Inverse problem for the subdiffusion equation with fractional Caputo derivative

TL;DR

This paper investigates the inverse problem of identifying a spatial function in a subdiffusion equation with a fractional Caputo derivative, proving existence and uniqueness under certain conditions and exploring cases of non-uniqueness.

Contribution

It provides a Fourier method-based proof of existence and uniqueness for the inverse problem and analyzes conditions leading to non-uniqueness for sign-changing functions.

Findings

Existence and uniqueness are established under specific conditions.

Non-uniqueness occurs for certain sign-changing functions g(t).

Orthogonality conditions are necessary for solutions in some cases.

Abstract

The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is 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 considering domain and $\psi (x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $\psi (x) $ the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. It is shown that for the existence of a solution to the inverse problem for such functions $g(t)$, certain orthogonality conditions for the given functions and some eigenfunctions of the elliptic part of the equation must be satisfied.