Research Article Exact Solutions of the Space Time ...metric regularized long wave (SRLW) equation. Conclusion is given in Section . 2. Jumarie s Modified Riemann-Liouville Derivative

Research ArticleExact Solutions of the Space Time Fractional SymmetricRegularized Long Wave Equation Using Different Methods

Oumlzkan Guumlner1 and Dursun Eser2

1 Department of Management Information Systems School of Applied Sciences Dumlupınar University 43100 Kutahya Turkey2Department of Mathematics-Computer Art-Science Faculty Eskisehir Osmangazi University 26480 Eskisehir Turkey

Correspondence should be addressed to Dursun Eser deseroguedutr

Received 4 April 2014 Accepted 22 June 2014 Published 22 July 2014

Academic Editor Hossein Jafari

Copyright copy 2014 O Guner and D Eser 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

We apply the functional variable method exp-function method and (1198661015840119866)-expansion method to establish the exact solutionsof the nonlinear fractional partial differential equation (NLFPDE) in the sense of the modified Riemann-Liouville derivative Asa result some new exact solutions for them are obtained The results show that these methods are very effective and powerfulmathematical tools for solving nonlinear fractional equations arising in mathematical physics As a result these methods can alsobe applied to other nonlinear fractional differential equations

1 Introduction

Fractional calculus is a field of mathematics that grows out ofthe traditional definitions of calculus Fractional calculus hasgained importance during the last decades mainly due to itsapplications in various areas of physics biologymathematicsand engineering Some of the current application fields offractional calculus include fluid flow dynamical process inself-similar and porous structures electrical networks prob-ability and statistics control theory of dynamical systemssystems identification acoustics viscoelasticity control the-ory electrochemistry of corrosion chemical physics financeoptics and signal processing [1ndash3]

There are several definitions of the fractional derivativewhich are generally not equivalent to each other Someof these definitions are Sun and Chenrsquos fractal derivative[4 5] Cressonrsquos derivative [6 7] Grunwald-Letnikovrsquos frac-tional derivative [8] Riemann-Liouvillersquos derivative [8] andCaputorsquos fractional derivative [9] But the Riemann-Liouvillederivative and the Caputo derivative are the most used ones

Lately both mathematicians and physicists have devotedconsiderable effort to the study of explicit solutions tononlinear fractional differential equations Many power-ful methods have been presented Among them are thefractional (1198661015840119866)-expansion method [10ndash13] the fractional

exp-function method [14ndash16] the fractional first integralmethod [17 18] the fractional subequation method [19ndash22]the fractional functional variable method [23] the fractionalmodified trial equation method [24 25]andthe fractionalsimplest equation method [26]

The paper suggests the functional variable method theexp-function method the (119866

1015840119866)-expansion method andfractional complex transform to find the exact solutions ofnonlinear fractional partial differential equation with themodified Riemann-Liouville derivative

This paper is organized as follows In Section 2 basicdefinitions of Jumariersquos Riemann-Liouville derivative aregiven in Section 3 description of the methods for FDEs isgiven Then in Section 4 these methods have been appliedto establish exact solutions for the space-time fractional sym-metric regularized long wave (SRLW) equation Conclusionis given in Section 5

2 Jumariersquos ModifiedRiemann-Liouville Derivative

Recently a new modified Riemann-Liouville derivative isproposed by Jumarie [27 28] This new definition of frac-tional derivative has two main advantages firstly comparing

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 456804 8 pageshttpdxdoiorg1011552014456804

2 Advances in Mathematical Physics

with the Caputo derivative the function to be differentiatedis not necessarily differentiable secondly different from theRiemann-Liouville derivative Jumariersquos modified Riemann-Liouville derivative of a constant is defined to be zeroJumariersquos modified Riemann-Liouville derivative of order 120572is defined by

119863120572

119909119891 (119909)

=

1

Γ (1 minus 120572)

times119889

119889119909int119909

0

(119909 minus 120585)minus120572

(119891 (120585) minus 119891 (0)) 119889120585

0 lt 120572 lt 1

(119891(119899) (119909))(120572minus119899)

119899 le 120572 lt 119899 + 1 119899 ge 1

(1)

where 119891 119877 rarr 119877 119909 rarr 119891(119909) denotes a continuous (but notnecessarily first-order-differentiable) function Some usefulformulas and results of JumariersquosmodifiedRiemann-Liouvillederivative can be found in [28 29]

119863120572

119909119909119903=

Γ (1 + 119903)

Γ (1 + 119903 minus 120572)119909119903minus120572

(2)

119863120572

119909(119906 (119909) V (119909)) = V (119909)119863120572

119909119906 (119909) + 119906 (119909)119863

120572

119909V (119909) (3)

119863120572

119909119891 [119906 (119909)] = 119891

1015840

119906(119906)119863120572

119909119906 (119909) (4)

119863120572

119909119891 [119906 (119909)] = 119863

120572

119906119891 (119906) (119906

1015840

(119909))120572

(5)

which are direct consequences of the equality

Γ (1 + 120572) 119889119909 = 119889120572119909 (6)

In the above formulas (3)ndash(5) 119906(119909) is nondifferentiablefunction in (3) and (4) and differentiable in (5) The functionV(119909) is nondifferentiable and 119891(119906) is differentiable in (4) andnondifferentiable in (5) Because of these the formulas (3)ndash(5) should be used carefully The above equations play animportant role in fractional calculus in Sections 3 and 4

3 Description of the Methods for FDEs

We consider the following general nonlinear FDEs of the type

119875 (119906119863120572

119905119906119863120573

119909119906119863120595

119910 119863120572

119905119863120572

119905119906119863120572

119905119863120573

119909119906

119863120573

119909119863120573

119909119906119863120573

119909119863120595

119910119906119863120595

119910119863120595

119910119906 ) = 0

0 lt 120572 120573 120595 lt 1

(7)

where 119906 is an unknown function 119875 is a polynomial of 119906 andits partial fractional derivatives in which the highest orderderivatives and the nonlinear terms are involved

The fractional complex transform [30ndash32] is the simplestapproach to convert the fractional differential equations

into ordinary differential equations This makes the solutionprocedure extremely simple The traveling wave variable is

119906 (119909 119910 119905) = 119880 (120585) (8)

where

120585 =120591119909120573

Γ (1 + 120573)+

120575119910120595

Γ (1 + 120595)+

120582119905120572

Γ (1 + 120572) (9)

where 120591 120575 and 120582 are nonzero arbitrary constants We canrewrite (7) in the following nonlinear ODE

119876(1198801198801015840 11988010158401015840 119880101584010158401015840 ) = 0 (10)

where the primedenotes the derivationwith respect to 120585 Nowwe consider three different methods

31 Basic Idea of Functional Variable Method The featuresof this method are presented in [33] We describe functionalvariable method to find exact solutions of nonlinear space-time fractional differential equations as follows

Let us make a transformation in which the unknownfunction 119880 is considered as a functional variable in the form

119880120585= 119865 (119880) (11)

and some successive derivatives of 119880 are

119880120585120585=

1

2(1198652)1015840

119880120585120585120585

=1

2(1198652)10158401015840radic1198652

119880120585120585120585120585

=1

2[(1198652)101584010158401015840

1198652+ (1198652)10158401015840

(1198652)1015840

]

(12)

where ldquo 1015840rdquo stands for 119889119889119880 The ODE (10) can be reduced interms of 119880 119865 and its derivatives by using the expressions of(12) into (10) as

