Approximate analytical solution for the fractional modified KdV by differential transform method

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Approximate analytical solution for the fractional modified KdV by differential transform method
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  Approximate analytical solution for the fractional modified KdVby differential transform method Muhammet Kurulay a, * , Mustafa Bayram b a Yildiz Technical University, Faculty of Art and Sciences, Department of Mathematics, 34210-Davutpasa- _ Istanbul, Turkey b Fatih Universty, Faculty of Arts and Science, Department of Mathematics, 34500 Büyükçekmece, Istanbul, Turkey a r t i c l e i n f o  Article history: Received 4 March 2009Received in revised form 6 April 2009Accepted 20 July 2009Available online 24 July 2009 Keywords: Fractional differential equationCaputo fractional derivativeDifferential transform methodfmKdVfKdV a b s t r a c t In this paper, the fractional modified Korteweg-de Vries equation (fmKdV) and fKdV areintroduced by fractional derivatives. The approach rest mainly on two-dimensional differ-entialtransformmethod(DTM)whichisoneoftheapproximatemethods.Themethodcaneasily be applied to many problems and is capable of reducing the size of computationalwork.ThefractionalderivativeisdescribedintheCaputosense.Someillustrativeexamplesare presented.Crown Copyright   2009 Published by Elsevier B.V. All rights reserved. 1. Introduction Fractional differential equations are studied in various fields of physics and engineering. The numerical and analyticalapproximations of such problems have been intensively studied since the work of Padovan[1]. Recently, several mathemat-ical methods including the Adomian decomposition method [4,5] variational iteration method [6,7] homotopy analysis method [18,20,21] and fractional method [2] have been developed to obtain exact and approximate analytic solutions. Among these solution techniques, the variational iteration method and the Adomian decomposition method are the mostclear methods of solution of fractional differential and integral equations, because they provide immediate and visible sym-bolictermsof analyticsolutions, as well as numericalapproximatesolutionstononlineardifferentialequationswithoutlin-earization or discretization.In this paper, we consider the generalized KdV equation of the form @  a u ð  x ; t  Þ @  t  a  þ e ð u ð  x ; t  ÞÞ m  @  b u ð  x ; t  Þ @   x b  þ v  @  3 u ð  x ; t  Þ @   x 3  ¼  g  ð  x : t  Þ :  ð 1 : 1 Þ For  t   > 0 ; 0 < a ; b 6 1, where  e ; v   are constants,  m  ¼  0 ; 1 ; 2 and  a  and  b  are parameters describing the order of the fractionaltime and space-derivatives. If   m  ¼  0 ; m  ¼  1 and  m  ¼  2, Eq. (1.1) becomes the linear fractional KdV, nonlinear fractional KdVandfractional modifiedKdV(fmKdV), respectively. Thefunction u ð  x ; t  Þ  is assumedtobe a causal functionof timeandspace.The fractional derivatives are considered in Caputo sense. In case of   a  ¼  b  ¼  1, Eq. (1) reduces to the classical mKdV.  _ In thispaper the application of DTM will be extended to obtain approximate solutions of fmKdV and fKdV  ð m  ¼  1 and  m  ¼  2 Þ . 1007-5704/$ - see front matter Crown Copyright   2009 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.cnsns.2009.07.014 *  Corresponding author. E-mail address:  muhammetkurulay@yahoo.com (M. Kurulay).Commun Nonlinear Sci Numer Simulat 15 (2010) 1777–1782 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns  In the last decades, fractional calculus has found diverse applications in various scientific and technological fields [2,3],such as thermal engineering, acoustics, fluid mechanics, biology, chemistry, electromagnetism, control, robotics, diffusion,edge detection, turbulence, signal processing and many other physical processes.Thedifferentialtransformmethodwasfirstappliedintheengineeringdomainin[8].Thedifferentialtransformmethodisa numerical method based on the Taylor series expansion which constructs an analytical solution in the form of a polyno-mial. The traditional high order Taylor series method requires symbolic computation. However, the differential transformmethod obtains a polynomial series solution by means of an iterative procedure. Recently, the application of differentialtransform method is successfully extended to obtain analytical approximate solutions to ordinary differential equationsof fractional order [9]. Application of fractional calculus in physics was presented in [3]. A comparison between the varia- tional iteration method and Adomian decomposition method for solving fractional differential equations is given in [11].Very recently, Hashim[12] demonstrated the application of homotopy-perturbation method for solving fmKdV. 2. Fractional calculus There are several definitions of a fractional derivative of order  a > 0 [2,22].e.g.Riemann–Liouville, Grunwald–Letnikow,Caputo and Generalized Functions Approach The most commonly used definitions are the Riemann–Liouville and Caputo.We give some basic definitions and properties of the fractional calculus theory which are used further in this paper. Definition 2.1.  A real function  f  ð  x Þ ;  x > 0, is said to be in the space  C  l ; l  2  R  if there exists a real number  p ð > l Þ , such that  f  ð  x Þ ¼  x  p  f  1 ð  x Þ , where  f  1 ð  x Þ 2  C  ½ 0 ; 1Þ , and it said to be in the space  C  m l  iff   f  m 2  C  l ; m  2  N  . Definition 2.2.  TheRiemann–Liouvillefractionalintegral operatoroforder  a P 0, of afunction  f   2  C  l ; l P  1, isdefinedas  J  v  0  f  ð  x Þ¼  1 C ð v  Þ Z   x 0 ð  x  t  Þ v   1  f  ð t  Þ dt  ;  v   >  0 ;  J  0  f  ð  x Þ¼  f  ð  x Þ : It has the following properties:For  f   2  C  l ;  l P  1 ;  a ;  b P 0 and  c > 1:1.  J  a  J  b  f  ð  x Þ ¼  J  a þ b  f  ð  x Þ ,2.  