numerical solution of two-dimensional nonlinear volterra integral equations by the legendre polynomials

the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. these properties together with the gauss-legendre nodes are then utilized to transform the given integral equation to the solution of nonlinear algebraic equations. also, an estimation of the error is presented. illustrative examples are included to demonstrate the validity and applicability of the new technique.