From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations

Angstmann, C. N. and Donnelly, I. C. and Henry, B. I. and Jacobs, B. A. and Langlands, T. A. M. and Nichols, J. A. (2016) From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations. Journal of Computational Physics, 307. pp. 508-534. ISSN 0021-9991

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Abstract

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Accepted version deposited in accordance with the copyright policy of the publisher.
Faculty / Department / School: Current - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences
Date Deposited: 21 Apr 2016 03:22
Last Modified: 09 Jan 2017 23:03
Uncontrolled Keywords: fractional diffusion; fractional reaction diffusion; anomalous diffusion; continuous time random walk; discrete time random walk; finite difference method
Fields of Research : 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations
01 Mathematical Sciences > 0104 Statistics > 010406 Stochastic Analysis and Modelling
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Funding Details:
Identification Number or DOI: 10.1016/j.jcp.2015.11.053
URI: http://eprints.usq.edu.au/id/eprint/28376

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