TY - JOUR
T1 - Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time
AU - Shen , Jin-Ye
AU - Sun , Zhi-zhong
AU - Du , Rui
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 834
EP - 858
PY - 2018
DA - 2018/10
SN - 8
DO - http://doi.org/10.4208/eajam.010418.020718 
UR - https://global-sci.org/intro/article_detail/eajam/12821.html
KW - Fractional differential equation, difference scheme, fast algorithm, singularity.
AB - <p style="text-align: justify;">A sharp estimate for the L1 formula on graded meshes, which approximates
the Caputo derivatives of functions with a weak singularity at t = 0 is obtained. Combining
such approximations with the sum-of-exponential approximations of the kernel, we
develop fast difference schemes for one- and two-dimensional fractional diffusion equations,
the solutions of which have a weak singularity at the starting time. The proof of
the stability and convergence is based on the maximum principle. Numerical examples
confirm theoretical estimates.</p>