A Kernel Compression Scheme for Fractional Differential Equations

Abstract

The nonlocal nature of the fractional integral makes the numerical treatment of fractional dierential equations expensive in terms of computational eort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables dened as solutions to standard ordinary dierential equations. We derive a priori error estimates, uniform in f, and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.