Two regularization methods for identifying the unknown source term of space-time fractional diffusion-wave equation

Huimin Heng; Fan Yang; Xiaoxiao Li; Zhenji Tian; Huimin Heng; Fan Yang; Xiaoxiao Li; Zhenji Tian

doi:10.3934/math.2025822

AIMS Mathematics

2025, Volume 10, Issue 8: 18398-18430. doi: 10.3934/math.2025822

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Research article

Two regularization methods for identifying the unknown source term of space-time fractional diffusion-wave equation

School of Science, Lanzhou University of Technology, Lanzhou 730050, Gansu, China

Received: 04 May 2025 Revised: 16 July 2025 Accepted: 24 July 2025 Published: 14 August 2025
MSC : 35R25, 35R30, 47A52

In this paper, the inverse problem of identifying the unknown source for the space-time fractional diffusion-wave equation is researched. This problem is ill-posed and needs the regularization approach to solve this inverse problem. The fractional Tikhonov regularization method and the quasi-inverse regularization method are used to obtain the fractional Tikhonov regularization solution and the quasi-inverse regularization solution, respectively. Under the a priori and the a posteriori regularization parameter selection rules, the error estimates of the regularization solutions and the exact solution are given. Finally, we provide several numerical examples to show the effectiveness of the approach.
- space-time fractional diffusion-wave equation,
- ill-posed,
- fractional Tikhonov regularization method,
- quasi-inverse regularization method
Citation: Huimin Heng, Fan Yang, Xiaoxiao Li, Zhenji Tian. Two regularization methods for identifying the unknown source term of space-time fractional diffusion-wave equation[J]. AIMS Mathematics, 2025, 10(8): 18398-18430. doi: 10.3934/math.2025822

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Abstract

In this paper, the inverse problem of identifying the unknown source for the space-time fractional diffusion-wave equation is researched. This problem is ill-posed and needs the regularization approach to solve this inverse problem. The fractional Tikhonov regularization method and the quasi-inverse regularization method are used to obtain the fractional Tikhonov regularization solution and the quasi-inverse regularization solution, respectively. Under the a priori and the a posteriori regularization parameter selection rules, the error estimates of the regularization solutions and the exact solution are given. Finally, we provide several numerical examples to show the effectiveness of the approach.

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