Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

Abstract

The solution of a Caputo time fractional diffusion equation of order 0<α<1 is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an N-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from O(N2) to O(Nα), given a precomputation of O(N1+αln⁡N). The mapping is applied successfully to the least squares fitting of a fractional advection–diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.