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Volume 41, Issue 2
Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation

Zewen Wang, Shufang Qiu, Shuang Yu, Bin Wu & Wen Zhang

J. Comp. Math., 41 (2023), pp. 173-190.

Published online: 2022-11

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  • Abstract

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter $\gamma$, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When $\gamma$ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of $\gamma$. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

  • AMS Subject Headings

35R30, 35R11, 65M32

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zwwang6@163.com (Zewen Wang)

shfqiu@ecut.edu.cn (Shufang Qiu)

yush29@mail2.sysu.edu.cn (Shuang Yu)

binwu@nuist.edu.cn (Bin Wu)

zhangw@ecut.edu.cn (Wen Zhang)

  • BibTex
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  • TXT
@Article{JCM-41-173, author = {Wang , ZewenQiu , ShufangYu , ShuangWu , Bin and Zhang , Wen}, title = {Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {2}, pages = {173--190}, abstract = {

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter $\gamma$, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When $\gamma$ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of $\gamma$. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2107-m2020-0133}, url = {http://global-sci.org/intro/article_detail/jcm/21175.html} }
TY - JOUR T1 - Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation AU - Wang , Zewen AU - Qiu , Shufang AU - Yu , Shuang AU - Wu , Bin AU - Zhang , Wen JO - Journal of Computational Mathematics VL - 2 SP - 173 EP - 190 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2107-m2020-0133 UR - https://global-sci.org/intro/article_detail/jcm/21175.html KW - Exponential regularization method, Inverse source problem, Fractional diffusion equation, Ill-posed problem, Convergence rate. AB -

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter $\gamma$, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When $\gamma$ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of $\gamma$. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

Zewen Wang, Shufang Qiu, Shuang Yu, Bin Wu & Wen Zhang. (2022). Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation. Journal of Computational Mathematics. 41 (2). 173-190. doi:10.4208/jcm.2107-m2020-0133
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