Inverse problems for a generalized fractional diffusion equation with unknown history
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911768974983168 |
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| author | Janno, Jaan |
| author_facet | Janno, Jaan |
| contents | Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are known for $t\in (t_0,T)$ where $t_0$ is some number in $(0,T)$. Provided that $f$ satisfies certain restrictions, it is proved that product of a kernel of the derivative with an elliptic operator as well as the history of $f$ for $t\in (0,t_0)$ are uniquely recovered. In case of less restrictions on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover, in a case when a functional of $u$ for $t\in (t_0,T)$ is given the uniqueness of the kernel is proved under unknown history of $f$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inverse problems for a generalized fractional diffusion equation with unknown history Janno, Jaan Mathematical Physics Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are known for $t\in (t_0,T)$ where $t_0$ is some number in $(0,T)$. Provided that $f$ satisfies certain restrictions, it is proved that product of a kernel of the derivative with an elliptic operator as well as the history of $f$ for $t\in (0,t_0)$ are uniquely recovered. In case of less restrictions on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover, in a case when a functional of $u$ for $t\in (t_0,T)$ is given the uniqueness of the kernel is proved under unknown history of $f$. |
| title | Inverse problems for a generalized fractional diffusion equation with unknown history |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2402.00482 |