A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations

This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractional-order. This technique is based on applying Muntz-Legendre polynomials and Petrov-Galerkin method. A new Riemann-Liouville operational matrix for Muntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, transforms the problem into a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. Moreover, some numerical examples are implemented in order to show the validity and accuracy of the suggested method.