Cent. Eur. J. Phys. • 11(10) • 2013 • 1523-1527 DOI: 10.2478/s11534-013-0209-1 Central European Journal of Physics Homotopy analysis method for solving Abel differential equation of fractional order Short Communication Hossein Jafari 1∗ , Khosro Sayevand 2† , Haleh Tajadodi 1‡ , Dumitru Baleanu 3,4,5§ 1 Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran 2 Department of Mathematics, Faculty of Basic Sciences, University of Malayer, P.O. Box 65719-95863, Malayer, Iran 3 Çankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, Ankara, Turkey 4 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia 5 Institute of Space Sciences, MG-23, R 76900, Magurele-Bucharest, Romania Received 03 February 2013; accepted 12 March 2013 Abstract: In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented. PACS (2008): 02.30.Mv, 04.20.Ex Keywords: Abel differential equation • fractional derivative • homotopy analysis method © Versita sp. z o.o. 1. Introduction Liao proposed the homotopy analysis method (HAM) in 1992, [1] and since then it has been used to obtain the analytical, and approximate analytical, solutions of many types ofnonlinear equations and systems of equations. It ∗ E-mail: jafari@umz.ac.ir (Corresponding author) † E-mail: ksayehvand@iust.ac.ir ‡ E-mail: tajadodi@umz.ac.ir § E-mail: dumitru@cankaya.edu.tr has also been applied to problems in engineering and science (see for example Refs. [1–7] and the references therein). With this method , we use a certain auxiliary parameter ¯ h to control and adjust the rate of convergence and the convergence region of the series solution. The valid regions of ¯ h are obtained by using an ¯ h-curve. The fractional calculus has been used extensively in basic sci- ences and engineering (see for example Refs. [8–22] and the references therein). A recent application has included numerically determining solutions for various classes of nonlinear fractional differential equations.Many engineer- ing and physical problems have been modelled using frac- 1523