## 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:

  Padovan, J., Computational algorithms for FE formulations involving fractional operators, Comput mech, 5, 271-287, (1987) · Zbl 0616.73066  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  ()  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  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  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  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  Zhou, J.K., Differential transformation and its applications for electrical circuits, (1986), Huazhong university Press Wuhan,China  Arikoglu, A.; Özkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons & fractals, 1473-1481, (2007) · Zbl 1152.34306  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  Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Appl math model, 32, 28-39, (2008) · Zbl 1133.65116  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  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  Odibat, Z.; Shawagfeh, N., Generalized taylor’s formula, Appl math comput, 186, 286-293, (2007) · Zbl 1122.26006  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  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  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  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  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  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  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  Caputo, M., Linear models of dissipation whose Q is almost frequency independent part II, J roy austral soc, 13, 529-539, (1967)  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  Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math comput simul, 70, 2, 110-118, (2005) · Zbl 1119.65394  Wang, Qi, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl math comput, 182, 2, 1048-1055, (2006) · Zbl 1107.65124  Wang, Qi, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos solitons & fractals, 35, 5, 843-850, (2008) · Zbl 1132.65118  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  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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