@Article{NMTMA-14-2,
author = {Wu, Chengda and Li, Dongfang and Weiwei, Sun 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 = {<p style="text-align: justify;">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&nbsp;$\mathcal{O}(N ^{−(2−α)})$ 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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2020-0129},
url = {https://global-sci.com/article/90302/a-novel-numerical-approach-to-time-fractional-parabolic-equations-with-nonsmooth-solutions}
}