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.

This paper presents a computational technique based on a special family of the Müntz–
Legendre polynomials to solve a class of Volterra–Fredholm integral equations. The
relationship between the Jacobi polynomials and Müntz–Legendre polynomials, in a
particular state, are expressed. The proposed method reduces the integral equation into
algebraic equations via the Chebyshev–Gauss–Lobatto points, so that the system matrix
coefficients are obtained by the least squares approximation method. The useful properties …