119877 (119880 119865 1198651015840 11986510158401015840 119865101584010158401015840 119865(4) ) = 0 (13)

The key idea of this particular form (13) is of specialinterest since it admits analytical solutions for a large classof nonlinear wave type equations Integrating (13) gives theexpression of 119865 This and (11) give the appropriate solutionsto the original problem

32 Basic Idea of Exp-Function Method According to exp-function method developed by He and Abdou [34] weassume that the wave solution can be expressed in thefollowing form

119880 (120585) =sum119889

119899=minus119888119886119899exp [119899120585]

sum119902

119898=minus119901119887119898exp [119898120585]

(14)

Advances in Mathematical Physics 3

where 119901 119902 119888 and 119889 are positive integers which are known tobe further determined and 119886

119899and 119887119898are unknown constants

We can rewrite (14) in the following equivalent form

119880 (120585) =119886minus119888exp [minus119888120585] + sdot sdot sdot + 119886

119889exp [119889120585]

119887minus119901

exp [minus119901120585] + sdot sdot sdot + 119887119902exp [119902120585]

(15)

This equivalent formulation plays an important andfundamental part in finding the analytic solution of problemsTo determine the value of 119888 and 119901 we balance the linear termof highest order of (10) with the highest order nonlinear termSimilarly to determine the value of 119889 and 119902 we balance thelinear termof lowest order of (10) with lowest order nonlinearterm [35ndash40]

33 Basic Idea of (1198661015840119866)-Expansion Method According to(1198661015840119866)-expansionmethod developed byWang et al [41] the

solution of (10) can be expressed by a polynomial in (1198661015840119866)

as

119880 (120585) =

119898

sum119894=0

119886119894(1198661015840

119866)

119894

119886119898

= 0 (16)

where 119886119894(119894 = 0 1 2 119898) are constants while 119866(120585) satisfies

the following second-order linear ordinary differential equa-tion

11986610158401015840

(120585) + 1205821198661015840

(120585) + 120583119866 (120585) = 0 (17)

where 120582 and 120583 are constants The positive integer 119898 canbe determined by considering the homogeneous balancebetween the highest order derivatives and the nonlinearterms appearing in (10) By substituting (16) into (10) andusing (17) we collect all terms with the same order of (1198661015840119866)Then by equating each coefficient of the resulting polynomialto zero we obtain a set of algebraic equations for 119886

119894(119894 =

0 1 2 119898) 120582 120583 120591 120575 and 120582 Finally solving the system ofequations and substituting 119886

119894(119894 = 0 1 2 119898) 120582 120583 120591 120575

120582 and the general solutions of (17) into (16) we can get avariety of exact solutions of (7) [42 43]

4 Exact Solutions of Space-TimeFractional Symmetric Regularized LongWave (SRLW) Equation

We consider the space-time fractional symmetric regularizedlong wave (SRLW) equation [44]

1198632120572

119905119906 + 119863

2120572

119909119906 + 119906119863

120572

119905(119863120572

119909119906)

+ 119863120572

119909119906119863120572

119905119906 + 119863

2120572

119905(1198632120572

119909119906) = 0

0 lt 120572 le 1

(18)

which arises in several physical applications including ionsound waves in plasma For 120572 = 1 it is shown thatthis equation describes weakly nonlinear ion acoustic andspace-charge waves and the real-valued 119906(119909 119905) corresponds

to the dimensionless fluid velocity with a decay condition[45]

We use the following transformations

119906 (119909 119905) = 119880 (120585) (19)

120585 =119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572) (20)

where 119896 and 119888 are nonzero constantsSubstituting (20) with (1) into (18) equation (18) can be

reduced into an ODE

(1198882+ 1198962)11988010158401015840+ 119888119896119880119880

10158401015840+ 119888119896(119880

1015840)2

+ 119888211989621198801015840101584010158401015840

= 0 (21)

where ldquo1198801015840rdquo = 119889119880119889120585

41 Exact Solutions by Functional Variable Method Integrat-ing (21) twice and setting the constants of integration to bezero we obtain

(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840= 0 (22)

or

119880120585120585= minus

1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896 (23)

Then we use the transformation (11) and (12) to convert(22) to

1

2(1198652)1015840

= minus1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896

119865 (119880) = ∓119880radicminus1198882 + 1198962

11988821198962minus

119880

3119888119896

(24)

The solution of (21) is obtained as

119880 (120585) = minus3 (1198882 + 1198962)

119888119896sec2 (

radic1198882 + 1198962

2119896119888120585) (25)

So we have

1199061(119909 119905) = minus

3 (1198882 + 1198962)

119888119896

times sec2 radic1198882 + 1198962

2119896119888(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(26)

which is the exact solution of space-time fractional symmet-ric regularized long wave (SRLW) equation One can see thatthe result is different than results of Alzaidy [44]


42 Exact Solutions by Exp-Function Method Balancing theorder of 11988010158401015840 and 1198802 in (22) we obtain

11988010158401015840=

1198881exp [minus (119888 + 3119901) 120585] + sdot sdot sdot

1198882exp [minus4119901120585] + sdot sdot sdot

1198802=



(27)

where 119888119894are determined coefficients only for simplicity

Balancing highest order of exp-function in (27) we have

minus (119888 + 3119901) = minus (2119888 + 2119901) (28)

which leads to the result

119901 = 119888 (29)

In the samewaywe balance the linear termof the lowest orderin (22) to determine the values of 119889 and 119902

11988010158401015840=

sdot sdot sdot + 1198891exp [(119889 + 3119902) 120585]

sdot sdot sdot + 1198892exp [4119902120585]

1198802=



(30)

where 119889119894are determined coefficients only for simplicity From

(30) we have

3119902 + 119889 = 2119889 + 2119902 (31)

and this gives

119902 = 119889 (32)

For simplicity we set 119901 = 119888 = 1 and 119902 = 119889 = 1 so (15)reduces to

119880 (120585) =1198861exp (120585) + 119886

0+ 119886minus1exp (minus120585)

1198871exp (120585) + 119887


(33)

Substituting (33) into (22) and using Maple we obtain

1

119860[1198773exp (3120585) + 119877

2exp (2120585) + 119877

1exp (120585) + 119877

0

+ 119877minus1exp (minus120585) + 119877

minus2exp (minus2120585) + 119877

minus3exp (minus3120585)] = 0

(34)

where

119860 = (119887minus1exp (minus120585) + 119887

0+ 1198871exp (120585))3

1198773= 119896211988611198872

1+ 119888211988611198872

1+1

21198881198961198862

11198871

1198772= 119896211988601198872

1+ 119888211988601198872

1minus 11988821198962119886111988711198870+ 119888119896119886111988601198871

+ 21198882119886111988711198870+1

21198881198961198862

11198870

+ 1198882119896211988601198872

1+ 21198962119886111988711198870

1198771= 21198962119886011988711198870+ 119888211988611198872

0+ 1198882119886minus11198872

1+ 1198962119886minus11198872

1

minus 11988821198962119886011988711198870+ 1198881198961198861119886minus11198871minus 41198882119896211988611198871119887minus1

