Inverse problem for subdiffusion equation with fractional Caputo derivative

R. R. Ashurov   ; M. D. Shakarova

Ufa mathematical journal, Tome 16 (2024) no. 1, pp. 112-126

Résumé

We consider an inverse problem on determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is the function $f(x)$. The condition $ u (x,t_0)= \psi (x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considered domain and $\psi (x) $ is a given function. By the Fourier method we show that under certain conditions on the functions $g(t)$ and $\psi (x) $ the solution of the inverse problem exists and is unique. We provide an example showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. For such functions $g(t)$ we find necessary and sufficient conditions on the initial function and on the function from the over-determination condition, which ensure the existence of a solution to the inverse problem.

Détail

Keywords: forward and inverse problems, the Caputo derivatives, Fourier method.
Mots-clés : subdiffusion equation

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