Numerical solution of Volterra–Fredholm integral equations using Legendre collocation method

A numerical method for solving the Volterra–Fredholm integral equations is presented. The method is based upon shifted Legendre polynomials approximation. The properties of shifted Legendre polynomials are first presented. These properties together with the shifted Gauss–Legendre nodes are then utilized to reduce the Volterra–Fredholm integral equations to the solution of a matrix equation. An estimation of the error is given. Illustrative examples are included to demonstrate the validity and applicability of the technique.