## Approximate analytical solution for the fractional modified Kdv by differential transform method.(English)Zbl 1222.35172

Summary: In this paper, the fractional modified Korteweg-de Vries equation (fmKdV) and fKdV are introduced by fractional derivatives. The approach rest mainly on two-dimensional differential transform method (DTM) which is one of the approximate methods. The method can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. Some illustrative examples are presented.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations)

BVPh
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### References:

 [1] Padovan, J., Computational algorithms for FE formulations involving fractional operators, Comput mech, 5, 271-287, (1987) · Zbl 0616.73066 [2] Podlubny, I., Fractional differential equations, An introduction to fractional derivatives fractional differential equations some methods of their solutionand some of their applications, (1999), Academic Press SanDiego · Zbl 0924.34008 [3] () [4] Hosseini, M.M.; Jafari, M., A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations, Commun nonlin sci numer simul, 14, 5, 1952-1957, (2009), May · Zbl 1221.65162 [5] Wu, Lei; Xie, Li-dan; Zhang, Jie-fang, Adomian decomposition method for nonlinear differential-difference equations, Commun nonlin sci numer simul, 14, 1, 12-18, (2009), January · Zbl 1221.65209 [6] Muhammad Aslam Noor, Khalida Inayat Noor, Syed Tauseef Mohyud-Din, Variational iteration method for solving sixth-order boundary value problems. Commun Nonlin Sci Numer Simul. Corrected Proof, Available online 31 October 2008. · Zbl 1153.49014 [7] Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int J nonlin sci numer simul, 7, 1, 15-27, (2006) · Zbl 1401.65087 [8] Zhou, J.K., Differential transformation and its applications for electrical circuits, (1986), Huazhong university Press Wuhan,China [9] Arikoglu, A.; Özkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons & fractals, 1473-1481, (2007) · Zbl 1152.34306 [10] Momani, S.; Odibat, Z.; Ertürk, V., Generalized differential transform method for solving a space and time fractional diffusion-wave equation, Phys lett A, 370, 5-6, 379-387, (2007), 29 October · Zbl 1209.35066 [11] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Appl math model, 32, 28-39, (2008) · Zbl 1133.65116 [12] Abdulaziz, O.; Hashim, I.; Ismail, E.S., Approximate analytical solution to fractional modified KdV equations, Math com model, 49, 136-145, (2009) · Zbl 1165.35441 [13] Odibat, Z.; Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl math comput, 181, 1, 767-774, (2006), 1 October · Zbl 1148.65100 [14] Odibat, Z.; Shawagfeh, N., Generalized taylor’s formula, Appl math comput, 186, 286-293, (2007) · Zbl 1122.26006 [15] Bildik, N.; Konuralp, A.; Bek, F.; Kucukarslan, S., Solution of differential type of the partial differential equation by differential transform method and adomian’s decomposition method, Appl math comput, 172, 551-567, (2006) · Zbl 1088.65085 [16] Abdel-Halim Hassan, I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos solitons & fractals, 36, 1, 53-65, (2008), April · Zbl 1152.65474 [17] Momani, S.; Odibat, Z., A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized taylor’s formula, J comput appl math, 220, 85-95, (2008) · Zbl 1148.65099 [18] Jafari, H.; Seifi, S., Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun nonlin sci numer simul, 14, 1962-1969, (2009) · Zbl 1221.35439 [19] Kangalgil, F.; Ayaz, F., Solitary wave solutions for the KdV and mkdv equations by differential transform method, Chaos, solitons & fractals, 41, 1, 464-472, (2009) · Zbl 1198.35222 [20] Jafari, H.; Seifi, S., Homotopy analysis method for solving linear nonlinear fractional diffusion-wave equation, Commun nonlin sci numer simul, 14, 5, 2006-2012, (2009), May · Zbl 1221.65278 [21] Xu, Hang; Liao, Shi-Jun; You, Xiang-Cheng, Analysis of nonlinear fractional partial differential equations with the homotopy analysis method, Commun nonlin sci numer simul, 14, 4, 1152-1156, (2009), April · Zbl 1221.65286 [22] Caputo, M., Linear models of dissipation whose Q is almost frequency independent part II, J roy austral soc, 13, 529-539, (1967) [23] Zhu, Yonggui; Chang, Qianshun; Wu, Shengchang, Exact solitary-wave solutions with compact support for the modified KdV equation, Chaos solitons & fractals, 24, 1, 365-369, (2005), April · Zbl 1067.35099 [24] Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math comput simul, 70, 2, 110-118, (2005) · Zbl 1119.65394 [25] Wang, Qi, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl math comput, 182, 2, 1048-1055, (2006) · Zbl 1107.65124 [26] Wang, Qi, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos solitons & fractals, 35, 5, 843-850, (2008) · Zbl 1132.65118 [27] Bataineh, A.S.; Alomari, A.K.; Noorani, M.S.M.; Hashim, I.; Nazar, R., Series solutions of systems of nonlinear fractional differential equations, Acta appl math int surv J appl math math appl, 105, 2, 189-198, (2009) · Zbl 1187.34007 [28] lynch, V.E.; Carreras, B.A.; del-Castillo-Negrete, D.; Ferriera-Mejias, K.M.; Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J comput phys, 192, 406-421, (2003) · Zbl 1047.76075
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