120578) (12)


1




0and 119860





119865 (120578) = minus3]119889

minus 6]119890119889 [cosh [119889 (120578+119862)] + sinh [119889 (120578+119862)]


(13)


120595 (119909 119910)

= minus3]11988911990913

minus 6]11989011988911990913

times [ (cosh [119889 (119909minus23

119910 + 119862)] + sinh [119889 (119909minus23

119910 + 119862)])


119910+119862)]

minus119890 sinh [119889 (119909minus23

119910+119862)])minus1

]

(14)


120595 (119909 119910) = 3]11990913 coth [

1

2(119909minus23

119910 + 119862)] (15)


120595 (119909 119910) = 3]11990913 tanh [

1

2(119909minus23

119910 + 119862)] (16)


119865 (120578) = 1198600+ 1198601119866 (120578) (17)


0and 119860

1 Solving this



2119890minus

120579

2119890tanh(

1

2120579 (120578 + 119862))]


2119890minus

120579

2119890tanh(

120578120579

2)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)]

(18)


120595 (119909 119910) = minus 3]11988911990913

minus 6]11989011990913

[minus119889

2119890minus

120579

2119890tanh(

1

2120579 (119909minus23

119910+119862))]

(19)


120595 (119909 119910) = minus3]11988911990913

minus 6]11989011990913

[ minus119889

2119890minus

120579

2119890tanh(

120579119909minus23119910

2)

+ (sech(120579119909minus23119910

2))

times (119862 cosh(120579119909minus23119910

2) minus

2119890

120579

times sinh(120579119909minus23119910

2))

minus1

]

(20)



120595 (119909 119910) = minus 9]11990913

+ 3]11990913


2(119909minus23

119910 + 119862))]

(21)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866 (120578) = minus119889

2119890minus

120579

2119890tanh [

1

2120579 (120578 + 119862)]

119866 (120578) = minus119889

2119890minus

120579

2119890tanh(

1

2120579120578)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)

(10)

where

1205792= 1198892minus 4119890119891 gt 0 (11)




119894 119889 119890 119891





119865 (120578) = 1198600+ 1198601119866 (120578) (12)


1




0and 119860





119865 (120578) = minus3]119889

minus 6]119890119889 [cosh [119889 (120578+119862)] + sinh [119889 (120578+119862)]


(13)


120595 (119909 119910)

= minus3]11988911990913

minus 6]11989011988911990913

times [ (cosh [119889 (119909minus23

119910 + 119862)] + sinh [119889 (119909minus23

119910 + 119862)])


119910+119862)]

minus119890 sinh [119889 (119909minus23

119910+119862)])minus1

]

(14)


120595 (119909 119910) = 3]11990913 coth [

1

2(119909minus23

119910 + 119862)] (15)


120595 (119909 119910) = 3]11990913 tanh [

1

2(119909minus23

119910 + 119862)] (16)


119865 (120578) = 1198600+ 1198601119866 (120578) (17)


0and 119860

1 Solving this



2119890minus

120579

2119890tanh(

1

2120579 (120578 + 119862))]


2119890minus

120579

2119890tanh(

120578120579

2)

+sech (1205791205782)

119862 cosh (1205791205782) minus (2119890120579) sinh (1205791205782)]

(18)


120595 (119909 119910) = minus 3]11988911990913

minus 6]11989011990913

[minus119889

2119890minus

120579

2119890tanh(

1

2120579 (119909minus23

119910+119862))]

(19)


120595 (119909 119910) = minus3]11988911990913

minus 6]11989011990913

[ minus119889

2119890minus

120579

2119890tanh(

120579119909minus23119910

2)

+ (sech(120579119909minus23119910

2))

times (119862 cosh(120579119909minus23119910

2) minus

2119890

120579

times sinh(120579119909minus23119910

2))

minus1

]

(20)



120595 (119909 119910) = minus 9]11990913

+ 3]11990913


2(119909minus23

119910 + 119862))]

(21)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120595 (119909 119910) = minus3]11988911990913

minus 6]11989011990913

[ minus119889

2119890minus

120579

2119890tanh(

120579119909minus23119910

2)

+ (sech(120579119909minus23119910

2))

times (119862 cosh(120579119909minus23119910

2) minus

2119890

120579

times sinh(120579119909minus23119910

2))

minus1

]

(20)



120595 (119909 119910) = minus 9]11990913

+ 3]11990913


2(119909minus23

119910 + 119862))]

(21)



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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