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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

DOI: 10.4208/jcm.2107-m2020-0133

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.

  • Keywords

Exponential regularization method, Inverse source problem, Fractional diffusion equation, Ill-posed problem, Convergence rate.

  • 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
  • RIS
  • 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.

Wang , ZewenQiu , ShufangYu , ShuangWu , Bin and Zhang , Wen. (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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