On the Superconvergence of Collocation Methods Based on Two Post-Processing Techniques for Fractional Integro-Differential Equations with Weakly Singular Kernels

Abstract

The purpose of this paper is to investigate the superconvergence of collocation methods for fractional integro-differential equations (FIDEs) with weakly singular kernels and Caputo derivative of order $0 < \alpha < 1$. First, the initial value problem of FIDEs is reformulated as a weakly singular Volterra integral equation (VIE), and the existence, uniqueness, and regularity of the exact solution for the original FIDE are obtained with the help of the resolvent theory of VIEs, and it is shown that the singularity of the exact solution is governed by the Caputo derivative, not the weakly singular kernel. Next, the piecewise polynomial collocation method is employed to solve the reformulated VIE numerically, and the optimal convergence order of the collocation solution is obtained on graded meshes. In order to improve the numerical accuracy, two types of postprocessing techniques are used – one is the classical iterated technique for VIEs and another one is the interpolation postprocessing technique. The superconvergence is thoroughly investigated and the optimal superconvergence orders are obtained for both of these two postprocessing techniques. Compared to the classical iterated collocation method, the interpolation postprocessing method has a lower calculation cost. The theoretical results are illustrated by numerical experiments.