Research ArticleDifferential Transform Method with Complex Transforms toSome Nonlinear Fractional Problems in Mathematical Physics

Syed Tauseef Mohyud-Din1 Farah Jabeen Awan1 Jamshad Ahmad1 and Saleh M Hassan23

1Faculty of Sciences HITEC University Taxila Cantonment 44000 Pakistan2Department of Mathematics College of Science King Saud University PO Box 2455 Riyadh 11451 Saudi Arabia3Department of Mathematics College of Science Ain Shams University Abbassia Cairo 11566 Egypt

Correspondence should be addressed to Syed Tauseef Mohyud-Din tauseefsyedngmailcom

Received 28 May 2015 Revised 31 August 2015 Accepted 28 September 2015

Academic Editor Fazal M Mahomed

Copyright copy 2015 Syed Tauseef Mohyud-Din et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transformwith fractional complex transform The proposed technique is applied on three mathematical models namely fractional Kaup-Kupershmidt equation generalized fractionalDrinfeld-Sokolov equations and systemof coupled fractional Sine-Gordon equationssubject to the appropriate initial conditions which arise frequently in mathematical physicsThe derivatives are defined in Jumariersquossense The accuracy efficiency and convergence of the proposed technique are demonstrated through the numerical examples Itis observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractionaldifferential equations The proposed technique is independent of complexities arising in the calculation of Lagrange multipliersAdomianrsquos polynomials linearization discretization perturbation and unrealistic assumptions and hence gives the solution in theform of convergent power series with elegantly computed components All the examples show that the proposed combination is apowerful mathematical tool to solve other nonlinear equations also

1 Introduction

Nonlinear partial differential equations (NLPDEs) aremathe-maticalmodels that are used to describe complex phenomenaand dynamic processes arising in the world around usThe NLPDEs appear in many applications of science andengineering such as fluid dynamics plasma physics hydro-dynamics solid state physics optical fibers and acousticsas well as other disciplines Recently lot of attention is paidto finding appropriate solutions of NLPDEs In the similarcontext various techniques including Adomianrsquos decom-position method (ADM) [1] Variational Iteration (VIM)[2] Homotopy Perturbation (HPM) [3] Homotopy Anal-ysis (HAM) [4] F-Expansion [5] Exp-function [6] sine-cosine [7] differential transform method (DTM) [8ndash11] andreduced differential transform [9 12ndash15] have been appliedon wide range of linear and nonlinear problems of diversifiedphysical nature Inspired and motivated by ongoing researchin this area we apply reduced differential transform method

(RDTM) [12ndash19] coupled with a complex transform to solvethree important mathematical models [20ndash27] namely frac-tional Kaup-Kupershmidt equation generalized fractionalDrinfeld-Sokolov equations and systemof coupled fractionalSine-Gordon equations subject to the appropriate initialconditions It is worthmentioning that derivatives are definedin Jumariersquos sense which is relatively a new approach and iseasier to handle however other approaches like Caputo andRiemann Liouville may also be utilized It is an establishedfact that models under discussion [20ndash23] are of extremeimportance and hence appear frequently in various physicalphenomena including nonlinear dispersive waves shallowwater waves ion acoustic plasma waves Lax pairs of a specialform four-reduction of KP hierarchy Frenkel-Kontorovadislocation model see [20ndash23] and the references therein Itis observed that the proposed technique is extremely simpleand user friendly and has shown very useful results It is tobe highlighted that the suggested modified version may beextended to some other important nonlinear problems which

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 364853 9 pageshttpdxdoiorg1011552015364853

2 Mathematical Problems in Engineering

have been solved by some other reliablemethods see [28ndash33]It is to be highlighted that Ganji et al have solved wide rangeof mathematical models [28ndash33] by making an appropriateuse of some recently developed schemes and hence giving anew avenue of research

2 Jumariersquos Fractional Derivative

Some useful formulas and results of Jumariersquos fractionalderivative were summarized [24]

119863120572

119909119888 = 0 120572 ge 0 119888 = constant (1)

119863120572

119909[119888119891 (119909)] = 119888119863

120572

119909119891 (119909) 120572 ge 0 119888 = constant (2)

119863120572

119909119909120573

=Γ (1 + 120573)

Γ (1 + 120573 minus 120572)119909120573minus120572

120573 ge 120572 ge 0 (3)

119863120572

119909[119891 (119909) 119892 (119909)] = [119863

120572

119909119891 (119909) 119892 (119909) + 119891 (119909)119863

120572

119909119892 (119909)] (4)

119863120572

119909119891 (119909 (119905)) = 119891

1015840

119909(119909) 119909120572

(119905) (5)

3 Fractional ComplexTransform Method (FCTM)

The fractional complex transform was first proposed in [25]and is defined as

119879 =119901119905120572

Γ (120572 + 1)

119883 =119902119909120573

Γ (120573 + 1)

119884 =119896119910120574

Γ (1 + 120574)

119885 =119897119911120582

Γ (1 + 120582)

(6)

where 119901 119902 119896 and 119897 are unknown constants 0 lt 120572 le 1 0 lt

120573 le 1 0 lt 120574 le 1 and 0 lt 120582 le 1

4 Reduced DifferentialTransform Method (RDTM)

To illustrate the basic idea of the DTMThe differential trans-form of 119896th derivative of a function 119906(119909 119905) which is analyticand differentiated continuously in the domain of interest isdefined as

119880119896(119909) =

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

(7)

The differential inverse transform of 119880119896(119909) is defined as fol-

lows

119906 (119909 119905) =

infin

sum

119896=0

119880119896(119909) (119905 minus 119905

0)119896

(8)

Equation (8) is known as the Taylor series expansion of 119906(119909 119905)around 119905 = 119905

0 Combining (7) and (8)

119906 (119909 119905) =

infin

sum

119896=0

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

(119905 minus 1199050)119896

(9)

when 1199050= 0 the above equation reduces to

119906 (119909 119905) =

infin

sum

119896=0

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

119905119896

(10)

and (2) reduces to

119906 (119909 119905) =

infin

sum

119896=0

119880119896(119909) 119905119896

(11)

Some properties of the reduced differential transform methodare as follows

(1) If the original function is 119906(119909 119905) = 119908(119909 119905) + V(119909 119905)then the transformed function is

