TY - JOUR
T1 - Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 845
EP - 866
PY - 2019
DA - 2019/04
SN - 12
DO - http://doi.org/10.4208/nmtma.OA-2018-0046
UR - https://global-sci.org/intro/article_detail/nmtma/13133.html
KW - Fractional diffusion equation, graded mesh, multi-term, variable coefficients, low regularity, stability and convergence analysis.
AB - <p style="text-align: justify;">Finite difference scheme for the variable coefficients subdiffusion equations
with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and $L$1 approximation is used for the discretization of the
fractional time derivative on a possibly graded mesh. Stability of the proposed scheme
is given using the discrete energy method. The numerical scheme is $\mathcal{O}$ ($N$<sup>−min{2−$α$,$rα$}</sup>)
accurate in time, where $α$ (0 &lt; $α$ &lt; 1) is the order of the fractional time derivative, $r$ is an index of the mesh partition, and it is second order accurate in space. Extension to
multi-term time-fractional problems with nonhomogeneous boundary conditions is also
discussed, with the stability and error estimate proved both in the discrete $l$<sup>2</sup>-norm and
the $l$<sup>∞</sup>-norm on the nonuniform temporal mesh. Numerical results are given for both
the two-dimensional single and multi-term time-fractional equations.</p>