Galerkin and Collocation Methods for Weakly Singular Fractional Integro-differential Equations

Abstract

This paper describes the collocation and Galerkin’s approaches for fractional integro-differential equations (FIDEs). We explain the application of Jacobi polynomials to solve the FIDEs which convert the problem into a system of algebraic equations. To approximate the solution of FIDEs by Jacobi polynomials, a suitable variable transformation is applied which assures that the solution of the transformed FIDEs is sufficiently smooth. This results in a rapid convergence of both the methods with Jacobi polynomials even when the solution is not smooth. The error estimate and convergence analysis for presented numerical methods are provided. To perform the numerical simulations, two test examples (linear and nonlinear) are considered with non-smooth solutions, and numerical results are presented. Further, the comparative study of the presented schemes with some existing numerical schemes is provided.