119880119896(119909) = 119882

119896(119909) + 119881

119896(119909) (12)

(2) If 119906(119909 119905) = 120572119908(119909 119905) then 119880119896(119909) = 120572119882

119896(119909)

(3) If 119906(119909 119905) = 120597119898

119908(119909 119905)120597119905119898 then 119880

119896(119909) = ((119896 + 119898)

119896)119882119896(119909)

(4) If 119906(119909 119905) = 120597119908(119909 119905)120597119909 then 119880119896(119909) = (120597120597119909)119882

119896(119909)

(5) If 119906(119909 119910 119905) = 120597119908(119909 119910 119905)120597119909 then 119880119896(119909 119910) = (120597

120597119909)119882119896(119909 119910)

(6) If 119906(119909 119910 119911 119905) = 120597119908(119909 119910 119911 119905)120597119909 then 119880119896(119909 119910 119911) =

(120597120597119909)119882119896(119909 119910 119911)

(7) If 119906(119909 119905) = 119909119898

119905119899

119908(119909 119905) then 119880119896(119909) = 119909

119898

119882119896minus119899

(119909)(8) If 119906(119909 119905) = 119908

2

(119909 119905) then119880119896(119909) = sum

119896

119903=0119882119903(119909)119882119896minus119903

(119909)

5 Numerical Applications

In this section we apply the proposed fractional complextransform method coupled with reduced differential trans-formmethod to solve three important mathematical modelsNumerical results are highly encouraging For details aboutsuch equations readers are referred to study [22 23]

51 Fractional Kaup-Kupershmidt (FKK) Equation It is anestablished fact that fractional Kaup-Kupershmidt (FKK)equation plays a major role in the study of nonlinear disper-sive waves Moreover it describes a large number of impor-tant physical phenomena such as shallow water waves andion acoustic plasma waves

Consider the nonlinear KK equation [22 23]

120597120572

119906

120597119905120572minus1205975

119906

1205971199095minus 5119906

1205973

119906

1205971199093minus25

3

120597119906

120597119909

1205972

119906

1205971199092minus 51199062120597119906

120597119909= 0 (13)

with the initial condition

119906 (119909 0) = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2 (14)

where 119896 is an arbitrary constant

Mathematical Problems in Engineering 3

Applying the transformation [25] we get the followingpartial differential equation

120597119906

120597119879minus1205975

119906

1205971199095minus 5119906

1205973

119906

1205971199093minus25

3

120597119906

120597119909

1205972

119906

1205971199092minus 51199062120597119906

120597119909= 0 (15)

Applying the differential transform to (15) and (14) we obtainthe following recursive formula

(119896 + 1)119880119896+1

(119909)

=1205975

119880119896(119909)

1205971199095+ 5

119896

sum

119903=0

119880119896minus119903

(119909)1205973

119880119903(119909)

1205971199093

+25

3

119896

sum

119903=0

119880119896minus119903

(119909)1205972

119880119903(119909)

1205971199092

+ 5

119896

sum

119903=0

119903

sum

119904=119896

119880119896minus119903

(119909)119880119903minus119904

(119909)120597119880119904(119909)

120597119909

(16)

Using the initial condition we have

1198800(119909) = minus2119896

2

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2 (17)

Now substituting (17) into (16) and by straightforwarditerative steps yields

1198801(119909) = minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

1198802(119909) = minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198803(119909) =

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

(18)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

119879

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198792

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

1198793

+ sdot sdot sdot

(19)

The inverse transformation will yield

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(119890119896119909

minus 1)

(1 + 119890119896119909)3

119905120572

Γ (120572 + 1)

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1199052120572

Γ2 (120572 + 1)

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

sdot1199053120572

Γ3 (120572 + 1)+ sdot sdot sdot

(20)

This solution is convergent to the exact solution

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909+111198965119905

minus241198962

(1 + 119890119896119909+111198965119905)2 (21)

In Figures 1(a) and 1(b) we have presented approximatesolution at 120572 = 1 and exact solutions

52 Generalized Fractional Drinfeld-Sokolov (GFDS) Equa-tions [20 21] This system was introduced independently byDrinfeld and Sokolov [20 21]This coupled system was givenas one of the numerous examples of nonlinear equationspossessing Lax pairs of a special form Also the coupledsystem was found as a special case of the four-reduction ofthe KP hierarchy see [20 21] and the references therein

We consider the system of generalized fractionalDrinfeld-Sokolov (GFDS) equations [20 21]

120597120573

119906

120597119905120573+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597120573V120597119905120573

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

0 lt 119909 119905 lt 120587 0 lt 120573 le 1

(22)

with the initial conditions

119906 (119909 0) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

V (119909 0) = 119887 tanh (119896119909) (23)

where 120572 is a constantApplying the transformation [25] we get the following

partial differential equations

120597119906

120597119879+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597V120597119879

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

(24)

4 Mathematical Problems in Engineering

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(a)

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(b)

Figure 1 (a) Approximate solution (b) Exact solution

Applying the differential transform to (24) and (23) weobtain the following recursive formula

(119896 + 1)119880119896+1

(119909) = minus1205973

119880119896(119909)

1205971199093+ 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119880119903(119909)

120597119909

+ 6120597119881120572

119896(119909)

120597119909

(119896 + 1)119881119896+1

(119909) = 21205973

119881119896(119909)

1205971199093minus 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119881119903(119909)

120597119909

(25)

Using the initial condition we have

1198800(119909) =

minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

1198810(119909) = 119887 tanh (119896119909)

(26)

Now substituting (26) into (25) when (120572 = 2) and bystraightforward iterative steps yields

1198801(119909) =

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

1198811(119909) =

1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896

1198802(119909) = minus

1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)4

1198812(119909) = minus

1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198803(119909)

=1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

1198813(119909) =

1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)4

(27)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119879

minus1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41198792

+1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)51198793

+ sdot sdot sdot

Mathematical Problems in Engineering 5

V (119909 119879)

= 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119879

minus1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198792

+1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41198793

+ sdot sdot sdot

(28)

The inverse transformation will yields

119906 = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119905120573

Γ (120573 + 1)minus1

2

sdot

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41199052120573

Γ2 (120573 + 1)+1

3

sdot

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

sdot1199053120573

Γ3 (120573 + 1)+ sdot sdot sdot

V = 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119905120573

Γ (120573 + 1)minus1

4

sdot

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1199052120573

Γ2 (120573 + 1)+

1

24

sdot

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41199053120573

Γ3 (120573 + 1)

