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

Stokes, Peter W.;Philippa, Bronson;Read, Wayne;White, Ronald D.

Abstract

The solution of a Caputo time fractional diffusion equation of order 0<α<10<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)O(N2) to O(Nα)O(Nα), given a precomputation of O(N1+αln⁡N)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.

Journal

JOURNAL OF COMPUTATIONAL PHYSICS

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282

ISBN/ISSN

1090-2716

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Publisher

Academic Press

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DOI

10.1016/j.jcp.2014.11.023