Stokes, Peter W., Philippa, Bronson, Read, Wayne, and White, Ronald D. (2015) Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart. Journal of Computational Physics, 282. pp. 334-344.

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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.

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