Finite difference methods and a fourier analysis for the fractional reaction–subdiffusion equation

Chen, Chang-ming, Liu, Fawang, & Burrage, Kevin (2008) Finite difference methods and a fourier analysis for the fractional reaction–subdiffusion equation. Applied Mathematics and Computation, 198(2), pp. 754-769.

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Various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations (FPDE), for example, systems biology, chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments. However, effective numerical methods and numerical analysis for FPDE are still in their infancy. In this paper, we consider a fractional reaction–subdiffusion equation (FR-subDE) in which both the motion and the reaction terms are affected by the subdiffusive character of the process. Using the relationship between the Riemann–Liouville and Grünwald–Letnikov definitions of fractional derivatives, an implicit and an explicit difference methods for the FR-subDE are presented. The stability and the convergence of the two numerical methods are investigated by a Fourier analysis. The solvability of the implicit finite difference method is also proved. The high-accuracy algorithm is structured using Richardson extrapolation. Finally, a comparison between the exact solution and the two numerical solutions is given. The numerical results are in excellent agreement with our theoretical analysis.

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90 citations in Scopus
85 citations in Web of Science®
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ID Code: 14755
Item Type: Journal Article
Refereed: Yes
Keywords: Space–time fractional derivatives, Fractional advection–diffusion equation, Implicit difference method, Explicit difference method, Stability, Convergence, Fourier analysis
DOI: 10.1016/j.amc.2007.09.020
ISSN: 0096-3003
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Analysis (010301)
Australian and New Zealand Standard Research Classification > BIOLOGICAL SCIENCES (060000) > BIOCHEMISTRY AND CELL BIOLOGY (060100) > Biochemistry and Cell Biology not elsewhere classified (060199)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100) > Ordinary Differential Equations Difference Equations and Dynamical Systems (010109)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2008 Elsevier
Copyright Statement: Reproduced in accordance with the copyright policy of the publisher.
Deposited On: 09 Sep 2008 00:00
Last Modified: 28 Aug 2014 03:19

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