J  a  J  b  f  ð  x Þ ¼  J  b  J  a  f  ð  x Þ ,3.  J  a  x c ¼  C ð c þ 1 Þ C ð a þ c þ 1 Þ  x a þ c .The Riemann–Liouville fractional derivative is mostly used by mathematicians but this approach is not suitable for thephysicalproblemsoftherealworldsinceitrequiresthedefinitionoffractionalorderinitialconditions,whichhavenophys-ically meaningful explanation yet. Caputo introduced an alternative definition, which has the advantage of defining integerorder initial conditions for fractional order differential equations. Definition 2.3.  The fractional derivative of   f  ð  x Þ  in the caputo sense is defined as D v    f  ð  x Þ¼  J  m  v  a  D m  f  ð  x Þ¼  1 C ð m  v  Þ Z   x 0 ð  x  t  Þ m  v   1  f  ð m Þ ð t  Þ dt  ;  for  m  1  <  v   <  m ;  m 2 N  ;  x  >  0 ;  f   2 C  m  1 : Lemma 2.1.  If m  1 < a < m ; m  2  N and f   2  C  m l ;  l P  1 , then D a   J  a  f  ð  x Þ¼  f  ð  x Þ ;  J  a D v    f  ð  x Þ¼  f  ð  x Þ X m  1 k ¼ 0  f  k ð 0 þ Þ  x k k !  ;  x  >  0 : TheCaputofractional derivative is consideredherebecause it allows traditional initial and boundaryconditions tobe in-cluded in the formulation of the problem. In this paper, we have considered fmKdV and fKdV, where the unknown function u  ¼  u ð  x ; t  Þ  is a assumed to be a causal function of fractional derivatives are taken in Caputo sense as follows: Definition 2.4.  Formtobethesmallestintegerthatexceeds a ,theCaputotime-fractionalderivativeoperatoroforder a > 0is defined as D a  t  u ð  x ; t  Þ¼ @  a u ð  x ; t  Þ @  t  a  ¼ 1 C ð m  a Þ R  t  0 ð t   n Þ m  a  1  @  m u ð  x ; n Þ @  n m  d n ;  for  m  1  <  a  <  m ; @  m u ð  x ; t  Þ @  t  m  ;  for  a ¼ m 2 N  : 8<: 1778  M. Kurulay, M. Bayram/Commun Nonlinear Sci Numer Simulat 15 (2010) 1777–1782  3. Generalized two-dimensional DTM TheDTMisappliedtothesolutionofelectriccircuitproblems.TheDTMisanumericalmethodbasedontheTaylorseriesexpansion which constructs an analytical solution in the form of a polynomial. The traditional high order Taylor seriesmethod requires symbolic computation. However, the DTM obtains a polynomial series solution by means of an iterativeprocedure. The method is well addressed in [10]. The proposed method is based on the combination of the classical two-dimensional DTM [13] and generalized Taylor’s Formula [14]. Consider a function of two variables  u ð  x ;  y Þ , and suppose that it can be represented as a product of two single-variablefunctions, i.e.,  u ð  x ;  y Þ ¼  f  ð  x Þ  g  ð  y Þ . Based on the properties of generalized two-dimensional differential transform [15,16],the function  u ð  x ;  y Þ  can be represented as u ð  x ;  y Þ¼ X 1 k ¼ 0 F  a ð k Þð  x   x 0 Þ k a X 1 h ¼ 0 G b ð h Þð  y   y 0 Þ h b ¼ X 1 k ¼ 0 X 1 h ¼ 0 U  a b ð k ; h Þð  x   x 0 Þ k a ð  y   y 0 Þ h b ;  ð 3 : 1 Þ where 0 < a ; b 6 1 ; U  a b ð k ; h Þ ¼  F  a ð k Þ G b ð h Þ  is called the spectrum of   u ð  x ;  y Þ . The generalized