Year: 2021
Author: Chengda Wu, Dongfang Li, Weiwei Sun, Chengda Wu
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2020-0129
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 2 : pp. 355–376
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Time-fractional differential equations nonsmooth solution finite difference methods $L1$ approximation.