Research Article Differential Transform Method with...
Transcript of Research Article Differential Transform Method with...
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
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OptimizationJournal of
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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
International Journal of Mathematics and Mathematical Sciences
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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
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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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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
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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
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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