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 - <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>