two-dimensional differentialtransform of the function  u ð  x ;  y Þ  is given by U  a ; b ð k ; h Þ¼  1 C ð a k þ 1 Þ C ð b h þ 1 Þ ð D a   x 0 Þ k ð D b   y 0 Þ h u ð  x ;  y Þ h i ð  x 0 ;  y 0 Þ ;  ð 3 : 2 Þ where  D a  x 0   k ¼  D a  x 0 D a  x 0   D a  x 0 ; k -times. In case of   a  ¼  1 and  b  ¼  1 the generalized two-dimensional differential transform(3.1) reduces to the classical two-dimensional differential transform. [17]. The operators in two-dimensional differential transformation Method [17]:Let  U  a ; b ð k ; h Þ ; V  a ; b ð k ; h Þ  and  W  a ; b ð k ; h Þ  be the differential transformations of the functions  u ð  x ;  y Þ ; v  ð  x ;  y Þ  and  w ð  x ;  y Þ :( a ) If   u ð  x ;  y Þ ¼  v  ð  x ;  y Þ w ð  x ;  y Þ , then  U  a ; b ð k ; h Þ ¼  V  a ; b ð k ; h Þ W  a ; b ð k ; h Þ .( b ) If   u ð  x ;  y Þ ¼  a v  ð  x ;  y Þ ; a  2  R , then  U  a ; b ð k ; h Þ ¼  aV  a ; b ð k ; h Þ .( c ) If   u ð  x ;  y Þ ¼  v  ð  x ;  y Þ w ð  x ;  y Þ , then  U  a ; b ð k ; h Þ ¼  P kr  ¼ 0 P hs ¼ 0 V  a ; b ð r  ; h  s Þ W  a ; b ð k  r  ; s Þ .( d ) If   u ð  x ;  y Þ ¼ ð  x   x 0 Þ n a ð  y   y 0 Þ m b , then  U  a ; b ð k ; h Þ ¼  d ð k  n Þ d ð h  m Þ .( e ) If   u ð  x ;  y Þ ¼  v  ð  x ;  y Þ w ð  x ;  y Þ q ð  x ;  y Þ , then  U  a ; b ð k ; h Þ ¼  P kr  ¼ 0 P k  r t  ¼ 0 P ht  ¼ 0 V  a ; b ð r  ; h  s   p Þ W  a ; b ð t  ; s Þ Q  a ; b ð k  r   t  ;  p Þ .( f  ) If   u ð  x ;  y Þ ¼  D a  x 0 v  ð  x ;  y Þ ; 0 < a 6 1, then  U  a ; b ð k ; h Þ ¼  C ð a ð k þ 1 Þþ 1 Þ C ð a k þ 1 Þ  V  a ; b ð k þ 1 ; h Þ .( g  ) If   u ð  x ;  y Þ ¼  f  ð  x Þ  g  ð  y Þ  and the function  f  ð  x Þ ¼  x k h ð  x Þ , where  k >  1 ; h ð  x Þ  has the generalized Taylor series expansion h ð  x Þ ¼  P 1 n ¼ 0 a n ð  x   x 0 Þ a k , and [17].( i )  b < k þ 1 and  a  arbitrary or( ii )  b P k þ 1 ; a  arbitrary and  a n  ¼  0 for  n  ¼  0 ; 1 ; . . . m  1 ;  where  m  1 < b 6 m .Then the generalized differential transform (3.2) becomes U  a ; b ð k ; h Þ¼  1 C ð a k þ 1 Þ C ð b h þ 1 Þ  D a k   x 0 ð D b   y 0 Þ h u ð  x ;  y Þ h i ð  x 0 ;  y 0 Þ ; ( h ) If   u ð  x ;  y Þ ¼  D c  x 0 v  ð  x ;  y Þ ; m  1 < c 6 m  and  v  ð  x ;  y Þ ¼  f  ð  x Þ  g  ð  y Þ , then U  a ; b ð k ; h Þ¼ C ð a k þ c þ 1 Þ C ð a k þ 1 Þ  V  a ; b ð k þ c = a ; h Þ : The proofs of the some properties can be found in [17]. 4. ExamplesExample 4.1.  We consider the following fKdV equation: D a t   u  6 uu  x þ u  xxx  ¼ 0 ;  0  <  a 6 1 ; t   >  0 ;  ð 4 : 1 Þ with initial conditions as u ð  x ; 0 Þ¼ 2  k 2 e kx ð 1 þ e kx Þ 2  :  ð 4 : 2 Þ The exact solution of  (4.1), for the special case  a  ¼  1 given in [19], is u ð  x ; t  Þ¼ 2  k 2 e k ð  x  k 2 t  Þ ð 1 þ e k ð  x  k 2 t  Þ Þ 2  :  ð 4 : 3 Þ M. Kurulay, M. Bayram/Commun Nonlinear Sci Numer Simulat 15 (2010) 1777–1782  1779  where u  ¼  u ð  x ; t  Þ isasufficientlyoftendifferentiablefunction.Weshallassumethatthesolution u ð  x ; t  Þ , alongwithitsderiv-atives, tends to zero as  j  x j ! 