TIKHONOV REGULARIZATION METHOD OF AN INVERSE SPACE-DEPENDENT SOURCE PROBLEM FOR A TIME-SPACE FRACTIONAL DIFFUSION EQUATION

Jing Li; Gongsheng Tong; Rouzi Duan; Shanlin Qin

doi:10.11948/20200397

2021 Volume 11 Issue 5

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Jing Li, Gongsheng Tong, Rouzi Duan, Shanlin Qin. TIKHONOV REGULARIZATION METHOD OF AN INVERSE SPACE-DEPENDENT SOURCE PROBLEM FOR A TIME-SPACE FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2387-2401. doi: 10.11948/20200397

Citation:

Jing Li, Gongsheng Tong, Rouzi Duan, Shanlin Qin. TIKHONOV REGULARIZATION METHOD OF AN INVERSE SPACE-DEPENDENT SOURCE PROBLEM FOR A TIME-SPACE FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(5): 2387-2401. doi: 10.11948/20200397

TIKHONOV REGULARIZATION METHOD OF AN INVERSE SPACE-DEPENDENT SOURCE PROBLEM FOR A TIME-SPACE FRACTIONAL DIFFUSION EQUATION

1.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China
2.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering
3.
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, China

Corresponding author: Email: lijingnew@126.com (J. Li)
Fund Project: Jing Li is supported by Hunan Provincial Natural Science Foundation of China (2021JJ30697) and the Scientific Research Project of the Hunan Provincial office of Education (20A022). Shanlin Qin is supported by the National Natural Science Foundation of China (11801543)

Abstract

Abstract

The aim of this paper is to identify a space-dependent source term in the time-space fractional diffusion equation with an initial-boundary data and an additional measurement data at the final time point. A series expression for the solution of the direct problem is used to transfer the inverse problem into the first type of Fredholm integral equation. Before solving the inverse problem, the uniqueness of its solution is proved. We then use the Tikhonov regularization method to deal with the integral equation and obtain a series expression for the regularized solution of the inverse problem. Moreover, according to the prior and the posterior regularization parameter selection rules, we prove the convergence rates of the regularization solution. Finally, we provide some numerical experiments to show the effectiveness of our method.
- Inverse space-dependent source problem /
- time-space fractional diffusion equation /
- Tikhonov regularization
MSC: 65M32, 65N20, 47A52

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References

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