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 twodimensionalnonlinear volterra integral equations using legendre orthogonal polynomials.first, the two-dimensional shifted legendre orthogonal polynomials are defined and theproperties of these polynomials are presented. the operational matrix of integration and theproduct operational matrix are introduced. these properties together with the gauss-legendre nodes are then utilized to transform the given integral equation to the solution ofnonlinear algebraic equations. also, an estimation of the error is presented. illustrativeexamples are included to demonstrate the validity and applicability of the new technique