Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time

被引：54
|
作者：
Shen, Jin-ye [1 ]
Sun, Zhi-zhong [1 ]
Du, Rui [1 ]
机构：
[1] Southeast Univ, Sch Math, Nanjing 210096, Jiangsu, Peoples R China
来源：
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS | 2018年 / 8卷 / 04期
基金：
中国国家自然科学基金;
关键词：
Fractional differential equation; difference scheme; fast algorithm; singularity; NONUNIFORM TIMESTEPS; STEPPING METHOD; DYNAMICS; MESHES;
D O I：
10.4208/eajam.010418.020718
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
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.
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页码：834 / 858
页数：25
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