+ 119888119896119886111988601198870+ 119896211988611198872

0+ 21198882119886011988711198870+1

21198881198961198862

01198871

+ 2119896211988611198871119887minus1

+ 1198882119896211988611198872

0+ 411988821198962119886minus11198872

1

+ 2119888211988611198871119887minus1

+1

21198881198961198862

1119887minus1

1198770= 2119896211988601198871119887minus1

+ 21198882119886minus111988711198870+ 2119896211988611198870119887minus1

+ 21198962119886minus111988711198870+ 2119888211988611198870119887minus1

+ 2119888211988601198871119887minus1

+ 311988821198962119886minus111988701198871+ 1198881198961198861119886minus11198870+ 1198881198961198860119886minus11198871

+ 31198882119896211988611198870119887minus1

+1

21198881198961198862

01198870minus 61198882119896211988601198871119887minus1

+ 119896211988601198872

0+ 119888211988601198872

0+ 11988811989611988611198860119887minus1

119877minus1

= 119896211988611198872

minus1+ 119888211988611198872

minus1+ 1198882119886minus11198872

0+ 1198962119886minus11198872

0

minus 411988821198962119886minus11198871119887minus1

minus 119888211989621198860119887minus11198870+ 1198881198961198861119886minus1119887minus1

+ 1198881198961198860119886minus11198870+1

21198881198961198862

minus11198871+ 21198962119886minus11198871119887minus1

+ 2119896211988601198870119887minus1

+ 11988821198962119886minus11198872

0+1

21198881198961198862

0119887minus1

+ 2119888211988601198870119887minus1

+ 21198882119886minus11198871119887minus1

+ 41198882119896211988611198872

minus1

119877minus2

= 119896211988601198872

minus1+ 119888211988601198872

minus1minus 11988821198962119886minus11198870119887minus1

+ 1198881198961198860119886minus1119887minus1

+1

21198881198961198862

minus11198870+ 1198882119896211988601198872


+ 21198962119886minus11198870119887minus1

119877minus3

= 1198882119886minus11198872

minus1+ 1198962119886minus11198872

minus1+1

21198881198961198862

minus1119887minus1

(35)


Solving this system of algebraic equations by usingMaplewe get the following results

1198861= 0 119886

0= ∓

611989621198870

radicminus1198962 minus 1 119886

minus1= 0

1198871=

11988720

4119887minus1

1198870= 1198870 119887

minus1= 119887minus1

119888 = ∓radic1198962

minus1198962 minus 1 119896 = 119896

(36)

where 1198870and 119887minus1

are arbitrary parameters Substituting theseresults into (33) we get the following exact solution

119880 (120585) = ∓611989621198870radicminus1198962 minus 1

(119887204119887minus1) exp (120585) + 119887


(37)

where 1198870and 1198871are arbitrary parameters and 120585 = (119896119909120572Γ(1 +

120572)) ∓ radic1198962(minus1198962 minus 1)(119905120572Γ(1 + 120572))Finally if we take 119887

minus1= 1 and 119887

0= 2 (37) becomes

119906 (119909 119905)

= ∓61198962

radicminus1198962 minus 1

times1

1 + cosh ((119896119909120572Γ (1 + 120572))∓radic1198962 (minus1198962 minus 1) (119905120572Γ (1 + 120572)))

(38)

and we obtain the hyperbolic function solution of the space-time fractional symmetric regularized long wave (SRLW)equation Comparing our result to the results in [46] it canbe seen that our solution has never been obtained

43 Exact Solutions by (1198661015840119866)-Expansion Method RecentlyZayed et al [47] obtained solitary wave solutions to SRLWequation by means of improved (119866

1015840119866)-expansion methodBut they applied this method to (22) Namely they took theconstants of integration as zero

In our study we integrate (21) twice with respect to 120585 andwe get

(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840+ 1205850119880 + 1205851= 0 (39)

where 1205850and 1205851are constants of integration

Use ansatz (39) for the linear term of highest order11988010158401015840 with the highest order nonlinear term 119880

2 By simplecalculation balancing 11988010158401015840 with 1198802 in (39) gives

119898 + 2 = 2119898 (40)

so that

119898 = 2 (41)

Suppose that the solutions of (41) can be expressed by apolynomial in (119866

1015840119866) as follows

119880 (120585) = 1198860+ 1198861(1198661015840

119866) + 1198862(1198661015840

119866)

2

1198862

= 0 (42)

By using (17) and (42) we have

11988010158401015840

(120585) = 61198872(1198661015840

119866)

4

+ (21198871+ 101198872120582)(

1198661015840

119866)

3

+ (81198872120583 + 3119887

1120582 + 4119887

21205822) (

1198661015840

119866)

2

+ (61198872120582120583 + 2119887

1120583 + 11988711205822)(

1198661015840

119866)

+ 211988721205832+ 1198871120582120583

1198802

(120585) = 1198872

2(1198661015840

119866)

4

+ 211988711198872(1198661015840

119866)

3

+ 211988701198872(1198661015840

119866)

2

+ 1198872

1(1198661015840

119866)

2

+ 211988701198871(1198661015840

119866) + 1198872

0

(43)

Substituting (42) and (43) into (39) collecting the coef-ficients of (1198661015840119866)119894 (119894 = 0 4) and setting it to zero weobtain the following system

minus1

21198881198961198862

2+ 6119888211989621198862= 0

2119888211989621198861minus 11988811989611988611198862+ 10119888211989621198862120582 = 0

minus1

21198881198961198862

1minus 11988811989611988601198862+ 8119888211989621198862120583 + 3119888

211989621198861120582

+ 12058501198862+ 41198882119896211988621205822+ 11989621198862+ 11988821198862= 0

minus 11988811989611988601198861+ 1198882119896211988611205822+ 11989621198861+ 12058501198861+ 6119888211989621198862120582120583

+ 11988821198861+ 2119888211989621198861120583 = 0

minus1

21198881198961198862

0+ 21198882119896211988621205832+ 11988821198860+ 12058501198860+ 11989621198860

+ 119888211989621198861120582120583 + 120585

1= 0

(44)

Solving this system by using Maple gives

1198860=

1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896 119886

1= 12119888119896120582

1198862= 12119888119896 119888 = 119888

119896 = 119896 1205850= 1205850

1205851= (minus2119888

21198962minus 8119888411989641205822120583 + 16119888

411989641205832+ 119888411989641205824minus 1198964

minus1198884minus 211989621205850minus 211988821205850minus 1205852

0) (2119888119896)

minus1

(45)

where 120582 120583 1205850 and 120585

1are arbitrary constants


By using (42) expression (45) can be written as

119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)

Substituting general solutions of (17) into (46) we havethree types of travelling wave solutions of space-time frac-tional symmetric regularized long wave (SRLW) equationThese are the following

When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882

119888119896minus 2119888119896 (1205822 minus 4120583) + 3119888119896 (1205822 minus 4120583)

times(1198621sinh (12)radic1205822 minus 4120583120585 + 119862

2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)

where 120585 = (119896119909120572Γ(1 + 120572)) + (119888119905120572Γ(1 + 120572))When 1205822 minus 4120583 lt 0

1198802(120585)

=1205850+ 1198962+ 1198882


times(minus1198621sin (12)radic4120583 minus 1205822120585 + 119862

2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)

where 120585 = (119896119909120572Γ(1 + 120572)) + (119888119905120572Γ(1 + 120572))When 1205822 minus 4120583 = 0

1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)

In particular if 1198621

= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)

Comparing our results to Zayedrsquos results [47] it can beseen that these results are new

