A Laplace transform based kernel reduction scheme for fractional differential equations
The nonlocal nature of the fractional integral makes the numerical treatment of fractional differential equations expensive in terms of computational effort 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 that the history term be replaced by a linear combination of auxiliary variables defined as solutions to standard ordinary differential 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.