Forward and backward problems for coupled subdiffusion systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912539430879232 |
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| author | Feng, Dian Liu, Yikan Lu, Shuai |
| author_facet | Feng, Dian Liu, Yikan Lu, Shuai |
| contents | In this article, we investigate both forward and backward problems for coupled systems of time-fractional diffusion equations, encompassing scenarios of strong coupling. For the forward problem, we establish the well-posedness of the system, leveraging the eigensystem of the corresponding elliptic system as the foundation. When considering the backward problem, specifically the determination of initial values through final time observations, we demonstrate a Lipschitz stability estimate, which is consistent with the stability observed in the case of a single equation. To numerically address this backward problem, we refer to the explicit formulation of Tikhonov regularization to devise a multi-channel neural network architecture. This innovative architecture offers a versatile approach, exhibiting its efficacy in multidimensional settings through numerical examples and its robustness in handling initial values that have not been trained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Forward and backward problems for coupled subdiffusion systems Feng, Dian Liu, Yikan Lu, Shuai Analysis of PDEs Numerical Analysis 35R11, 35K58, 35B44 In this article, we investigate both forward and backward problems for coupled systems of time-fractional diffusion equations, encompassing scenarios of strong coupling. For the forward problem, we establish the well-posedness of the system, leveraging the eigensystem of the corresponding elliptic system as the foundation. When considering the backward problem, specifically the determination of initial values through final time observations, we demonstrate a Lipschitz stability estimate, which is consistent with the stability observed in the case of a single equation. To numerically address this backward problem, we refer to the explicit formulation of Tikhonov regularization to devise a multi-channel neural network architecture. This innovative architecture offers a versatile approach, exhibiting its efficacy in multidimensional settings through numerical examples and its robustness in handling initial values that have not been trained. |
| title | Forward and backward problems for coupled subdiffusion systems |
| topic | Analysis of PDEs Numerical Analysis 35R11, 35K58, 35B44 |
| url | https://arxiv.org/abs/2407.00588 |