5 Conclusion

In this paper the functional variable method the exp-function method and (1198661015840119866)-expansion method have beensuccessfully employed to obtain solution of the space-timefractional symmetric regularized long wave (SRLW) equa-tionThese solutions include the generalized hyperbolic func-tion solutions generalized trigonometric function solutionsand rational function solutions which may be very useful tounderstand the nonlinear FDEs and our result can turn intohyperbolic solution when suitable parameters are chosen Tothe best of our knowledge the solutions obtained in this paperhave not been reported in literature Maple has been used forprogramming and computations in this work

Conflict of Interests

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

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations North-Holland Mathematics Studies Elsevier Science BV Amster-dam Netherlands 2006

[4] H Sun and W Chen ldquoFractal derivative multi-scale modelof fluid particle transverse accelerations in fully developedturbulencerdquo Science in China Series E Technological Sciencesvol 52 no 3 pp 680ndash683 2009

[5] W Chen and H Sun ldquoMultiscale statistical model of fully-developed turbulence particle accelerationsrdquo Modern PhysicsLetters B vol 23 no 3 pp 449ndash452 2009

[6] J Cresson ldquoScale calculus and the Schrodinger equationrdquoJournal of Mathematical Physics vol 44 no 11 pp 4907ndash49382003

[7] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[8] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[9] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[10] B Zheng ldquo(1198661015840

119866)-expansion method for solving fractionalpartial differential equations in the theory of mathematicalphysicsrdquo Communications in Theoretical Physics vol 58 no 5pp 623ndash630 2012

[11] K A Gepreel and S Omran ldquoExact solutions for nonlinearpartial fractional differential equationsrdquo Chinese Physics B vol21 no 11 Article ID 110204 2012


[12] A Bekir and O Guner ldquoExact solutions of nonlinear fractionaldifferential equations by (119866

1015840

119866)-expansion methodrdquo ChinesePhysics B vol 22 no 11 Article ID 110202 2013

[13] N Shang and B Zheng ldquoExact solutions for three fractionalpartial differential equations by the (1198661015840119866) methodrdquo IAENGInternational Journal of Applied Mathematics vol 43 no 3 pp114ndash119 2013

[14] 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 no 1 pp 48ndash512010

[15] A Bekir O Guner and A C Cevikel ldquoFractional complextransform and exp-function methods for fractional differentialequationsrdquo Abstract and Applied Analysis vol 2013 Article ID426462 8 pages 2013

[16] B Zheng ldquoExp-function method for solving fractional partialdifferential equationsrdquo The Scientific World Journal vol 2013Article ID 465723 8 pages 2013

[17] B Lu ldquoThe first integral method for some time fractionaldifferential equationsrdquo Journal of Mathematical Analysis andApplications vol 395 no 2 pp 684ndash693 2012

[18] M Eslami B FVajargahMMirzazadeh andA Biswas ldquoAppli-cation of first integral method to fractional partial differentialequationsrdquo Indian Journal of Physics vol 88 no 2 pp 177ndash1842014

[19] 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

[20] B Zheng and CWen ldquoExact solutions for fractional partial dif-ferential equations by a new fractional sub-equation methodrdquoAdvances in Difference Equations vol 2013 article 199 2013

[21] J F Alzaidy ldquoFractional sub-equation method and its appli-cations to the spacemdashtime fractional differential equationsin mathematical physicsrdquo British Journal of Mathematics ampComputer Science vol 3 no 2 pp 153ndash163 2013

[22] 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

[23] ANazarzadehM Eslami andMMirzazadeh ldquoExact solutionsof somenonlinear partial differential equations using functionalvariablemethodrdquoPramanamdashJournal of Physics vol 81 no 2 pp225ndash236 2013

[24] H Bulut M Baskonus H and Y Pandir ldquoThe modifiedtrial equation method for fractional wave equation and timefractional generalized burgers equationrdquo Abstract and AppliedAnalysis vol 2013 Article ID 636802 8 pages 2013

[25] Y Pandir Y Gurefe and E Misirli ldquoNew exact solutionsof the time-fractional nonlinear dispersive KdV equationrdquoInternational Journal of Modeling and Optimization vol 3 no4 pp 349ndash352 2013

[26] N Taghizadeh M Mirzazadeh M Rahimian and M AkbarildquoApplication of the simplest equation method to some time-fractional partial differential equationsrdquoAin Shams EngineeringJournal vol 4 no 4 pp 897ndash902 2013

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

[28] G Jumarie ldquoTable of some basic fractional calculus formulaederived from amodified Riemann-Liouville derivative for non-differentiable functionsrdquo Applied Mathematics Letters vol 22no 3 pp 378ndash385 2009

[29] G Jumarie ldquoLaplacersquos transform of fractional order viathe Mittag-Leffler function and modified RiemannmdashLiouvillederivativerdquoAppliedMathematics Letters vol 22 no 11 pp 1659ndash1664 2009

[30] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 5 pp 970ndash973 2010

[31] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letters A Mathematics Physics and Mechanics vol 2 pp121ndash126 2011

[32] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[33] A Zerarka S Ouamane and A Attaf ldquoOn the functionalvariable method for finding exact solutions to a class of waveequationsrdquo Applied Mathematics and Computation vol 217 no7 pp 2897ndash2904 2010

[34] J He and M A Abdou ldquoNew periodic solutions for nonlinearevolution equations using Exp-function methodrdquo Chaos Soli-tons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[35] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons and Fractals vol 30 no 3 pp 700ndash708 2006

[36] A Bekir and A Boz ldquoApplication of Hersquos exp-function methodfor nonlinear evolution equationsrdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2286ndash2293 2009

[37] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[38] I Aslan ldquoOn the application of the Exp-functionmethod to theKP equation for119873-soliton solutionsrdquo Applied Mathematics andComputation vol 219 no 6 pp 2825ndash2828 2012

[39] S Yu ldquo119873-soliton solutions of the KP equation by Exp-functionmethodrdquoAppliedMathematics and Computation vol 219 no 8pp 3420ndash3424 2012

[40] S Zhang and H Zhang ldquoAn Exp-function method for a new$N$-soliton solutions with arbitrary functions of a $(2+1)$-dimensional vcBK systemrdquo Computers amp Mathematics withApplications vol 61 no 8 pp 1923ndash1930 2011

[41] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[42] A Bekir ldquoApplication of the (119866Ecircź119866)-expansion method fornonlinear evolution equationsrdquo Physics Letters A vol 372 no19 pp 3400ndash3406 2008

[43] H Jafari N Kadkhoda and A Biswas ldquoThe 119866Ecircź119866-expansionmethod for solutions of evolution equations from isothermalmagnetostatic atmospheresrdquo Journal of King Saud University-Science vol 25 no 1 pp 57ndash62 2013

[44] J F Alzaidy ldquoThe fractional sub-equation method and exactanalytical solutions for some nonlinear fractional PDEsrdquoAmer-ican Journal of Mathematical Analysis vol 1 no 1 pp 14ndash192013


[45] H Jafari A Borhanifar and S A Karimi ldquoNew solitarywave solutions for generalized regularized long-wave equationrdquoInternational Journal of Computer Mathematics vol 87 no 1ndash3pp 509ndash514 2010

[46] F Xu ldquoApplication of Exp-function method to symmetricregularized long wave (SRLW) equationrdquo Physics Letters A vol372 no 3 pp 252ndash257 2008