1 .Taking the two-dimensional transform of Eq. (4.1) by using the related theorem, we have C ð a ð h þ 1 Þþ 1 Þ C ð a h þ 1 Þ  U  a ; 1 ð k ; h þ 1 Þ¼ 6 X kr  ¼ 0 X hs ¼ 0 ð k þ 1  r  Þ U  ð r  ; h  s Þ U  ð k þ 1  r  ; s Þ : ð k þ 1 Þð k þ 2 Þð k þ 3 Þ U  ð k þ 3 ; h Þ ð 4 : 4 Þ The generalized two-dimensional differential transform of the initial condition can be obtained follows: U  ð k ; 0 Þ¼ 0 if   k ¼ 1 ; 3 ; 5 . . . ; U  ð 0 ; 0 Þ¼ 12 k 2 ;  U  ð 2 ; 0 Þ¼ 18 k 4 ;  U  ð 4 ; 0 Þ¼  148 k 6 ; . . .  ð 4 : 5 Þ By applying Eq. (4.5) into Eq. (4.4) we can obtain some value of   U  ð k ; h Þ  a follows: k ¼ 0 ; 2 ; 4 ; . . .  if   U  ð k ; 1 Þ¼ 0 k ¼ 1 ; 3 ; 5 ; . . .  if   U  ð k ; 2 Þ¼ 0 k ¼ 1 ; 3 ; 5 ; . . .  if   U  ð k ; 3 Þ¼ 0 Consequently substituting all  U  ð k ; h Þ  into Eq. (3.1) and, we obtain the series form solutions of Eq. (4.1) and (4.2) as u ð  x ; t  Þ¼  12 k 2 þ 18 k 4  x 2   148 k 6  x 4 þ  175760 k 8  x 6   3180640 k 10  x 8 þ   þ  14 k 6  x t  a C ð a þ 1 Þþ  112 k 8  x 3  t  a C ð a þ 1 Þ  17960 k 10  x 5  t  a C ð a þ 1 Þþ  3110080 k 12  x 7 t    6911451520 k 14  x 9  t  a C ð a þ 1 Þþ   þ  14 k 8  t  2 a C ð 2 a þ 1 Þ 14 k 10  x 2  t  2 a C ð 2 a þ 1 Þþ  17192 k 12  x 4  t  2 a C ð 2 a þ 1 Þ  311440 k 14  x 6  t  2 a C ð 2 a þ 1 Þþ  691161280 k 16  x 8  t  2 a C ð 2 a þ 1 Þþ   þ  10 k 11  t  3 a C ð 3 a þ 1 Þþð 184 k 12 e k þ 174 k 12 ð e k Þ 2 þ 237 k 13 Þ  x 2  t  3 a C ð 3 a þ 1 Þþ   ð 4 : 6 Þ The closed form of the first, second third and fourth curly brackets are u 1 ð  x ; t  Þ¼ 2  k 2 e kx ð 1 þ e kx Þ 2 ; u 2 ð  x ; t  Þ¼  4 k 5 ð e kx Þ 2 ð 1 þ e kx Þ 3 þ  2 k 5 e kx ð 1 þ e kx Þ 2 " #  t  a C ð a þ 1 Þ u 3 ð  x ; t  Þ¼  4 k 8 ð e kx Þ 2 ð 1 þ e kx Þ 4  4 k 8 ð e kx Þ 3 ð 1 þ e kx Þ 4   2 k 8 e kx ð 1 þ e kx Þ 2 þ 8 k 8 ð e kx Þ 8 ð 1 þ e kx Þ 3 " #  t  2 a C ð 2 a þ 1 Þ u 4 ð  x ; t  Þ¼  24 k 11 ð e kx Þ 3 ð 1 þ e kx Þ 4   12 k 11 ð e kx Þ 2 ð 1 þ e kx Þ 4   12 k 11 ð e kx Þ 2 ð 1 þ e kx Þ 3  þ 28 k 11 ð e kx Þ 3 ð 1 þ e kx Þ 5   16 k 11 ð e kx Þ 4 ð 1 þ e kx Þ 5   4 k 11 ð e kx Þ 2 ð 1 þ e kx Þ 5  þ  2 k 11 e kx ð 1 þ e kx Þ " #  t  3 a C ð 3 a þ 1 Þ ð 4 : 7 Þ For the special case  a  ¼  1 is u ð  x ; t  Þ¼ 2  k 2 e k ð  x  k 2 t  Þ ð 1 þ e k ð  x  k 2 t  Þ Þ 2  :  ð 4 : 8 Þ Example 4.2.  We consider the following time-fractional fmKdV equations: 0 < a 6 1 ; t   > 0, D a t   u þ u 2 u  x þ u  xxx  ¼ 0  ð 4 : 9 Þ with the initial conditions as u ð  x ; 0 Þ¼ 4  ffiffiffi 2 p   k sin 2 ð kx Þ 3  sin 2 ð kx Þ ;  ð 4 : 10 Þ where  k  is an arbitrary constant. The transformed version of Eq. (4.9) is C ð a ð h þ 1 Þþ 1 Þ C ð a h þ 1 Þ  U  ð k ; h þ 1 Þ¼ X kr  ¼ 0 X k  r t  ¼ 0 X hs ¼ 0 X h  s p ¼ 0 ð k  r   t  þ 1 Þ U  ð r  ; h  s   p Þ U  ð t  ; s Þ U  ð k  r   t  þ 1 ;  p Þð k þ 1 Þð k þ 2 Þð k þ 3 Þ U  ð k þ 3 ; h Þ :  ð 4 : 11 Þ 1780  M. Kurulay, M. Bayram/Commun