Volume 14, Issue 2
A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 355-376.

Published online: 2021-01

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• Abstract

This paper is concerned with numerical solutions of time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are usually singular near the initial time $t = 0$ even for a smooth setting. Based on a simple change of variable $s = t^β$, an equivalent $s$-fractional differential equation is derived and analyzed. Two types of finite difference methods based on linear and quadratic approximations in the $s$-direction are presented, respectively, for solving the $s$-fractional differential equation. We show that the method based on the linear approximation provides the optimal accuracy $\mathcal{O}(N ^{−(2−α)})$ where $N$ is the number of grid points in temporal direction. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with $L1$ methods on uniform meshes and graded meshes, respectively. Our numerical results show clearly the accuracy and efficiency of the proposed methods.

• Keywords

Time-fractional differential equations, nonsmooth solution, finite difference methods, $L1$ approximation.

35A35, 35R11, 65M12

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@Article{NMTMA-14-355, author = {Li , Dongfang and Sun , Weiwei and Wu , Chengda}, title = {A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {355--376}, abstract = {

This paper is concerned with numerical solutions of time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are usually singular near the initial time $t = 0$ even for a smooth setting. Based on a simple change of variable $s = t^β$, an equivalent $s$-fractional differential equation is derived and analyzed. Two types of finite difference methods based on linear and quadratic approximations in the $s$-direction are presented, respectively, for solving the $s$-fractional differential equation. We show that the method based on the linear approximation provides the optimal accuracy $\mathcal{O}(N ^{−(2−α)})$ where $N$ is the number of grid points in temporal direction. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with $L1$ methods on uniform meshes and graded meshes, respectively. Our numerical results show clearly the accuracy and efficiency of the proposed methods.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0129}, url = {http://global-sci.org/intro/article_detail/nmtma/18603.html} }
TY - JOUR T1 - A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions AU - Li , Dongfang AU - Sun , Weiwei AU - Wu , Chengda JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 355 EP - 376 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0129 UR - https://global-sci.org/intro/article_detail/nmtma/18603.html KW - Time-fractional differential equations, nonsmooth solution, finite difference methods, $L1$ approximation. AB -

This paper is concerned with numerical solutions of time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are usually singular near the initial time $t = 0$ even for a smooth setting. Based on a simple change of variable $s = t^β$, an equivalent $s$-fractional differential equation is derived and analyzed. Two types of finite difference methods based on linear and quadratic approximations in the $s$-direction are presented, respectively, for solving the $s$-fractional differential equation. We show that the method based on the linear approximation provides the optimal accuracy $\mathcal{O}(N ^{−(2−α)})$ where $N$ is the number of grid points in temporal direction. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with $L1$ methods on uniform meshes and graded meshes, respectively. Our numerical results show clearly the accuracy and efficiency of the proposed methods.

Dongfang Li, Weiwei Sun & Chengda Wu. (2021). A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 355-376. doi:10.4208/nmtma.OA-2020-0129
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