Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911478946201600 |
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| author | Ashurov, Ravshan Shamuratov, Damir |
| author_facet | Ashurov, Ravshan Shamuratov, Damir |
| contents | This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01833 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives Ashurov, Ravshan Shamuratov, Damir Analysis of PDEs 35R30, 35R11, 26A33, 33E12, 35K05 This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions. |
| title | Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives |
| topic | Analysis of PDEs 35R30, 35R11, 26A33, 33E12, 35K05 |
| url | https://arxiv.org/abs/2603.01833 |