[47] E M E Zayed Y A Amer and R M A Shohib ldquoExact trav-eling wave solutions for nonlinear fractional partial differen-tial equations using the improved (G1015840G)-expansion methodrdquoInternational Journal of Engineering and Applied Science vol 7pp 18ndash31 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with the Caputo derivative the function to be differentiatedis not necessarily differentiable secondly different from theRiemann-Liouville derivative Jumariersquos modified Riemann-Liouville derivative of a constant is defined to be zeroJumariersquos modified Riemann-Liouville derivative of order 120572is defined by

119863120572

119909119891 (119909)

=

1

Γ (1 minus 120572)

times119889

119889119909int119909

0

(119909 minus 120585)minus120572

(119891 (120585) minus 119891 (0)) 119889120585

0 lt 120572 lt 1

(119891(119899) (119909))(120572minus119899)

119899 le 120572 lt 119899 + 1 119899 ge 1

(1)

where 119891 119877 rarr 119877 119909 rarr 119891(119909) denotes a continuous (but notnecessarily first-order-differentiable) function Some usefulformulas and results of JumariersquosmodifiedRiemann-Liouvillederivative can be found in [28 29]

119863120572

119909119909119903=

Γ (1 + 119903)

Γ (1 + 119903 minus 120572)119909119903minus120572

(2)

119863120572

119909(119906 (119909) V (119909)) = V (119909)119863120572

119909119906 (119909) + 119906 (119909)119863

120572

119909V (119909) (3)

119863120572

119909119891 [119906 (119909)] = 119891

1015840

119906(119906)119863120572

119909119906 (119909) (4)

119863120572

119909119891 [119906 (119909)] = 119863

120572

119906119891 (119906) (119906

1015840

(119909))120572

(5)

which are direct consequences of the equality

Γ (1 + 120572) 119889119909 = 119889120572119909 (6)

In the above formulas (3)ndash(5) 119906(119909) is nondifferentiablefunction in (3) and (4) and differentiable in (5) The functionV(119909) is nondifferentiable and 119891(119906) is differentiable in (4) andnondifferentiable in (5) Because of these the formulas (3)ndash(5) should be used carefully The above equations play animportant role in fractional calculus in Sections 3 and 4

3 Description of the Methods for FDEs

We consider the following general nonlinear FDEs of the type

119875 (119906119863120572

119905119906119863120573

119909119906119863120595

119910 119863120572

119905119863120572

119905119906119863120572

119905119863120573

119909119906

119863120573

119909119863120573

119909119906119863120573

119909119863120595

119910119906119863120595

119910119863120595

119910119906 ) = 0

0 lt 120572 120573 120595 lt 1

(7)

where 119906 is an unknown function 119875 is a polynomial of 119906 andits partial fractional derivatives in which the highest orderderivatives and the nonlinear terms are involved

The fractional complex transform [30ndash32] is the simplestapproach to convert the fractional differential equations

into ordinary differential equations This makes the solutionprocedure extremely simple The traveling wave variable is

119906 (119909 119910 119905) = 119880 (120585) (8)

where

120585 =120591119909120573

Γ (1 + 120573)+

120575119910120595

Γ (1 + 120595)+

120582119905120572

Γ (1 + 120572) (9)

where 120591 120575 and 120582 are nonzero arbitrary constants We canrewrite (7) in the following nonlinear ODE

119876(1198801198801015840 11988010158401015840 119880101584010158401015840 ) = 0 (10)

where the primedenotes the derivationwith respect to 120585 Nowwe consider three different methods

31 Basic Idea of Functional Variable Method The featuresof this method are presented in [33] We describe functionalvariable method to find exact solutions of nonlinear space-time fractional differential equations as follows

Let us make a transformation in which the unknownfunction 119880 is considered as a functional variable in the form

119880120585= 119865 (119880) (11)

and some successive derivatives of 119880 are

119880120585120585=

1

2(1198652)1015840

119880120585120585120585

=1

2(1198652)10158401015840radic1198652

119880120585120585120585120585

=1

2[(1198652)101584010158401015840

1198652+ (1198652)10158401015840

(1198652)1015840

]

(12)

where ldquo 1015840rdquo stands for 119889119889119880 The ODE (10) can be reduced interms of 119880 119865 and its derivatives by using the expressions of(12) into (10) as

119877 (119880 119865 1198651015840 11986510158401015840 119865101584010158401015840 119865(4) ) = 0 (13)

The key idea of this particular form (13) is of specialinterest since it admits analytical solutions for a large classof nonlinear wave type equations Integrating (13) gives theexpression of 119865 This and (11) give the appropriate solutionsto the original problem

32 Basic Idea of Exp-Function Method According to exp-function method developed by He and Abdou [34] weassume that the wave solution can be expressed in thefollowing form

119880 (120585) =sum119889

119899=minus119888119886119899exp [119899120585]

sum119902

119898=minus119901119887119898exp [119898120585]

(14)






119889exp [119889120585]

119887minus119901


(15)




as

119880 (120585) =

119898

sum119894=0

119886119894(1198661015840

119866)

119894

119886119898

= 0 (16)



11986610158401015840

(120585) + 1205821198661015840

(120585) + 120583119866 (120585) = 0 (17)


119894(119894 =


119894(119894 = 0 1 2 119898) 120582 120583 120591 120575




1198632120572

119905119906 + 119863

2120572

119909119906 + 119906119863

120572

119905(119863120572

119909119906)

+ 119863120572

119909119906119863120572

119905119906 + 119863

2120572

119905(1198632120572

119909119906) = 0

0 lt 120572 le 1

(18)




119906 (119909 119905) = 119880 (120585) (19)

120585 =119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572) (20)


reduced into an ODE

(1198882+ 1198962)11988010158401015840+ 119888119896119880119880

10158401015840+ 119888119896(119880

1015840)2

+ 119888211989621198801015840101584010158401015840

= 0 (21)



(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840= 0 (22)

or

119880120585120585= minus

1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896 (23)


1

2(1198652)1015840

= minus1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896

119865 (119880) = ∓119880radicminus1198882 + 1198962

11988821198962minus

119880

3119888119896

(24)


119880 (120585) = minus3 (1198882 + 1198962)

119888119896sec2 (

radic1198882 + 1198962

2119896119888120585) (25)

So we have

1199061(119909 119905) = minus

3 (1198882 + 1198962)

119888119896

times sec2 radic1198882 + 1198962

2119896119888(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(26)




11988010158401015840=



1198802=



(27)



minus (119888 + 3119901) = minus (2119888 + 2119901) (28)


119901 = 119888 (29)


11988010158401015840=



1198802=



(30)


(30) we have

3119902 + 119889 = 2119889 + 2119902 (31)

and this gives

119902 = 119889 (32)


119880 (120585) =1198861exp (120585) + 119886


1198871exp (120585) + 119887


(33)


