Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations

Abstract

A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time \(t=0\) . A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.