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Volume 16, Issue 6
Nonstandard Finite Difference Method for Nonlinear Riesz Space Fractional Reaction-Diffusion Equation

Li Cai, Meifang Guo, Yiqiang Li, Wenjun Ying, Hao Gao & Xiaoyu Luo

Int. J. Numer. Anal. Mod., 16 (2019), pp. 925-938.

Published online: 2019-08

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  • Abstract

In this paper, a modified nonstandard finite difference method for the two-dimensional Riesz space fractional reaction-diffusion equations is developed. The space fractional derivative is discretized by the shifted Grünwald-Letnikov method and the nonlinear reaction term is approximated by Taylor formula instead of Micken's. Multigrid method is introduced to reduce the computation time of the traditional Gauss-Seidal method. The stability and convergence of the nonstandard implicit difference scheme are strictly proved. The method is extended to simulate the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

  • Keywords

Riesz fractional derivative, nonstandard finite difference method, shifted Grünwald-Letnikov method.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

caili@nwpu.edu.cn (Li Cai)

meifang@mail.nwpu.edu.cn (Meifang Guo)

liyiqiang@mail.nwpu.edu.cn (Yiqiang Li)

wying@sjtu.edu.cn (Wenjun Ying)

hao.gao@glasgow.ac.uk (Hao Gao)

xiaoyu.luo@glasgow.ac.uk (Xiaoyu Luo)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-925, author = {Cai , LiGuo , MeifangLi , YiqiangYing , WenjunGao , Hao and Luo , Xiaoyu}, title = {Nonstandard Finite Difference Method for Nonlinear Riesz Space Fractional Reaction-Diffusion Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {6}, pages = {925--938}, abstract = {

In this paper, a modified nonstandard finite difference method for the two-dimensional Riesz space fractional reaction-diffusion equations is developed. The space fractional derivative is discretized by the shifted Grünwald-Letnikov method and the nonlinear reaction term is approximated by Taylor formula instead of Micken's. Multigrid method is introduced to reduce the computation time of the traditional Gauss-Seidal method. The stability and convergence of the nonstandard implicit difference scheme are strictly proved. The method is extended to simulate the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13260.html} }
TY - JOUR T1 - Nonstandard Finite Difference Method for Nonlinear Riesz Space Fractional Reaction-Diffusion Equation AU - Cai , Li AU - Guo , Meifang AU - Li , Yiqiang AU - Ying , Wenjun AU - Gao , Hao AU - Luo , Xiaoyu JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 925 EP - 938 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13260.html KW - Riesz fractional derivative, nonstandard finite difference method, shifted Grünwald-Letnikov method. AB -

In this paper, a modified nonstandard finite difference method for the two-dimensional Riesz space fractional reaction-diffusion equations is developed. The space fractional derivative is discretized by the shifted Grünwald-Letnikov method and the nonlinear reaction term is approximated by Taylor formula instead of Micken's. Multigrid method is introduced to reduce the computation time of the traditional Gauss-Seidal method. The stability and convergence of the nonstandard implicit difference scheme are strictly proved. The method is extended to simulate the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

Cai , LiGuo , MeifangLi , YiqiangYing , WenjunGao , Hao and Luo , Xiaoyu. (2019). Nonstandard Finite Difference Method for Nonlinear Riesz Space Fractional Reaction-Diffusion Equation. International Journal of Numerical Analysis and Modeling. 16 (6). 925-938. doi:
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