1

119860[1198773exp (3120585) + 119877

2exp (2120585) + 119877

1exp (120585) + 119877

0

+ 119877minus1exp (minus120585) + 119877

minus2exp (minus2120585) + 119877


(34)

where

119860 = (119887minus1exp (minus120585) + 119887

0+ 1198871exp (120585))3

1198773= 119896211988611198872

1+ 119888211988611198872

1+1

21198881198961198862

11198871

1198772= 119896211988601198872

1+ 119888211988601198872

1minus 11988821198962119886111988711198870+ 119888119896119886111988601198871

+ 21198882119886111988711198870+1

21198881198961198862

11198870

+ 1198882119896211988601198872

1+ 21198962119886111988711198870

1198771= 21198962119886011988711198870+ 119888211988611198872

0+ 1198882119886minus11198872

1+ 1198962119886minus11198872

1


+ 119888119896119886111988601198870+ 119896211988611198872

0+ 21198882119886011988711198870+1

21198881198961198862

01198871

+ 2119896211988611198871119887minus1

+ 1198882119896211988611198872

0+ 411988821198962119886minus11198872

1

+ 2119888211988611198871119887minus1

+1

21198881198961198862

1119887minus1

1198770= 2119896211988601198871119887minus1

+ 21198882119886minus111988711198870+ 2119896211988611198870119887minus1

+ 21198962119886minus111988711198870+ 2119888211988611198870119887minus1

+ 2119888211988601198871119887minus1


+ 31198882119896211988611198870119887minus1

+1

21198881198961198862

01198870minus 61198882119896211988601198871119887minus1

+ 119896211988601198872

0+ 119888211988601198872

0+ 11988811989611988611198860119887minus1

119877minus1

= 119896211988611198872

minus1+ 119888211988611198872

minus1+ 1198882119886minus11198872

0+ 1198962119886minus11198872

0



+ 1198881198961198860119886minus11198870+1

21198881198961198862


+ 2119896211988601198870119887minus1

+ 11988821198962119886minus11198872

0+1

21198881198961198862

0119887minus1

+ 2119888211988601198870119887minus1

+ 21198882119886minus11198871119887minus1

+ 41198882119896211988611198872

minus1

119877minus2

= 119896211988601198872

minus1+ 119888211988601198872


+ 1198881198961198860119886minus1119887minus1

+1

21198881198961198862

minus11198870+ 1198882119896211988601198872


+ 21198962119886minus11198870119887minus1

119877minus3

= 1198882119886minus11198872

minus1+ 1198962119886minus11198872

minus1+1

21198881198961198862

minus1119887minus1

(35)



1198861= 0 119886

0= ∓

611989621198870


minus1= 0

1198871=

11988720

4119887minus1

1198870= 1198870 119887


119888 = ∓radic1198962

minus1198962 minus 1 119896 = 119896

(36)




(119887204119887minus1) exp (120585) + 119887


(37)




0= 2 (37) becomes

119906 (119909 119905)

= ∓61198962


times1


(38)





(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840+ 1205850119880 + 1205851= 0 (39)




119898 + 2 = 2119898 (40)

so that

119898 = 2 (41)


1015840119866) as follows

119880 (120585) = 1198860+ 1198861(1198661015840

119866) + 1198862(1198661015840

119866)

2

1198862

= 0 (42)


11988010158401015840

(120585) = 61198872(1198661015840

119866)

4

+ (21198871+ 101198872120582)(

1198661015840

119866)

3

+ (81198872120583 + 3119887

1120582 + 4119887

21205822) (

1198661015840

119866)

2

+ (61198872120582120583 + 2119887

1120583 + 11988711205822)(

1198661015840

119866)

+ 211988721205832+ 1198871120582120583

1198802

(120585) = 1198872

2(1198661015840

119866)

4

+ 211988711198872(1198661015840

119866)

3

+ 211988701198872(1198661015840

119866)

2

+ 1198872

1(1198661015840

119866)

2

+ 211988701198871(1198661015840

119866) + 1198872

0

(43)


minus1

21198881198961198862

2+ 6119888211989621198862= 0

2119888211989621198861minus 11988811989611988611198862+ 10119888211989621198862120582 = 0

minus1

21198881198961198862

1minus 11988811989611988601198862+ 8119888211989621198862120583 + 3119888

211989621198861120582

+ 12058501198862+ 41198882119896211988621205822+ 11989621198862+ 11988821198862= 0

minus 11988811989611988601198861+ 1198882119896211988611205822+ 11989621198861+ 12058501198861+ 6119888211989621198862120582120583

+ 11988821198861+ 2119888211989621198861120583 = 0

minus1

21198881198961198862

0+ 21198882119896211988621205832+ 11988821198860+ 12058501198860+ 11989621198860

+ 119888211989621198861120582120583 + 120585

1= 0

(44)


1198860=

1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896 119886

1= 12119888119896120582

1198862= 12119888119896 119888 = 119888

119896 = 119896 1205850= 1205850

1205851= (minus2119888

21198962minus 8119888411989641205822120583 + 16119888

411989641205832+ 119888411989641205824minus 1198964


0) (2119888119896)

minus1

(45)

where 120582 120583 1205850 and 120585




119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)


When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882



2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)


1198802(120585)

=1205850+ 1198962+ 1198882



2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)


1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)


= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)


5 Conclusion




References










[10] B Zheng ldquo(1198661015840





1015840













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119889exp [119889120585]

119887minus119901


(15)




as

119880 (120585) =

119898

sum119894=0

119886119894(1198661015840

119866)

119894

119886119898

= 0 (16)



11986610158401015840

(120585) + 1205821198661015840

(120585) + 120583119866 (120585) = 0 (17)


119894(119894 =


119894(119894 = 0 1 2 119898) 120582 120583 120591 120575




1198632120572

119905119906 + 119863

2120572

119909119906 + 119906119863

120572

119905(119863120572

119909119906)

+ 119863120572

119909119906119863120572

119905119906 + 119863

2120572

119905(1198632120572

119909119906) = 0

0 lt 120572 le 1

(18)




119906 (119909 119905) = 119880 (120585) (19)

120585 =119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572) (20)


reduced into an ODE

(1198882+ 1198962)11988010158401015840+ 119888119896119880119880

10158401015840+ 119888119896(119880

1015840)2

+ 119888211989621198801015840101584010158401015840

= 0 (21)



(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840= 0 (22)

or

119880120585120585= minus

1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896 (23)


1

2(1198652)1015840

= minus1198882 + 1198962

11988821198962119880 minus

1198802

2119888119896

119865 (119880) = ∓119880radicminus1198882 + 1198962

11988821198962minus

119880

3119888119896

(24)


119880 (120585) = minus3 (1198882 + 1198962)

119888119896sec2 (

radic1198882 + 1198962

2119896119888120585) (25)

So we have

1199061(119909 119905) = minus

3 (1198882 + 1198962)

119888119896

times sec2 radic1198882 + 1198962

2119896119888(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(26)




11988010158401015840=



1198802=



(27)



minus (119888 + 3119901) = minus (2119888 + 2119901) (28)


119901 = 119888 (29)


11988010158401015840=



1198802=



(30)


(30) we have

3119902 + 119889 = 2119889 + 2119902 (31)

and this gives

119902 = 119889 (32)


119880 (120585) =1198861exp (120585) + 119886


1198871exp (120585) + 119887


(33)


