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Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Authors
  • Liu, Fawang
  • Chen, Shiping
  • Turner, Ian
  • Burrage, Kevin
  • Anh, Vo
Type
Published Article
Journal
Open Physics
Publisher
Versita
Publication Date
Oct 01, 2013
Volume
11
Issue
10
Pages
1221–1232
Identifiers
DOI: 10.2478/s11534-013-0296-z
Source
De Gruyter
Keywords
License
Green

Abstract

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

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