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

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.

Item ID: 42388
Item Type: Article (Research - C1)
ISSN: 1090-2716
Keywords: Caputo time fractional advection–diffusion equation; finite difference methods; anomalous diffusion mapping; time of flight experiment
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A version of this publication was included as Chapter 2 of the following PhD thesis: Stokes, Peter (2018) Anomalous charged-particle transport in organic and soft-condensed matter. PhD thesis, James Cook University, which is available Open Access in ResearchOnline@JCU. Please see the Related URLs for access.

Date Deposited: 04 Feb 2016 03:04
FoR Codes: 01 MATHEMATICAL SCIENCES > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations @ 100%
SEO Codes: 97 EXPANDING KNOWLEDGE > 970101 Expanding Knowledge in the Mathematical Sciences @ 100%
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