Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations
This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet bound...
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MDPI AG
2025-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/13/2123 |
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| author | Yushan Li Yuxuan Yang Nanbo Chen |
| author_facet | Yushan Li Yuxuan Yang Nanbo Chen |
| author_sort | Yushan Li |
| collection | DOAJ |
| description | This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology. |
| format | Article |
| id | doaj-art-b949be6a46cd43db8da9a4c130474efe |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-b949be6a46cd43db8da9a4c130474efe2025-08-20T02:35:44ZengMDPI AGMathematics2227-73902025-06-011313212310.3390/math13132123Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion EquationsYushan Li0Yuxuan Yang1Nanbo Chen2School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, ChinaSchool of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, ChinaSchool of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, ChinaThis paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology.https://www.mdpi.com/2227-7390/13/13/2123Caputo fractional derivativefractional Laplacianinverse source problemmulti-term time–space fractional diffusion equationoptimal perturbation algorithm |
| spellingShingle | Yushan Li Yuxuan Yang Nanbo Chen Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations Mathematics Caputo fractional derivative fractional Laplacian inverse source problem multi-term time–space fractional diffusion equation optimal perturbation algorithm |
| title | Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations |
| title_full | Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations |
| title_fullStr | Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations |
| title_full_unstemmed | Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations |
| title_short | Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations |
| title_sort | identification of a time dependent source term in multi term time space fractional diffusion equations |
| topic | Caputo fractional derivative fractional Laplacian inverse source problem multi-term time–space fractional diffusion equation optimal perturbation algorithm |
| url | https://www.mdpi.com/2227-7390/13/13/2123 |
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