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Volume 5, Issue 1
Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Hongxing Rui & Jian Huang

DOI: 10.4208/eajam.030614.051114a

East Asian J. Appl. Math., 5 (2015), pp. 29-47.

Published online: 2018-02

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  • Abstract

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

  • Keywords

Finite difference scheme, fractional diffusion equation, uniformly stable, explicitly solvable method, asymmetric technique, error estimate.

  • AMS Subject Headings

65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-5-29, author = {Hongxing Rui and Jian Huang}, title = {Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {5}, number = {1}, pages = {29--47}, abstract = {

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.030614.051114a}, url = {http://global-sci.org/intro/article_detail/eajam/10776.html} }
TY - JOUR T1 - Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations AU - Hongxing Rui & Jian Huang JO - East Asian Journal on Applied Mathematics VL - 1 SP - 29 EP - 47 PY - 2018 DA - 2018/02 SN - 5 DO - http://doi.org/10.4208/eajam.030614.051114a UR - https://global-sci.org/intro/article_detail/eajam/10776.html KW - Finite difference scheme, fractional diffusion equation, uniformly stable, explicitly solvable method, asymmetric technique, error estimate. AB -

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

Hongxing Rui and Jian Huang. (2018). Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations. East Asian Journal on Applied Mathematics. 5 (1). 29-47. doi:10.4208/eajam.030614.051114a
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