Iranian Journal of Numerical Analysis and Optimization Vol 4, No. 2, (2014), pp 73-83 Solving nonlinear Volterra integro-differential equation by using Legendre polynomial approximations M. Gachpazan ∗ , M. Erfanian and H. Beiglo Abstract In this paper, we construct a new iterative method for solving nonlinear Volterra Integral Equation of the second kind, by approximating the Legendre polynomial basis. Error analysis is worked using property of interpolation. Finally, some examples are given to compare the results with some of the existing methods. Keywords: Nonlinear Volterra integro-differential equation; Legendre poly- nomial; Error analysis. 1 Introduction The area of orthogonal polynomials is an active research area in mathe- matics as well as with applications in mathematical physics, engineering, and computer science [6, 16]. Several numerical methods were used to solve integro-differential equations such as successive approximation method, Adomian decomposition method, Chebyshev and Taylor collocation meth- ods, Haar Wavelet method, Wavelet Galerkin method, monotone iterative technique, Tau method, Walsh series method and Bezier curves method [2, 3, 4, 6, 13, 22]. One of the most common set of orthogonal polynomi- als is the set of the Legendre polynomials L 0 (x),L 1 (x), ..., L M (x), which are orthogonal on [−1, 1] with respect to the weight function w(x) = 1. The * Corresponding author Received 22 April 2014; revised 19 August 2014; accepted 23 August 2014 M. Gachpazan Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi Univer- sity of Mashhad, Mashhad, Iran. e-mail: gachpazan@um.ac.ir M. Erfanian Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi Univer- sity of Mashhad, Mashhad, Iran. e-mail: erfaniyan@uoz.ac.ir H. Beiglo Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi Univer- sity of Mashhad, Mashhad, Iran. e-mail: h.beiglo@gmail.com 73