From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations
Article
Article Title | From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations |
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ERA Journal ID | 35103 |
Article Category | Article |
Authors | Angstmann, C. N. (Author), Donnelly, I. C. (Author), Henry, B. I. (Author), Jacobs, B. A. (Author), Langlands, T. A. M. (Author) and Nichols, J. A. (Author) |
Journal Title | Journal of Computational Physics |
Journal Citation | 307, pp. 508-534 |
Number of Pages | 27 |
Year | 2016 |
Place of Publication | Netherlands |
ISSN | 0021-9991 |
1090-2716 | |
Digital Object Identifier (DOI) | https://doi.org/10.1016/j.jcp.2015.11.053 |
Web Address (URL) | http://www.sciencedirect.com/science/article/pii/S0021999115007937 |
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. |
Keywords | fractional diffusion; fractional reaction diffusion; anomalous diffusion; continuous time random walk; discrete time random walk; finite difference method |
Contains Sensitive Content | Does not contain sensitive content |
ANZSRC Field of Research 2020 | 490303. Numerical solution of differential and integral equations |
490510. Stochastic analysis and modelling | |
Public Notes | File reproduced in accordance with the copyright policy of the publisher/author. |
Byline Affiliations | University of New South Wales |
University of the Witwatersrand, South Africa | |
Computational Engineering and Science Research Centre | |
Institution of Origin | University of Southern Queensland |
Funding source | Australian Research Council (ARC) Grant ID DP140101193 |
https://research.usq.edu.au/item/q33yv/from-stochastic-processes-to-numerical-methods-a-new-scheme-for-solving-reaction-subdiffusion-fractional-partial-differential-equations
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