+ sdot sdot sdot

(29)

This solution is convergent to the exact solution

119906 (119909 119905) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909 +

31198872

+ 41198964

2119896119905)

V (119909 119905) = 119887 tanh(119896119909 +31198872

+ 41198964

2119896119905)

(30)

In Figures 2(a)ndash2(d) we have presented approximate 119906(119909 119905)and V(119909 119905) at 120572 = 1 and exact solutions

53 System of Coupled Fractional Sine-Gordon Equations [2627] Coupled Sine-Gordon equations were introduced byRay et al [26 27] The coupled Sine-Gordon equations gen-eralize the Frenkel-Kontorova dislocation model see [26 27]and the references therein

We now consider a system of coupled Sine-Gordon equa-tions [26 27]

1205972120572

119906

1205971199052120572minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972120572V1205971199052120572

minus 11988821205972V

1205971199092= sin (119906 minus V)

0 lt 119909 119905 lt 120587 0 lt 120572 le 1

(31)

with the initial conditions

119906 (119909 0) = 119860 cos (119896119909)

119906119905(119909 0) = 0

V (119909 0) = 0

V119905(119909 0) = 0

(32)

where 119888 is the ratio of the acoustic velocities of the compo-nents 119906 and V

Applying the transformation [25] to (31) we get the fol-lowing partial differential equations

1205972

119906

1205971198792minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972V

1205971198792minus 11988821205972V

1205971199092= sin (119906 minus V)

(33)

Applying the differential transform to (33) and (32) we obtainthe following recursive formula

(119896 + 2)

119896119880119896+2

(119909) =1205972

119880119896(119909)

1205971199092minus 1198862

119873119896(119909)

(119896 + 2)

119896119881119896+2

(119909) = 11988821205972

119880119896(119909)

1205971199092+ 119873119896(119909)

(34)

Using the initial condition we have

1198800(119909) = 119860 cosh (119896119909)

1198801(119909) = 0

1198810(119909) = 0

1198811(119909) = 0

(35)

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

have been solved by some other reliablemethods see [28ndash33]It is to be highlighted that Ganji et al have solved wide rangeof mathematical models [28ndash33] by making an appropriateuse of some recently developed schemes and hence giving anew avenue of research

2 Jumariersquos Fractional Derivative

Some useful formulas and results of Jumariersquos fractionalderivative were summarized [24]

119863120572

119909119888 = 0 120572 ge 0 119888 = constant (1)

119863120572

119909[119888119891 (119909)] = 119888119863

120572

119909119891 (119909) 120572 ge 0 119888 = constant (2)

119863120572

119909119909120573

=Γ (1 + 120573)

Γ (1 + 120573 minus 120572)119909120573minus120572

120573 ge 120572 ge 0 (3)

119863120572

119909[119891 (119909) 119892 (119909)] = [119863

120572

119909119891 (119909) 119892 (119909) + 119891 (119909)119863

120572

119909119892 (119909)] (4)

119863120572

119909119891 (119909 (119905)) = 119891

1015840

119909(119909) 119909120572

(119905) (5)

3 Fractional ComplexTransform Method (FCTM)

The fractional complex transform was first proposed in [25]and is defined as

119879 =119901119905120572

Γ (120572 + 1)

119883 =119902119909120573

Γ (120573 + 1)

119884 =119896119910120574

Γ (1 + 120574)

119885 =119897119911120582

Γ (1 + 120582)

(6)

where 119901 119902 119896 and 119897 are unknown constants 0 lt 120572 le 1 0 lt

120573 le 1 0 lt 120574 le 1 and 0 lt 120582 le 1

4 Reduced DifferentialTransform Method (RDTM)

To illustrate the basic idea of the DTMThe differential trans-form of 119896th derivative of a function 119906(119909 119905) which is analyticand differentiated continuously in the domain of interest isdefined as

119880119896(119909) =

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

(7)

The differential inverse transform of 119880119896(119909) is defined as fol-

lows

119906 (119909 119905) =

infin

sum

119896=0

119880119896(119909) (119905 minus 119905

0)119896

(8)

Equation (8) is known as the Taylor series expansion of 119906(119909 119905)around 119905 = 119905

0 Combining (7) and (8)

119906 (119909 119905) =

infin

sum

119896=0

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

(119905 minus 1199050)119896

(9)

when 1199050= 0 the above equation reduces to

119906 (119909 119905) =

infin

sum

119896=0

1

119896[120597119896

119906 (119909 119905)

120597119905119896]

119905=1199050

119905119896

(10)

and (2) reduces to

119906 (119909 119905) =

infin

sum

119896=0

119880119896(119909) 119905119896

(11)

Some properties of the reduced differential transform methodare as follows

(1) If the original function is 119906(119909 119905) = 119908(119909 119905) + V(119909 119905)then the transformed function is

119880119896(119909) = 119882

119896(119909) + 119881

119896(119909) (12)

(2) If 119906(119909 119905) = 120572119908(119909 119905) then 119880119896(119909) = 120572119882

119896(119909)

(3) If 119906(119909 119905) = 120597119898

119908(119909 119905)120597119905119898 then 119880

119896(119909) = ((119896 + 119898)

119896)119882119896(119909)

(4) If 119906(119909 119905) = 120597119908(119909 119905)120597119909 then 119880119896(119909) = (120597120597119909)119882

119896(119909)

(5) If 119906(119909 119910 119905) = 120597119908(119909 119910 119905)120597119909 then 119880119896(119909 119910) = (120597

120597119909)119882119896(119909 119910)

(6) If 119906(119909 119910 119911 119905) = 120597119908(119909 119910 119911 119905)120597119909 then 119880119896(119909 119910 119911) =

(120597120597119909)119882119896(119909 119910 119911)

(7) If 119906(119909 119905) = 119909119898

119905119899

119908(119909 119905) then 119880119896(119909) = 119909

119898

119882119896minus119899

(119909)(8) If 119906(119909 119905) = 119908

2

(119909 119905) then119880119896(119909) = sum

119896

119903=0119882119903(119909)119882119896minus119903

(119909)

