Adomian decomposition method for solving fractional nonlinear differential equations S.A. El-Wakil, A. Elhanbaly, M.A. Abdou * Physics Department, Faculty of Science, Theoretical Research Group, Mansoura University, Mansoura, Egypt Abstract In this article, we have discussed a new application of Adomian decomposition method on time fractional nonlinear fractional differential equations. Three models with fractional-time derivative of order a,0< a < 1 are considered and solved by means of Adomian decomposition method. The behaviour of Adomian solutions and the effects of different val- ues of a are investigated. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithm. Ó 2006 Elsevier Inc. All rights reserved. 1. Introduction Up to now more and more nonlinear partial differential equations were presented, which describe the motion of isolated waves, localized in a small part of space, in many fields such as hydrodynamics, plasma physics and nonlinear optics, etc. Various fields of science and engineering, nolinear evaluation equation, as well as their analytic and numer- ical solutions, are of fundamental importance. One of the most attractive and surprising wave phenomenum is the creation of solitary waves or solitons. The solitary waves are often modeled by the bell shaped sech solu- tions and kink shaped tanh solution. Various methods for seeking explicit traveling solutions to nonlinear partial differential equations are pro- posed [1–6]. In the beginning of the 1980, a so-called Adomian decomposition (ADM) method [7–23] has been used to solve effectively, easily, and accurately a large class of linear and nonlinear equations, solutions partial, deterministic or stochastic differential equations with approximates which converge rapidly. Unlike classical techniques, the nonlinear equations are solved easily and elegantly without transforming the equation by using the ADM. The technique has many advantages over the classical techniques, mainly, it avoids linearization and perturbation in order to find explicit solutions of a given nonlinear equations. 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.02.055 * Corresponding author. E-mail address: m_abdou_eg@yahoo.com (M.A. Abdou). Applied Mathematics and Computation 182 (2006) 313–324 www.elsevier.com/locate/amc