TY - JOUR
T1 - Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation
AU - Mokhtari , Reza
AU - Ramezani , Mohadese
AU - Haase , Gundolf
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 945
EP - 971
PY - 2021
DA - 2021/09
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2021-0020
UR - https://global-sci.org/intro/article_detail/nmtma/19525.html
KW - Stability analysis, order of convergence, Caputo derivative, L1 formula, L1-2 formula, L1-2-3 formula, subdiffusion equation, Gronwall inequality.
AB - <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>