5 Numerical Applications

In this section we apply the proposed fractional complextransform method coupled with reduced differential trans-formmethod to solve three important mathematical modelsNumerical results are highly encouraging For details aboutsuch equations readers are referred to study [22 23]

51 Fractional Kaup-Kupershmidt (FKK) Equation It is anestablished fact that fractional Kaup-Kupershmidt (FKK)equation plays a major role in the study of nonlinear disper-sive waves Moreover it describes a large number of impor-tant physical phenomena such as shallow water waves andion acoustic plasma waves

Consider the nonlinear KK equation [22 23]

120597120572

119906

120597119905120572minus1205975

119906

1205971199095minus 5119906

1205973

119906

1205971199093minus25

3

120597119906

120597119909

1205972

119906

1205971199092minus 51199062120597119906

120597119909= 0 (13)

with the initial condition

119906 (119909 0) = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2 (14)

where 119896 is an arbitrary constant

Mathematical Problems in Engineering 3

Applying the transformation [25] we get the followingpartial differential equation

120597119906

120597119879minus1205975

119906

1205971199095minus 5119906

1205973

119906

1205971199093minus25

3

120597119906

120597119909

1205972

119906

1205971199092minus 51199062120597119906

120597119909= 0 (15)

Applying the differential transform to (15) and (14) we obtainthe following recursive formula

(119896 + 1)119880119896+1

(119909)

=1205975

119880119896(119909)

1205971199095+ 5

119896

sum

119903=0

119880119896minus119903

(119909)1205973

119880119903(119909)

1205971199093

+25

3

119896

sum

119903=0

119880119896minus119903

(119909)1205972

119880119903(119909)

1205971199092

+ 5

119896

sum

119903=0

119903

sum

119904=119896

119880119896minus119903

(119909)119880119903minus119904

(119909)120597119880119904(119909)

120597119909

(16)

Using the initial condition we have

1198800(119909) = minus2119896

2

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2 (17)

Now substituting (17) into (16) and by straightforwarditerative steps yields

1198801(119909) = minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

1198802(119909) = minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198803(119909) =

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

(18)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

119879

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198792

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

1198793

+ sdot sdot sdot

(19)

The inverse transformation will yield

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(119890119896119909

minus 1)

(1 + 119890119896119909)3

119905120572

Γ (120572 + 1)

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1199052120572

Γ2 (120572 + 1)

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

sdot1199053120572

Γ3 (120572 + 1)+ sdot sdot sdot

(20)

This solution is convergent to the exact solution

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909+111198965119905

minus241198962

(1 + 119890119896119909+111198965119905)2 (21)

In Figures 1(a) and 1(b) we have presented approximatesolution at 120572 = 1 and exact solutions

52 Generalized Fractional Drinfeld-Sokolov (GFDS) Equa-tions [20 21] This system was introduced independently byDrinfeld and Sokolov [20 21]This coupled system was givenas one of the numerous examples of nonlinear equationspossessing Lax pairs of a special form Also the coupledsystem was found as a special case of the four-reduction ofthe KP hierarchy see [20 21] and the references therein

We consider the system of generalized fractionalDrinfeld-Sokolov (GFDS) equations [20 21]

120597120573

119906

120597119905120573+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597120573V120597119905120573

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

0 lt 119909 119905 lt 120587 0 lt 120573 le 1

(22)

with the initial conditions

119906 (119909 0) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

V (119909 0) = 119887 tanh (119896119909) (23)

where 120572 is a constantApplying the transformation [25] we get the following

partial differential equations

120597119906

120597119879+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597V120597119879

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

(24)

4 Mathematical Problems in Engineering

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(a)

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(b)

Figure 1 (a) Approximate solution (b) Exact solution

Applying the differential transform to (24) and (23) weobtain the following recursive formula

(119896 + 1)119880119896+1

(119909) = minus1205973

119880119896(119909)

1205971199093+ 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119880119903(119909)

120597119909

+ 6120597119881120572

119896(119909)

120597119909

(119896 + 1)119881119896+1

(119909) = 21205973

119881119896(119909)

1205971199093minus 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119881119903(119909)

120597119909

(25)

Using the initial condition we have

1198800(119909) =

minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

1198810(119909) = 119887 tanh (119896119909)

(26)

Now substituting (26) into (25) when (120572 = 2) and bystraightforward iterative steps yields

1198801(119909) =

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

1198811(119909) =

1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896

1198802(119909) = minus

1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)4

1198812(119909) = minus

1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198803(119909)

=1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

1198813(119909) =

1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)4

(27)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119879

minus1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41198792

+1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)51198793

+ sdot sdot sdot

Mathematical Problems in Engineering 5

V (119909 119879)

= 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119879

minus1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198792

+1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41198793

+ sdot sdot sdot

(28)

The inverse transformation will yields

119906 = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119905120573

Γ (120573 + 1)minus1

2

sdot

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41199052120573

Γ2 (120573 + 1)+1

3

sdot

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

sdot1199053120573

Γ3 (120573 + 1)+ sdot sdot sdot

V = 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119905120573

Γ (120573 + 1)minus1

4

sdot

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1199052120573

Γ2 (120573 + 1)+

1

24

sdot

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41199053120573

Γ3 (120573 + 1)

+ sdot sdot sdot

(29)

This solution is convergent to the exact solution

119906 (119909 119905) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909 +

31198872

+ 41198964

2119896119905)

V (119909 119905) = 119887 tanh(119896119909 +31198872

+ 41198964

2119896119905)

(30)

In Figures 2(a)ndash2(d) we have presented approximate 119906(119909 119905)and V(119909 119905) at 120572 = 1 and exact solutions

53 System of Coupled Fractional Sine-Gordon Equations [2627] Coupled Sine-Gordon equations were introduced byRay et al [26 27] The coupled Sine-Gordon equations gen-eralize the Frenkel-Kontorova dislocation model see [26 27]and the references therein

We now consider a system of coupled Sine-Gordon equa-tions [26 27]

1205972120572

119906

1205971199052120572minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972120572V1205971199052120572

minus 11988821205972V

1205971199092= sin (119906 minus V)

0 lt 119909 119905 lt 120587 0 lt 120572 le 1

(31)

with the initial conditions

119906 (119909 0) = 119860 cos (119896119909)

