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Volume 8, Issue 4
Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time

Jin-Ye Shen, Zhi-zhong Sun & Rui Du

DOI: 10.4208/eajam.010418.020718

East Asian J. Appl. Math., 8 (2018), pp. 834-858.

Published online: 2018-10

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  • Abstract

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.

  • Keywords

Fractional differential equation, difference scheme, fast algorithm, singularity.

  • AMS Subject Headings

65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-8-834, author = {Shen , Jin-YeSun , Zhi-zhong and Du , Rui}, title = {Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {8}, number = {4}, pages = {834--858}, abstract = {

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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.010418.020718 }, url = {http://global-sci.org/intro/article_detail/eajam/12821.html} }
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 -

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.

Shen , Jin-YeSun , Zhi-zhong and Du , Rui. (2018). Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time. East Asian Journal on Applied Mathematics. 8 (4). 834-858. doi:10.4208/eajam.010418.020718
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