Solution of weakly singular fractional integro-differential equations by using a new operational approach

Abstract

In this study, an operational approach, based on the shifted Jacobi polynomials, is presented to solve a class of weakly singular fractional integro-differential equations. The fractional derivative operators are considered as the Caputo sense. In addition to finding the operational matrices of integration and product, a new operational matrix is derived to be approximated the singular kernels of this class of the functional equations. These operational matrices together with the collocation method are applied to convert the main equation into a linear or non-linear system of algebraic equations. By presenting some theorems, conditions of the existence and uniqueness of the solution of the main equation and its corresponding algebraic system are investigated. Also, the stability and convergence of the proposed method are studied and an error bound is obtained in a Jacobi-weighted Sobolev space. Finally, some illustrative examples are provided to demonstrate the efficiency and simplicity of the suggested approach.