Nonlinear Sci Numer Simulat 15 (2010) 1777–1782  Following the same procedures which are outlined for the previous problems the transformation coefficient can easily beevaluated. As a result, by substituting  U  ð k ; h Þ  into Eq. (3.1), we have series solution as follows: u ð  x ; t  Þ¼  43  ffiffiffi 2 p   k 3  x 2 þ 49  ffiffiffi 2 p   k 5  x 4 þ  8135  ffiffiffi 2 p   k 7  x 6   682835  ffiffiffi 2 p   k 9  x 8   884725  ffiffiffi 2 p   k 11  x 10 þ   þ  323  ffiffiffi 2 p   k 5  x t  a C ð a þ 1 Þ 649  ffiffiffi 2 p   k 7  x 3  t  a C ð a þ 1 Þ 6445  ffiffiffi 2 p   k 9  x 5  t  a C ð a þ 1 Þþ 21762835  ffiffiffi 2 p   k 11  x 7  t  a C ð a þ 1 Þþ 704945  ffiffiffi 2 p   k 13  x 9  t  a C ð a þ 1 Þþ   þ  1283  ffiffiffi 2 p   k 7  t  2 a C ð 2 a þ 1 Þþ 2563  ffiffiffi 2 p   k 9  x 2  t  2 a C ð 2 a þ 1 Þþ 2569  ffiffiffi 2 p   k 11  x 4  t  2 a C ð 2 a þ 1 Þ 8704405  ffiffiffi 2 p   k 13  x 6  t  2 a C ð 2 a þ 1 Þþ   : ð 4 : 12 Þ For the special case  a  ¼  1 given in [23], Using Taylor series into (4.12), we find the closed form solution u ð x ; t Þ¼ 4  ffiffi 2 p   k sin 2 ð k ð  x  4 k 2 t  ÞÞ 3  2sin 2 ð k ð  x  4 k 2 t  ÞÞ ;  0 6 k ð  x  4 k 2 t  Þ 6 p 0 ;  otherwise (  ð 4 : 13 Þ It is an exact solitary solution with compact support for the fmKdV Eq. (4.9) [23]. 5. Conclusions In this work, differential transform method is extended to solve the fmKdV and fKdV. Several mathematical methodsincludingtheAdomiandecompositionmethod[5,11,13,24,25],variationaliterationmethod[6,7,11],homotopy-perturbation method [12,18,26,27] and fractional difference method [28] have been developed to obtain exact and approximate analytic solutions.Someofthesemethodsusetransformationinordertoreduceequationsintosimplerequationsorsystemsofequa-tionsandsomeothermethodsgivethesolutioninaseriesformwhichconvergestotheexactsolution.Theworkemphasizedourbeliefthatthemethodisareliabletechniquetohandlelinearandnonlinearfractionaldifferentialequationsandthedif-ferentialtransformmethodoffersignificantadvantagesintermsofitsstraightforwardapplicability, itscomputationaleffec-tiveness and its accuracy. In general, the proposed method is promising and applicable to a broad class of linear andnonlinear problems in the theory of fractional calculus. References [1] Padovan J. Computational algorithms for FE formulations involving fractional operators. Comput Mech 1987;5:271–87.[2] Podlubny I. Fractional differential equations. An introduction to fractional derivatives fractional differential equations some methods of theirsolutionand some of their applications. SanDiego: Academic Press; 1999.[3] Hilfer R, editor. Applications of fractional calculus in physics. Orlando: Academic press; 1999.[4] Hosseini MM, Jafari M. A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations. CommunNonlin Sci Numer Simul 2009;14(5):1952–7. May.