Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order Zaid Odibat a, * , Shaher Momani b a Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa’ Applied University, Salt 19117, Jordan b Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak, Jordan Accepted 7 June 2006 Abstract In this paper, a modification of He’s homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p 2 [0, 1] is constructed. The pro- posed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction Although fractional derivatives have a long mathematical history, for many years they were not used in physics. One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional deriv- atives [1]. Another difficulty is that fractional derivatives have no evident geometrical interpretation because of their nonlocal character [2]. However, during the last 10 years fractional calculus starts to attract much more attention of physicists and mathematicians. It was found that various, especially interdisciplinary applications can be elegantly mod- eled with the help of the fractional derivatives. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [3], and the fluid-dynamic traffic model with fractional derivatives [4] can eliminate the defi- ciency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in Ref. [5], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [1,6]. A review of some appli- cations of fractional derivatives in continuum and statistical mechanics is given by Mainardi [7]. In this paper, we present numerical and analytical solutions for the fractional Riccati differential equation d a y dt a ¼ AðtÞþ BðtÞy þ CðtÞy 2 ; t > 0; m  1 < a 6 m ð1:1Þ 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.041 * Corresponding author. Tel.: +962 79 6669951; fax: +962 5 3530462. E-mail addresses: odibat@bau.edu.jo (Z. Odibat), shahermm@yahoo.com (S. Momani). Available online at www.sciencedirect.com Chaos, Solitons and Fractals 36 (2008) 167–174 www.elsevier.com/locate/chaos