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A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions

A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions
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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.

Author Details

Chengda Wu

Dongfang Li

Weiwei Sun

Chengda Wu

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