[5] Wu Lei, Xie Li-dan, Zhang Jie-fang. Adomian decomposition method for nonlinear differential-difference equations. Commun Nonlin Sci Numer Simul2009;14(1):12–8. January.[6] Muhammad Aslam Noor, Khalida Inayat Noor, Syed Tauseef Mohyud-Din, Variational iteration method for solving sixth-order boundary valueproblems. Commun Nonlin Sci Numer Simul. Corrected Proof, Available online 31 October 2008.[7] Odibat Z, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlin Sci Numer Simul2006;7(1):15–27.[8] Zhou JK. Differential transformation and its applications for electrical circuits. Wuhan,China: Huazhong university Press; 1986.[9] Arikoglu A, Özkol I. Solution of fractional differential equations by using differential transform method. Chaos Solitons & Fractals 2007:1473–81.[10] Momani S, Odibat Z, Ertürk V. Generalized differential transform method for solving a space and time fractional diffusion-wave equation. Phys Lett A2007;370(5-6):379–87. 29 October.[11] Odibat Z, Momani S. Numerical methods for nonlinear partial differential equations of fractional order. Appl Math Model 2008;32:28–39.[12] Abdulaziz O, Hashim I, Ismail ES. Approximate analytical solution to fractional modified KdV equations. Math Com Model 2009;49:136–45.[13] OdibatZ,MomaniS.Approximatesolutionsforboundaryvalueproblemsoftime-fractionalwaveequation.ApplMathComput2006;181(1):767–74.1October.[14] Odibat Z, Shawagfeh N. Generalized Taylor’s Formula. Appl Math Comput 2007;186:286–93.[15] Bildik N, Konuralp A, Bek F, Kucukarslan S. Solution of differential type of the partial differential equation by differential transform method andAdomian’s decomposition method. Appl Math Comput 2006;172:551–67.[16] Abdel-HalimHassanIH. Comparisondifferential transformationtechniquewithAdomiandecompositionmethodfor linear andnonlinear initial valueproblems. Chaos Solitons & Fractals 2008;36(1):53–65. April.[17] Momani S, Odibat Z. A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula. JComput Appl Math 2008;220:85–95.[18] Jafari H, Seifi S. Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. Commun Nonlin Sci NumerSimul 2009;14:1962–9.[19] Kangalgil F, Ayaz F. Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals2009;41(1):464–72.[20] Jafari H, Seifi S. Homotopy analysis method for solving linear nonlinear fractional diffusion-wave equation. Commun Nonlin Sci Numer Simul2009;14(5):2006–12. May.[21] Xu Hang, Liao Shi-Jun, You Xiang-Cheng. Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. CommunNonlin Sci Numer Simul 2009;14(4):1152–6. April.[22] Caputo M. Linear models of dissipation whose Q is almost frequency independent Part II. J Roy Austral Soc 1967;13:529–39.[23] Zhu Yonggui, Chang Qianshun, Wu Shengchang. Exact solitary-wave solutions with compact support for the modified KdV equation. Chaos Solitons &Fractals 2005;24(1):365–9. April. M. Kurulay, M. Bayram/Commun Nonlinear Sci Numer Simulat 15 (2010) 1777–1782  1781
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