119906119905(119909 0) = 0

V (119909 0) = 0

V119905(119909 0) = 0

(32)

where 119888 is the ratio of the acoustic velocities of the compo-nents 119906 and V

Applying the transformation [25] to (31) we get the fol-lowing partial differential equations

1205972

119906

1205971198792minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972V

1205971198792minus 11988821205972V

1205971199092= sin (119906 minus V)

(33)

Applying the differential transform to (33) and (32) we obtainthe following recursive formula

(119896 + 2)

119896119880119896+2

(119909) =1205972

119880119896(119909)

1205971199092minus 1198862

119873119896(119909)

(119896 + 2)

119896119881119896+2

(119909) = 11988821205972

119880119896(119909)

1205971199092+ 119873119896(119909)

(34)

Using the initial condition we have

1198800(119909) = 119860 cosh (119896119909)

1198801(119909) = 0

1198810(119909) = 0

1198811(119909) = 0

(35)

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Applying the transformation [25] we get the followingpartial differential equation

120597119906

120597119879minus1205975

119906

1205971199095minus 5119906

1205973

119906

1205971199093minus25

3

120597119906

120597119909

1205972

119906

1205971199092minus 51199062120597119906

120597119909= 0 (15)

Applying the differential transform to (15) and (14) we obtainthe following recursive formula

(119896 + 1)119880119896+1

(119909)

=1205975

119880119896(119909)

1205971199095+ 5

119896

sum

119903=0

119880119896minus119903

(119909)1205973

119880119903(119909)

1205971199093

+25

3

119896

sum

119903=0

119880119896minus119903

(119909)1205972

119880119903(119909)

1205971199092

+ 5

119896

sum

119903=0

119903

sum

119904=119896

119880119896minus119903

(119909)119880119903minus119904

(119909)120597119880119904(119909)

120597119909

(16)

Using the initial condition we have

1198800(119909) = minus2119896

2

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2 (17)

Now substituting (17) into (16) and by straightforwarditerative steps yields

1198801(119909) = minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

1198802(119909) = minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198803(119909) =

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

(18)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(minus1 + 119890119896119909

)

(1 + 119890119896119909)3

119879

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1198792

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

1198793

+ sdot sdot sdot

(19)

The inverse transformation will yield

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

minus

2641198967

119890119896119909

(119890119896119909

minus 1)

(1 + 119890119896119909)3

119905120572

Γ (120572 + 1)

minus

145211989612

119890119896119909

(4119890119896119909

minus 1198902119896119909

minus 1)

(1 + 119890119896119909)4

1199052120572

Γ2 (120572 + 1)

+

352411989617

119890119896119909

(minus11119890119896119909

+ 111198902119896119909

minus 1198903119896119909

+ 1)

(1 + 119890119896119909)5

sdot1199053120572

Γ3 (120572 + 1)+ sdot sdot sdot

(20)

This solution is convergent to the exact solution

119906 (119909 119905) = minus21198962

+241198962

1 + 119890119896119909+111198965119905

minus241198962

(1 + 119890119896119909+111198965119905)2 (21)

In Figures 1(a) and 1(b) we have presented approximatesolution at 120572 = 1 and exact solutions

52 Generalized Fractional Drinfeld-Sokolov (GFDS) Equa-tions [20 21] This system was introduced independently byDrinfeld and Sokolov [20 21]This coupled system was givenas one of the numerous examples of nonlinear equationspossessing Lax pairs of a special form Also the coupledsystem was found as a special case of the four-reduction ofthe KP hierarchy see [20 21] and the references therein

We consider the system of generalized fractionalDrinfeld-Sokolov (GFDS) equations [20 21]

120597120573

119906

120597119905120573+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597120573V120597119905120573

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

0 lt 119909 119905 lt 120587 0 lt 120573 le 1

(22)

with the initial conditions

119906 (119909 0) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

V (119909 0) = 119887 tanh (119896119909) (23)

where 120572 is a constantApplying the transformation [25] we get the following

partial differential equations

120597119906

120597119879+1205973

119906

1205971199093minus 6119906

120597119906

120597119909minus 6

120597V120572

120597119909= 0

120597V120597119879

minus 21205973V

1205971199093+ 6119906

120597V120572

120597119909= 0

(24)

4 Mathematical Problems in Engineering

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(a)

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(b)

Figure 1 (a) Approximate solution (b) Exact solution

Applying the differential transform to (24) and (23) weobtain the following recursive formula

(119896 + 1)119880119896+1

(119909) = minus1205973

119880119896(119909)

1205971199093+ 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119880119903(119909)

120597119909

+ 6120597119881120572

119896(119909)

120597119909

(119896 + 1)119881119896+1

(119909) = 21205973

119881119896(119909)

1205971199093minus 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119881119903(119909)

120597119909

(25)

Using the initial condition we have

1198800(119909) =

minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

1198810(119909) = 119887 tanh (119896119909)

(26)

Now substituting (26) into (25) when (120572 = 2) and bystraightforward iterative steps yields

1198801(119909) =

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

1198811(119909) =

1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896

1198802(119909) = minus

1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)4

1198812(119909) = minus

1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198803(119909)

=1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

1198813(119909) =

1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)4

(27)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119879

minus1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41198792

+1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)51198793

+ sdot sdot sdot

Mathematical Problems in Engineering 5

V (119909 119879)

= 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119879

minus1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198792

+1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41198793

+ sdot sdot sdot

(28)

The inverse transformation will yields

119906 = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119905120573

Γ (120573 + 1)minus1

2

sdot

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41199052120573

Γ2 (120573 + 1)+1

3

sdot

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

sdot1199053120573

Γ3 (120573 + 1)+ sdot sdot sdot

V = 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119905120573

Γ (120573 + 1)minus1

4

sdot

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1199052120573

Γ2 (120573 + 1)+

1

24

sdot

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41199053120573

Γ3 (120573 + 1)

+ sdot sdot sdot

(29)

This solution is convergent to the exact solution

119906 (119909 119905) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909 +

31198872

+ 41198964

2119896119905)

V (119909 119905) = 119887 tanh(119896119909 +31198872

+ 41198964

2119896119905)

(30)

In Figures 2(a)ndash2(d) we have presented approximate 119906(119909 119905)and V(119909 119905) at 120572 = 1 and exact solutions