1

119860[1198773exp (3120585) + 119877

2exp (2120585) + 119877

1exp (120585) + 119877

0

+ 119877minus1exp (minus120585) + 119877

minus2exp (minus2120585) + 119877


(34)

where

119860 = (119887minus1exp (minus120585) + 119887

0+ 1198871exp (120585))3

1198773= 119896211988611198872

1+ 119888211988611198872

1+1

21198881198961198862

11198871

1198772= 119896211988601198872

1+ 119888211988601198872

1minus 11988821198962119886111988711198870+ 119888119896119886111988601198871

+ 21198882119886111988711198870+1

21198881198961198862

11198870

+ 1198882119896211988601198872

1+ 21198962119886111988711198870

1198771= 21198962119886011988711198870+ 119888211988611198872

0+ 1198882119886minus11198872

1+ 1198962119886minus11198872

1


+ 119888119896119886111988601198870+ 119896211988611198872

0+ 21198882119886011988711198870+1

21198881198961198862

01198871

+ 2119896211988611198871119887minus1

+ 1198882119896211988611198872

0+ 411988821198962119886minus11198872

1

+ 2119888211988611198871119887minus1

+1

21198881198961198862

1119887minus1

1198770= 2119896211988601198871119887minus1

+ 21198882119886minus111988711198870+ 2119896211988611198870119887minus1

+ 21198962119886minus111988711198870+ 2119888211988611198870119887minus1

+ 2119888211988601198871119887minus1


+ 31198882119896211988611198870119887minus1

+1

21198881198961198862

01198870minus 61198882119896211988601198871119887minus1

+ 119896211988601198872

0+ 119888211988601198872

0+ 11988811989611988611198860119887minus1

119877minus1

= 119896211988611198872

minus1+ 119888211988611198872

minus1+ 1198882119886minus11198872

0+ 1198962119886minus11198872

0



+ 1198881198961198860119886minus11198870+1

21198881198961198862


+ 2119896211988601198870119887minus1

+ 11988821198962119886minus11198872

0+1

21198881198961198862

0119887minus1

+ 2119888211988601198870119887minus1

+ 21198882119886minus11198871119887minus1

+ 41198882119896211988611198872

minus1

119877minus2

= 119896211988601198872

minus1+ 119888211988601198872


+ 1198881198961198860119886minus1119887minus1

+1

21198881198961198862

minus11198870+ 1198882119896211988601198872


+ 21198962119886minus11198870119887minus1

119877minus3

= 1198882119886minus11198872

minus1+ 1198962119886minus11198872

minus1+1

21198881198961198862

minus1119887minus1

(35)



1198861= 0 119886

0= ∓

611989621198870


minus1= 0

1198871=

11988720

4119887minus1

1198870= 1198870 119887


119888 = ∓radic1198962

minus1198962 minus 1 119896 = 119896

(36)




(119887204119887minus1) exp (120585) + 119887


(37)




0= 2 (37) becomes

119906 (119909 119905)

= ∓61198962


times1


(38)





(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840+ 1205850119880 + 1205851= 0 (39)




119898 + 2 = 2119898 (40)

so that

119898 = 2 (41)


1015840119866) as follows

119880 (120585) = 1198860+ 1198861(1198661015840

119866) + 1198862(1198661015840

119866)

2

1198862

= 0 (42)


11988010158401015840

(120585) = 61198872(1198661015840

119866)

4

+ (21198871+ 101198872120582)(

1198661015840

119866)

3

+ (81198872120583 + 3119887

1120582 + 4119887

21205822) (

1198661015840

119866)

2

+ (61198872120582120583 + 2119887

1120583 + 11988711205822)(

1198661015840

119866)

+ 211988721205832+ 1198871120582120583

1198802

(120585) = 1198872

2(1198661015840

119866)

4

+ 211988711198872(1198661015840

119866)

3

+ 211988701198872(1198661015840

119866)

2

+ 1198872

1(1198661015840

119866)

2

+ 211988701198871(1198661015840

119866) + 1198872

0

(43)


minus1

21198881198961198862

2+ 6119888211989621198862= 0

2119888211989621198861minus 11988811989611988611198862+ 10119888211989621198862120582 = 0

minus1

21198881198961198862

1minus 11988811989611988601198862+ 8119888211989621198862120583 + 3119888

211989621198861120582

+ 12058501198862+ 41198882119896211988621205822+ 11989621198862+ 11988821198862= 0

minus 11988811989611988601198861+ 1198882119896211988611205822+ 11989621198861+ 12058501198861+ 6119888211989621198862120582120583

+ 11988821198861+ 2119888211989621198861120583 = 0

minus1

21198881198961198862

0+ 21198882119896211988621205832+ 11988821198860+ 12058501198860+ 11989621198860

+ 119888211989621198861120582120583 + 120585

1= 0

(44)


1198860=

1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896 119886

1= 12119888119896120582

1198862= 12119888119896 119888 = 119888

119896 = 119896 1205850= 1205850

1205851= (minus2119888

21198962minus 8119888411989641205822120583 + 16119888

411989641205832+ 119888411989641205824minus 1198964


0) (2119888119896)

minus1

(45)

where 120582 120583 1205850 and 120585




119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)


When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882



2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)


1198802(120585)

=1205850+ 1198962+ 1198882



2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)


1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)


= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)


5 Conclusion




References










[10] B Zheng ldquo(1198661015840





1015840













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11988010158401015840=



1198802=



(27)



minus (119888 + 3119901) = minus (2119888 + 2119901) (28)


119901 = 119888 (29)


11988010158401015840=



1198802=



(30)


(30) we have

3119902 + 119889 = 2119889 + 2119902 (31)

and this gives

119902 = 119889 (32)


119880 (120585) =1198861exp (120585) + 119886


1198871exp (120585) + 119887


(33)


1

119860[1198773exp (3120585) + 119877

2exp (2120585) + 119877

1exp (120585) + 119877

0

+ 119877minus1exp (minus120585) + 119877

minus2exp (minus2120585) + 119877


(34)

where

119860 = (119887minus1exp (minus120585) + 119887

0+ 1198871exp (120585))3

1198773= 119896211988611198872

1+ 119888211988611198872

1+1

21198881198961198862

11198871

1198772= 119896211988601198872

1+ 119888211988601198872

1minus 11988821198962119886111988711198870+ 119888119896119886111988601198871

+ 21198882119886111988711198870+1

21198881198961198862

11198870

+ 1198882119896211988601198872

1+ 21198962119886111988711198870

1198771= 21198962119886011988711198870+ 119888211988611198872

0+ 1198882119886minus11198872

1+ 1198962119886minus11198872

1


+ 119888119896119886111988601198870+ 119896211988611198872

0+ 21198882119886011988711198870+1

21198881198961198862

01198871

+ 2119896211988611198871119887minus1

+ 1198882119896211988611198872

0+ 411988821198962119886minus11198872

1

+ 2119888211988611198871119887minus1

+1

21198881198961198862

1119887minus1

1198770= 2119896211988601198871119887minus1

+ 21198882119886minus111988711198870+ 2119896211988611198870119887minus1

+ 21198962119886minus111988711198870+ 2119888211988611198870119887minus1

+ 2119888211988601198871119887minus1


+ 31198882119896211988611198870119887minus1

+1

21198881198961198862

01198870minus 61198882119896211988601198871119887minus1

+ 119896211988601198872

0+ 119888211988601198872

0+ 11988811989611988611198860119887minus1

119877minus1

= 119896211988611198872

minus1+ 119888211988611198872

minus1+ 1198882119886minus11198872

0+ 1198962119886minus11198872

0



+ 1198881198961198860119886minus11198870+1

21198881198961198862


+ 2119896211988601198870119887minus1

+ 11988821198962119886minus11198872

0+1

21198881198961198862

0119887minus1

+ 2119888211988601198870119887minus1

+ 21198882119886minus11198871119887minus1

+ 41198882119896211988611198872

minus1

119877minus2

= 119896211988601198872

minus1+ 119888211988601198872


+ 1198881198961198860119886minus1119887minus1

+1

21198881198961198862

minus11198870+ 1198882119896211988601198872


+ 21198962119886minus11198870119887minus1

119877minus3

= 1198882119886minus11198872

minus1+ 1198962119886minus11198872

minus1+1

21198881198961198862

minus1119887minus1

(35)



