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Volume 11, Issue 2
Finite Difference Methods for Fractional Differential Equations on Non-Uniform Meshes

Haili Qiao & Aijie Cheng

East Asian J. Appl. Math., 11 (2021), pp. 255-275.

Published online: 2021-02

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  • Abstract

The solutions of fractional equations with Caputo derivative often have a singularity at the initial time. Therefore, for numerical methods on uniform meshes it is difficult to achieve optimal convergence rates. To improve the convergence, Liu et al. [10] considered a finite difference method on non-uniform meshes. Following the ideas of [10], we introduce two more sets of non-uniform meshes and show that the corresponding discrete models have higher convergence rates. Besides, we apply the trapezoidal rule in the case of linear fractional partial differential equations. The results of numerical experiments are consistent with the theoretical analysis.

  • AMS Subject Headings

65M06, 65M12, 65M15

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-255, author = {Qiao , Haili and Cheng , Aijie}, title = {Finite Difference Methods for Fractional Differential Equations on Non-Uniform Meshes}, journal = {East Asian Journal on Applied Mathematics}, year = {2021}, volume = {11}, number = {2}, pages = {255--275}, abstract = {

The solutions of fractional equations with Caputo derivative often have a singularity at the initial time. Therefore, for numerical methods on uniform meshes it is difficult to achieve optimal convergence rates. To improve the convergence, Liu et al. [10] considered a finite difference method on non-uniform meshes. Following the ideas of [10], we introduce two more sets of non-uniform meshes and show that the corresponding discrete models have higher convergence rates. Besides, we apply the trapezoidal rule in the case of linear fractional partial differential equations. The results of numerical experiments are consistent with the theoretical analysis.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.190520.111020 }, url = {http://global-sci.org/intro/article_detail/eajam/18634.html} }
TY - JOUR T1 - Finite Difference Methods for Fractional Differential Equations on Non-Uniform Meshes AU - Qiao , Haili AU - Cheng , Aijie JO - East Asian Journal on Applied Mathematics VL - 2 SP - 255 EP - 275 PY - 2021 DA - 2021/02 SN - 11 DO - http://doi.org/10.4208/eajam.190520.111020 UR - https://global-sci.org/intro/article_detail/eajam/18634.html KW - Fractional differential equation, weak singularity, finite difference, convergence analysis, non-uniform meshes. AB -

The solutions of fractional equations with Caputo derivative often have a singularity at the initial time. Therefore, for numerical methods on uniform meshes it is difficult to achieve optimal convergence rates. To improve the convergence, Liu et al. [10] considered a finite difference method on non-uniform meshes. Following the ideas of [10], we introduce two more sets of non-uniform meshes and show that the corresponding discrete models have higher convergence rates. Besides, we apply the trapezoidal rule in the case of linear fractional partial differential equations. The results of numerical experiments are consistent with the theoretical analysis.

Haili Qiao & Aijie Cheng. (2021). Finite Difference Methods for Fractional Differential Equations on Non-Uniform Meshes. East Asian Journal on Applied Mathematics. 11 (2). 255-275. doi:10.4208/eajam.190520.111020
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