53 System of Coupled Fractional Sine-Gordon Equations [2627] Coupled Sine-Gordon equations were introduced byRay et al [26 27] The coupled Sine-Gordon equations gen-eralize the Frenkel-Kontorova dislocation model see [26 27]and the references therein

We now consider a system of coupled Sine-Gordon equa-tions [26 27]

1205972120572

119906

1205971199052120572minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972120572V1205971199052120572

minus 11988821205972V

1205971199092= sin (119906 minus V)

0 lt 119909 119905 lt 120587 0 lt 120572 le 1

(31)

with the initial conditions

119906 (119909 0) = 119860 cos (119896119909)

119906119905(119909 0) = 0

V (119909 0) = 0

V119905(119909 0) = 0

(32)

where 119888 is the ratio of the acoustic velocities of the compo-nents 119906 and V

Applying the transformation [25] to (31) we get the fol-lowing partial differential equations

1205972

119906

1205971198792minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972V

1205971198792minus 11988821205972V

1205971199092= sin (119906 minus V)

(33)

Applying the differential transform to (33) and (32) we obtainthe following recursive formula

(119896 + 2)

119896119880119896+2

(119909) =1205972

119880119896(119909)

1205971199092minus 1198862

119873119896(119909)

(119896 + 2)

119896119881119896+2

(119909) = 11988821205972

119880119896(119909)

1205971199092+ 119873119896(119909)

(34)

Using the initial condition we have

1198800(119909) = 119860 cosh (119896119909)

1198801(119909) = 0

1198810(119909) = 0

1198811(119909) = 0

(35)

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(a)

024

022

020

018

016

014

minus4

minus2

0

2

4

minus4

minus2

0

2

4

t x

(b)

Figure 1 (a) Approximate solution (b) Exact solution

Applying the differential transform to (24) and (23) weobtain the following recursive formula

(119896 + 1)119880119896+1

(119909) = minus1205973

119880119896(119909)

1205971199093+ 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119880119903(119909)

120597119909

+ 6120597119881120572

119896(119909)

120597119909

(119896 + 1)119881119896+1

(119909) = 21205973

119881119896(119909)

1205971199093minus 6

119896

sum

119903=0

119880119896minus119903

(119909)120597119881119903(119909)

120597119909

(25)

Using the initial condition we have

1198800(119909) =

minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909)

1198810(119909) = 119887 tanh (119896119909)

(26)

Now substituting (26) into (25) when (120572 = 2) and bystraightforward iterative steps yields

1198801(119909) =

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

1198811(119909) =

1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896

1198802(119909) = minus

1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)4

1198812(119909) = minus

1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198803(119909)

=1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

1198813(119909) =

1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)4

(27)

and so onThe series solution is given by

119906 (119909 119879)

= minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119879

minus1

2

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41198792

+1

3

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)51198793

+ sdot sdot sdot

Mathematical Problems in Engineering 5

V (119909 119879)

= 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119879

minus1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198792

+1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41198793

+ sdot sdot sdot

(28)

The inverse transformation will yields

119906 = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119905120573

Γ (120573 + 1)minus1

2

sdot

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41199052120573

Γ2 (120573 + 1)+1

3

sdot

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

sdot1199053120573

Γ3 (120573 + 1)+ sdot sdot sdot

V = 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119905120573

Γ (120573 + 1)minus1

4

sdot

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1199052120573

Γ2 (120573 + 1)+

1

24

sdot

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41199053120573

Γ3 (120573 + 1)

+ sdot sdot sdot

(29)

This solution is convergent to the exact solution

119906 (119909 119905) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909 +

31198872

+ 41198964

2119896119905)

V (119909 119905) = 119887 tanh(119896119909 +31198872

+ 41198964

2119896119905)

(30)

In Figures 2(a)ndash2(d) we have presented approximate 119906(119909 119905)and V(119909 119905) at 120572 = 1 and exact solutions

53 System of Coupled Fractional Sine-Gordon Equations [2627] Coupled Sine-Gordon equations were introduced byRay et al [26 27] The coupled Sine-Gordon equations gen-eralize the Frenkel-Kontorova dislocation model see [26 27]and the references therein

We now consider a system of coupled Sine-Gordon equa-tions [26 27]

1205972120572

119906

1205971199052120572minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972120572V1205971199052120572

minus 11988821205972V

1205971199092= sin (119906 minus V)

0 lt 119909 119905 lt 120587 0 lt 120572 le 1

(31)

with the initial conditions

119906 (119909 0) = 119860 cos (119896119909)

119906119905(119909 0) = 0

V (119909 0) = 0

V119905(119909 0) = 0

(32)

where 119888 is the ratio of the acoustic velocities of the compo-nents 119906 and V

Applying the transformation [25] to (31) we get the fol-lowing partial differential equations

1205972

119906

1205971198792minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972V

1205971198792minus 11988821205972V

1205971199092= sin (119906 minus V)

(33)

Applying the differential transform to (33) and (32) we obtainthe following recursive formula

(119896 + 2)

119896119880119896+2

(119909) =1205972

119880119896(119909)

1205971199092minus 1198862

119873119896(119909)

(119896 + 2)

119896119881119896+2

(119909) = 11988821205972

119880119896(119909)

1205971199092+ 119873119896(119909)

(34)

Using the initial condition we have

1198800(119909) = 119860 cosh (119896119909)

1198801(119909) = 0

1198810(119909) = 0

1198811(119909) = 0

(35)

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

V (119909 119879)

= 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119879

minus1

4

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1198792

+1

24

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41198793

+ sdot sdot sdot

(28)

The inverse transformation will yields

119906 = minus21198962

+241198962

1 + 119890119896119909minus

241198962

(1 + 119890119896119909)2

+

2119896 (41198962

+ 31198872

) sinh (119896119909)cosh (119896119909)3

119905120573

Γ (120573 + 1)minus1

2

sdot

(2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)2

cosh (119896119909)41199052120573

Γ2 (120573 + 1)+1

3

sdot

sinh (119896119909) (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

119896 cosh (119896119909)5

sdot1199053120573

Γ3 (120573 + 1)+ sdot sdot sdot

V = 119887 tanh (119896119909) + 1

2

119887 (41198962

+ 31198872

)

cosh (119896119909)2 119896119905120573

Γ (120573 + 1)minus1

4

sdot

119887 (41198962

+ 31198872

)2

sinh (119896119909)cosh (119896119909)3 1198962

1199052120573

Γ2 (120573 + 1)+

1

24

sdot

119887 (2 cosh (119896119909)2 minus 3) (41198962

+ 31198872

)3

1198963 cosh (119896119909)41199053120573

Γ3 (120573 + 1)