1198861= 0 119886

0= ∓

611989621198870


minus1= 0

1198871=

11988720

4119887minus1

1198870= 1198870 119887


119888 = ∓radic1198962

minus1198962 minus 1 119896 = 119896

(36)




(119887204119887minus1) exp (120585) + 119887


(37)




0= 2 (37) becomes

119906 (119909 119905)

= ∓61198962


times1


(38)





(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840+ 1205850119880 + 1205851= 0 (39)




119898 + 2 = 2119898 (40)

so that

119898 = 2 (41)


1015840119866) as follows

119880 (120585) = 1198860+ 1198861(1198661015840

119866) + 1198862(1198661015840

119866)

2

1198862

= 0 (42)


11988010158401015840

(120585) = 61198872(1198661015840

119866)

4

+ (21198871+ 101198872120582)(

1198661015840

119866)

3

+ (81198872120583 + 3119887

1120582 + 4119887

21205822) (

1198661015840

119866)

2

+ (61198872120582120583 + 2119887

1120583 + 11988711205822)(

1198661015840

119866)

+ 211988721205832+ 1198871120582120583

1198802

(120585) = 1198872

2(1198661015840

119866)

4

+ 211988711198872(1198661015840

119866)

3

+ 211988701198872(1198661015840

119866)

2

+ 1198872

1(1198661015840

119866)

2

+ 211988701198871(1198661015840

119866) + 1198872

0

(43)


minus1

21198881198961198862

2+ 6119888211989621198862= 0

2119888211989621198861minus 11988811989611988611198862+ 10119888211989621198862120582 = 0

minus1

21198881198961198862

1minus 11988811989611988601198862+ 8119888211989621198862120583 + 3119888

211989621198861120582

+ 12058501198862+ 41198882119896211988621205822+ 11989621198862+ 11988821198862= 0

minus 11988811989611988601198861+ 1198882119896211988611205822+ 11989621198861+ 12058501198861+ 6119888211989621198862120582120583

+ 11988821198861+ 2119888211989621198861120583 = 0

minus1

21198881198961198862

0+ 21198882119896211988621205832+ 11988821198860+ 12058501198860+ 11989621198860

+ 119888211989621198861120582120583 + 120585

1= 0

(44)


1198860=

1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896 119886

1= 12119888119896120582

1198862= 12119888119896 119888 = 119888

119896 = 119896 1205850= 1205850

1205851= (minus2119888

21198962minus 8119888411989641205822120583 + 16119888

411989641205832+ 119888411989641205824minus 1198964


0) (2119888119896)

minus1

(45)

where 120582 120583 1205850 and 120585




119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)


When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882



2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)


1198802(120585)

=1205850+ 1198962+ 1198882



2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)


1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)


= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)


5 Conclusion




References










[10] B Zheng ldquo(1198661015840





1015840













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198861= 0 119886

0= ∓

611989621198870


minus1= 0

1198871=

11988720

4119887minus1

1198870= 1198870 119887


119888 = ∓radic1198962

minus1198962 minus 1 119896 = 119896

(36)




(119887204119887minus1) exp (120585) + 119887


(37)




0= 2 (37) becomes

119906 (119909 119905)

= ∓61198962


times1


(38)





(1198882+ 1198962)119880 + 119888119896

1198802

2+ 1198882119896211988010158401015840+ 1205850119880 + 1205851= 0 (39)




119898 + 2 = 2119898 (40)

so that

119898 = 2 (41)


1015840119866) as follows

119880 (120585) = 1198860+ 1198861(1198661015840

119866) + 1198862(1198661015840

119866)

2

1198862

= 0 (42)


11988010158401015840

(120585) = 61198872(1198661015840

119866)

4

+ (21198871+ 101198872120582)(

1198661015840

119866)

3

+ (81198872120583 + 3119887

1120582 + 4119887

21205822) (

1198661015840

119866)

2

+ (61198872120582120583 + 2119887

1120583 + 11988711205822)(

1198661015840

119866)

+ 211988721205832+ 1198871120582120583

1198802

(120585) = 1198872

2(1198661015840

119866)

4

+ 211988711198872(1198661015840

119866)

3

+ 211988701198872(1198661015840

119866)

2

+ 1198872

1(1198661015840

119866)

2

+ 211988701198871(1198661015840

119866) + 1198872

0

(43)


minus1

21198881198961198862

2+ 6119888211989621198862= 0

2119888211989621198861minus 11988811989611988611198862+ 10119888211989621198862120582 = 0

minus1

21198881198961198862

1minus 11988811989611988601198862+ 8119888211989621198862120583 + 3119888

211989621198861120582

+ 12058501198862+ 41198882119896211988621205822+ 11989621198862+ 11988821198862= 0

minus 11988811989611988601198861+ 1198882119896211988611205822+ 11989621198861+ 12058501198861+ 6119888211989621198862120582120583

+ 11988821198861+ 2119888211989621198861120583 = 0

minus1

21198881198961198862

0+ 21198882119896211988621205832+ 11988821198860+ 12058501198860+ 11989621198860

+ 119888211989621198861120582120583 + 120585

1= 0

(44)


1198860=

1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896 119886

1= 12119888119896120582

1198862= 12119888119896 119888 = 119888

119896 = 119896 1205850= 1205850

1205851= (minus2119888

21198962minus 8119888411989641205822120583 + 16119888

411989641205832+ 119888411989641205824minus 1198964


0) (2119888119896)

minus1

(45)

where 120582 120583 1205850 and 120585




119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)


When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882



2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)


1198802(120585)

=1205850+ 1198962+ 1198882



2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)


1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)


= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)


5 Conclusion




References










[10] B Zheng ldquo(1198661015840





1015840













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 (120585) =1205850+ 1198882 + 1198962 + 119888211989621205822 + 811988821198962120583

119888119896

+ 12119888119896120582(1198661015840

119866) + 12119888119896(

1198661015840

119866)

2

(46)


When 1205822 minus 4120583 gt 0

1198801(120585)

=1205850+ 1198962 + 1198882



2cosh (12)radic1205822 minus 4120583120585

1198621cosh (12)radic1205822 minus 4120583120585 + 119862

2sinh (12)radic1205822 minus 4120583120585

)

2

(47)


1198802(120585)

=1205850+ 1198962+ 1198882



2cos (12)radic4120583 minus 1205822120585

1198621cos (12)radic4120583 minus 1205822120585 + 119862

2sin (12)radic4120583 minus 1205822120585

)

2

(48)


1199063(119909 119905) =

1205850+ 1198962 + 1198882

119888119896+ 12119888119896

times (1198622

1198621+ 1198622((119896119909120572Γ (1 + 120572)) + (119888119905120572Γ (1 + 120572)))

)

2

(49)


= 0 1198622= 0 120582 gt 0 120583 = 0 then 119880

1

and 1198802become

1199061(119909 119905) =

1205850+ 1198962 + 1198882

119888119896minus 2119888119896120582

2

+ 31198881198961205822tanh2 120582

2(

119896119909120572

Γ (1 + 120572)+

119888119905120572

Γ (1 + 120572))

(50)


5 Conclusion




References










[10] B Zheng ldquo(1198661015840





1015840













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1015840













































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









Documents

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