Numerical comparison of methods for solving linear differential equations of fractional order Shaher Momani a, * , Zaid Odibat b a Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak, Jordan b Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa’ Applied University, Salt, Jordan Accepted 20 October 2005 Communicated by Prof. Ji-Huan He Abstract In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied math- ematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of frac- tional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented. Ó 2005 Elsevier Ltd. All rights reserved. 1. Introduction In this paper, we consider the numerical solution of linear fractional differential equation of the form d m u dt m  a d a u dt a  bu ¼ f ðtÞ; t > 0; m  1 < a 6 m; ð1:1Þ subject to the initial conditions u ðjÞ ð0Þ¼ c j ; j ¼ 1; 0; ... ; m  1; ð1:2Þ where c j , j = 0, 1, ... , m  1, are arbitrary constants and u(t) is assumed to be a causal function of time, i.e., vanishing for t < 0. The fractional derivatives is considered in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. We refer to Eq. (1.1) as to the composite fractional relaxation and to the composite fractional oscillation equation in the cases {0 < a 6 1, m = 1} and {1 < a 6 2, m = 2}, respectively. 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.068 * Corresponding author. Tel.: +962 7 77500326; fax: +962 6 4654061. E-mail addresses: shahermm@yahoo.com (S. Momani), odibat@bau.edu.jo (Z. Odibat). Chaos, Solitons and Fractals 31 (2007) 1248–1255 www.elsevier.com/locate/chaos