Homotopy analysis method for fractional IVPs I. Hashim a, * , O. Abdulaziz a , S. Momani b a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia b Department of Mathematics and Physics, College of Arts and Sciences, Qatar University, Qatar Received 17 September 2007; accepted 30 September 2007 Available online 16 October 2007 Abstract In this paper, the homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach. Ó 2007 Elsevier B.V. All rights reserved. PACS: 02.30.Hq; 02.30.Mv; 02.60.Cb Keywords: Fractional IVPs; Homotopy analysis method; Caputo’s fractional derivative 1. Introduction In recent years, fractional differential equations (FDEs) have found applications in many problems in physics and engineering [1–4]. Since most of the nonlinear FDEs cannot be solved exactly, approximate and numerical methods must be used. Some of the recent analytic methods for solving nonlinear problems include the Adomian decomposition method (ADM) [5–9], homotopy-perturbation method (HPM) [10,11], variational iteration method (VIM) [12,13] and homotopy analysis method (HAM) [14–19]. The HAM, first proposed in 1992 by Liao [14], has been successfully applied to solve many problems in physics and science [20–29]. Very recently, Song and Zhang [30] applied HAM to solve fractional KdV–Burgers–Kuramoto equation. In this paper, HAM is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). Some test examples shall be presented to show the efficiency and accuracy of HAM. Furthermore, the Taylor series expansion shall be employed to avoid the difficulties with radical nonlinear terms. 1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.09.014 * Corresponding author. Tel.: +603 8921 5758; fax: +603 8925 4519. E-mail address: ishak_h@ukm.my (I. Hashim). Available online at www.sciencedirect.com Communications in Nonlinear Science and Numerical Simulation 14 (2009) 674–684 www.elsevier.com/locate/cnsns