+ sdot sdot sdot

(29)

This solution is convergent to the exact solution

119906 (119909 119905) =minus1198872

minus 41198964

41198962+ 21198962tanh2 (119896119909 +

31198872

+ 41198964

2119896119905)

V (119909 119905) = 119887 tanh(119896119909 +31198872

+ 41198964

2119896119905)

(30)

In Figures 2(a)ndash2(d) we have presented approximate 119906(119909 119905)and V(119909 119905) at 120572 = 1 and exact solutions

53 System of Coupled Fractional Sine-Gordon Equations [2627] Coupled Sine-Gordon equations were introduced byRay et al [26 27] The coupled Sine-Gordon equations gen-eralize the Frenkel-Kontorova dislocation model see [26 27]and the references therein

We now consider a system of coupled Sine-Gordon equa-tions [26 27]

1205972120572

119906

1205971199052120572minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972120572V1205971199052120572

minus 11988821205972V

1205971199092= sin (119906 minus V)

0 lt 119909 119905 lt 120587 0 lt 120572 le 1

(31)

with the initial conditions

119906 (119909 0) = 119860 cos (119896119909)

119906119905(119909 0) = 0

V (119909 0) = 0

V119905(119909 0) = 0

(32)

where 119888 is the ratio of the acoustic velocities of the compo-nents 119906 and V

Applying the transformation [25] to (31) we get the fol-lowing partial differential equations

1205972

119906

1205971198792minus1205972

119906

1205971199092= minus1198862 sin (119906 minus V)

1205972V

1205971198792minus 11988821205972V

1205971199092= sin (119906 minus V)

(33)

Applying the differential transform to (33) and (32) we obtainthe following recursive formula

(119896 + 2)

119896119880119896+2

(119909) =1205972

119880119896(119909)

1205971199092minus 1198862

119873119896(119909)

(119896 + 2)

119896119881119896+2

(119909) = 11988821205972

119880119896(119909)

1205971199092+ 119873119896(119909)

(34)

Using the initial condition we have

1198800(119909) = 119860 cosh (119896119909)

1198801(119909) = 0

1198810(119909) = 0

1198811(119909) = 0

(35)

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

minus4

minus2

0

2

4

minus4

minus2

0

2

4

tx

4

minus2

minus

minus2

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

10minus6

(a)

minus4minus4

minus2minus2

00

22

4 4

t x

4minus

2

9997

09998

09999

10000

(b)

x

minus4

minus20

24

minus4minus2

02

4t

00002

00001

0

00001

00002

(c)

x

minus4minus4

minus2minus2

0 0

2 2

4 4t

00010

00005

0

00005

00010

(d)

Figure 2 (a) Approximate solution (b) Exact solution (c) Approximate solution (d) Exact solution

Now substituting (35) into (34) and by straightforwarditerative steps yields

1198802(119909) = minus

1198601198962 cosh (119896119909)

2minus1198862 sin (119860 cosh (119896119909))

2

1198812(119909) =

sin (119860 cosh (119896119909))2

1198803(119909) = 0

1198813(119909) = 0

1198804(119909) =

1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

1198814(119909) =

1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24

(36)

and so onThe series solution is given by

119906 (119909 119879) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)1198792

+ (1198601198964 cosh (119896119909)

24

+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)1198794

+ sdot sdot sdot

V (119909 119879) =sin (119860 cosh (119896119909))

21198792

+ (1198882

1198602

1198962 sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus1198882

1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus1198862 cos (119860 cosh (119896119909)) sin cosh (119896119909)

24

minuscos (119860 cosh (119896119909)) sin cosh (119896119909)

24)1198793

+ sdot sdot sdot

(37)

The inverse transformation will yield

119906 (119909 119905) = 119860 cosh (119896119909) minus (1198601198962 cosh (119896119909)

2

+1198862 sin (119860 cosh (119896119909))

2)

1199052120572

Γ2 (120572 + 1)

+ (1198601198964 cosh (119896119909)

24+1198862

1198602

1198962 sin (119860 cosh (119896119909))

24

minus1198862

1198602

1198962 sin (119860 cosh (119896119909)) cos2 (119896119909)

24

+1198862

1198601198962 cos (119860 cosh (119896119909)) cos (119896119909)

12

+1198864 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24

+1198862 cos (119860 cosh (119896119909)) sin (119860 cosh (119896119909))

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

V (119909 119905) =sin (119860 cosh (119896119909))

2

1199052120572

Γ2 (120572 + 1)

+ (1198882

1198602

1198962sin (119860 cosh (119896119909))

24

+1198882

1198602

1198962 sin (119860 cosh (119896119909)) cosh2 (119896119909)

24

minus (1 + 1198882

)1198601198962 cos (119860 cosh (119896119909)) cosh (119896119909)

24

minus (1 + 1198862

)cos (119860 cosh (119896119909)) sin cosh (119896119909)

24)

sdot1199054120572

Γ4 (120572 + 1)+ sdot sdot sdot

(38)

This solution is convergent to Adomianrsquos decompositionmethod solution [26 27]

In Figures 3(a) and 3(b) we have presented approximatesolutions 119906(119909 119905) and V(119909 119905) at 120572 = 1

6 Conclusion

In this research we present new applications of the fractionalcomplex transform method with coupling reduced differen-tial transform method (RDTM) by handling three nonlinearphysical fractional dynamical models This coupling is analternative approach to overcome the demerit of complexcalculation of fractional differential equations The proposed

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

x

minus2

0

2

4

minus4minus4

minus2

0

2

4t

minus2

0

minus4minus4

minus2

0

1000

800

600

200

400

0

(a)

xminus2

0

2

4

minus4

minus4

minus2

0

2

4

t2

0

2

minus4

minus4

minus2

0

2

t

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

0

(b)

Figure 3 (a) Approximate solution 119906(119909 119905) (b) Approximate solution V(119909 119905)

technique which does not require linearization discretiza-tion or perturbation gives the solution in the form of con-vergent power series with elegantly computed componentsAll the examples show that the proposed combination isa powerful mathematical tool for solving nonlinear equa-tions and hence may be extended to other nonlinear problemsalso

