@Article{NMTMA-14-945,
author = {Mokhtari , RezaRamezani , Mohadese and Haase , Gundolf},
title = {Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2021},
volume = {14},
number = {4},
pages = {945--971},
abstract = {<p style="text-align: justify;">Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae,
are usually employed to approximate the Caputo fractional derivative of order α ∈
(0, 1). In this paper, we aim to elaborate on the stability and convergence analyses
of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e.,
a diffusion equation which exploits the Caputo time-fractional derivative of order $α$.
In fact, the FDMs considered here are based on the usual central difference scheme
for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the
discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that
the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the
global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical
tests confirm the theoretical results. A brief conclusion finishes the paper.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0020},
url = {http://global-sci.org/intro/article_detail/nmtma/19525.html}
}