He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind J. Biazar a, * , H. Ghazvini a,b a Department of Mathematics, Faculty of Sciences, University of Guilan, P.O. Box 1914, P.C. 41938, Rasht, Iran b Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316, P.C. 3619995161, Shahrood, Iran Accepted 2 January 2007 Abstract In this paper, the He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such sys- tems. The results reveal that the method is very effective and simple. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction The Topic of the He’s homotopy perturbation method [1–6] has been rapidly growing in recent years [7–23]. In this method the solution of functional equations is considered as the summation of an infinite series usually converging to the solution. Using homotopy technique of topology, a homotopy is constructed with an embedding parameter p 2½0; 1 which is considered as a ‘‘small parameter’’. To illustrate the basic concept of homotopy perturbation method, consider the following non-linear functional equation: AðU Þ¼ f ðrÞ; r 2 X; ð1Þ with the following boundary conditions: B U ; oU on   ¼ 0; r 2 C; where A is a general functional operator, B is a boundary operator, f(r) is a known analytic function, and C is the boundary of the domain X. Generally speaking the operator A can be decomposed into two parts L and N, where L is a linear and N is a non-linear operator. Eq. (1), therefore, can be rewritten as the following: LðU Þþ N ðU Þ f ðrÞ¼ 0: 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.108 * Corresponding author. E-mail addresses: biazar@guilan.ac.ir, jbiazar@dal.ca (J. Biazar), hghazvini@guilan.ac.ir (H. Ghazvini). Chaos, Solitons and Fractals 39 (2009) 770–777 www.elsevier.com/locate/chaos