Conflict of Interests

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

Acknowledgment

The project was funded by the National Plan for ScienceTechnology and Innovation (MAARIFA) King Abdul AzizCity for Science amp Technology Kingdom of Saudi ArabiaAward no 15-MAT4688-02

References

[1] G Adomian ldquoA new approach to nonlinear partial differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 102 no 2 pp 420ndash434 1984

[2] S Abbasbandy ldquoNumerical solution of non-linear KleinndashGor-don equations by variational iteration methodrdquo InternationalJournal for Numerical Methods in Engineering vol 70 no 7 pp876ndash881 2007

[3] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solving thomas-fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009

[4] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[5] J-L Zhang M-L Wang Y-M Wang and Z-D Fang ldquoTheimproved F-expansion method and its applicationsrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol350 no 1-2 pp 103ndash109 2006

[6] S T Mohyud-Din M A Noor K Noor and M M HosseinildquoVariational iteration method for re-formulated partial differ-ential equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 87ndash92 2010

[7] A-M Wazwaz ldquoA sinendashcosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[8] J K Zhou Differential Transformation and Its Application forElectrical Circuits Huazhong University Press Wuhan China1986

[9] J Ahmad and S T Mohyud-Din ldquoAn efficient algorithmfor some highly nonlinear fractional PDEs in mathematicalphysicsrdquo PLoS ONE vol 9 no 12 Article ID e109127 2014

[10] A Arikoglu and I Ozkol ldquoSolution of fractional differentialequations by using differential transform methodrdquo ChaosSolitons amp Fractals vol 34 no 5 pp 1473ndash1481 2007

[11] A Kurnaz and G Oturanc ldquoThe differential transform approx-imation for the system of ordinary differential equationsrdquoInternational Journal of Computer Mathematics vol 82 no 6pp 709ndash719 2005

[12] A Saravanan and N Magesh ldquoA comparison between thereduced differential transform method and the Adomiandecomposition method for the Newell-Whitehead-Segel equa-tionrdquo Journal of the Egyptian Mathematical Society vol 21 no3 pp 259ndash265 2013

[13] R Abazari and M Abazari ldquoNumerical study of Burgers-Huxley equations via reduced differential transform methodrdquo

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Computational amp Applied Mathematics vol 32 no 1 pp 1ndash172013

[14] B Bis and M Bayram ldquoApproximate solutions for some non-linear evolutions equations by using the reduced differentialtransformmethodrdquo International Journal of Applied Mathemat-ical Research vol 1 no 3 pp 288ndash302 2012

[15] R Abazari and B Soltanalizadeh ldquoReduced differential trans-form method and its application on Kawahara equationsrdquoThaiJournal of Mathematics vol 11 no 1 pp 199ndash216 2013

[16] M A Abdou and A A Soliman ldquoNumerical simulations ofnonlinear evolution equations in mathematical physicsrdquo Inter-national Journal of Nonlinear Science vol 12 no 2 pp 131ndash1392011

[17] M A Abdou ldquoApproximate solutions of system of PDEEs aris-ing in physicsrdquo International Journal of Nonlinear Science vol12 no 3 pp 305ndash312 2011

[18] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers amp Mathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[19] R Abazari and M Abazari ldquoNumerical simulation of gener-alized HirotandashSatsuma coupled KdV equation by RDTM andcomparison with DTMrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 2 pp 619ndash629 2012

[20] Y Ugurlu and D Kaya ldquoExact and numerical solutions ofgeneralized Drinfeld-Sokolov equationsrdquo Physics Letters A vol372 no 16 pp 2867ndash2873 2008

[21] G A Afrouzi J Vahidi and M Saeidy ldquoNumerical solutionsof generalized Drinfeld-Sokolov equations using the homotopyanalysismethodrdquo International Journal ofNonlinear Science vol9 no 2 pp 165ndash170 2010

[22] A Mohebbi ldquoNumerical solution of nonlinear Kaup-Kuper-shmit equation KdV-KdV and hirota-satsuma systemsrdquo Inter-national Journal of Nonlinear Sciences and Numerical Simula-tion vol 13 no 7-8 pp 479ndash486 2012

[23] K A Gepreel S Omran and S K Elagan ldquoThe traveling wavesolutions for some nonlinear PDEs in mathematical physicsrdquoApplied Mathematics vol 2 no 3 pp 343ndash347 2011

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

[25] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[26] S S Ray ldquoA numerical solution of the coupled sine-Gordonequation using the modified decomposition methodrdquo AppliedMathematics and Computation vol 175 no 2 pp 1046ndash10542006

[27] A Sadighi D D Ganji and B Ganjavi ldquoTraveling wave solu-tions of the sine-gordon and the coupled sine-gordon equationsusing the homotopy-perturbation methodrdquo Scientia IranicaTransaction B Mechanical Engineering vol 16 no 2 pp 189ndash195 2009

[28] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[29] Z Z Ganji D D Ganji and Y Rostamiyan ldquoSolitary wave solu-tions for a time-fraction generalized HirotandashSatsuma coupled

KdV equation by an analytical techniquerdquo Applied Mathemati-cal Modelling vol 33 no 7 pp 3107ndash3113 2009

[30] S E Ghasemi A Zolfagharian and D D Ganji ldquoStudy onmotion of rigid rod on a circular surface using MHPMrdquoPropulsion and Power Research vol 3 no 3 pp 159ndash164 2014

[31] M Hatami and D D Ganji ldquoThermal and flow analysis ofmicrochannel heat sink (MCHS) cooled by Cu-water nanofluidusing porous media approach and least square methodrdquo EnergyConversion and Management vol 78 pp 347ndash358 2014

[32] D D Ganji A Sadighi and I Khatami ldquoAssessment of twoanalytical approaches in some nonlinear problems arising inengineering sciencesrdquo Physics Letters A vol 372 no 24 pp4399ndash4406 2008

[33] M Rafei D D Ganji H Daniali and H Pashaei ldquoThe varia-tional iteration method for nonlinear oscillators with disconti-nuitiesrdquo Journal of Sound and Vibration vol 305 no 4-5 pp614ndash620 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of