ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 3, No. 2, 2008, pp. 097-103 A Numerical Scheme to Solve Nonlinear Volterra Integral Equations System of the Second Kind Omid Solaymani Fard + , Ali Tahmasbi Department of Applied Mathematics, School of Mathematics and Computer Science, Damghan University of Basic Sciences, Damghan, Iran (Received March 11, 2008, accepted April 30, 2008) Abstract. In this study, we use a recursive method based upon power series to solve nonlinear Volterra integral equations system of the second kind. This method gives an approximate solution as the Taylor expansion for the solution of the system via some simple computations. Numerical examples illustrate the pertinent features of the method. Keywords: Nonlinear integral equations system, Numerical method, Taylor expansion. 1. Introduction In recent years, many different methods have been used to approximate the solution of linear or nonlinear Volterra integral equations system [1, 2, 3, 4, 5, and 7]. Tricomi, in his book [6], introduced the classical method of successive approximations for nonlinear Volterra integral equations. In [2], Brunner applied a collocation-type method to nonlinear Volterra equations and integro-differential equations and discussed its connection with the iterated collocation method. For Volterra-Hammerstein equations, the asymptotic error expansion of a collocation method was introduced in [3]. In general, most of numerical methods transform the integral equation to a linear or nonlinear system of algebraic equations which can be solved by direct or iterative methods. Yousefi and Razzaghi in [7], and also Maleknejad et al in [4] used Legendre wavelets to numerical solution of linear and nonlinear Volterra integral equations. Recently, In [5], Chebyshev polynomials are applied for solving of nonlinear Volterra integral equations of the second kind. In the present article, we consider the second kind Volterra integral equations system of the form: ( ) ( ) ( ) 1 2 1 1 1 1 1 0 2 2 2 2 1 0 1 0 () () (,) () , () () (,) () , () () (,) () , j j nj s P n j j j j s P n j j j j s P n n n nj nj j j y s g s k st y t dt y s g s k st y t dt y s g s k st y t dt λ λ λ = = = ⎧ = + ⎪ ⎪ = + ⎪ ⎨ ⎪ ⎪ ⎪ = + ⎩ ∑ ∫ ∑ ∫ ∑ ∫ # (1) where, 0 , 1, 0, , 1, 2,..., ij ts ij n λ ≤ ≤ ≠ = is a real constant, and , ,. 1, 2,..., ij P ij n = is a nonnegative integer. Moreover, in Eq.(1) the function and the kernel are given and assumed to be sufficiently differentiable with respect to all their arguments on the interval 0 , , for all . Also, is the solution to be determined. () i g s , (,) ij k st 1 ts ≤ ≤ , 1, 2,..., ij n = 1 2 () [ ( ), ( ),..., ( )] T n Ys y s y s y s = To avoid of complesity, we simplify Eq.(1) by using the following notations, + Corresponding author. Tel.: +98-232-5233056; fax: +98-232-5235316. E-mail address: osfard@dubs.ac.ir . Published by World Academic Press, World Academic Union