Inverse problem for fractional order subdiffusion equation

TL;DR

This paper investigates the inverse problem of determining the unknown spatial component in a fractional subdiffusion equation with Caputo derivative, establishing conditions for existence and uniqueness of solutions under various sign conditions of the coefficient function.

Contribution

It provides new theoretical results on existence and uniqueness for the inverse problem in fractional subdiffusion equations, including necessary and sufficient conditions.

Findings

Existence and uniqueness proved when g(t) has a constant sign.

Conditions for existence when g(t) changes sign are established.

Additional integral condition ensures solvability of the inverse problem.

Abstract

The study examines the inverse problem of finding the appropriate right-hand side for the subdiffusion equation with the Caputo fractional derivative in a Hilbert space represented by $H$. The right-hand side of the equation has the form $g(t)f$ and an element $f\in H$ is unknown. If the sign of $g(t)$ is a constant, then the existence and uniqueness of the solution is proved. When $g(t)$ changes sign, then in some cases, the existence and uniqueness of the solution is proved, in other cases, we found the necessary and sufficient condition for a solution to exist. Obviously, we need an extra condition to solve this inverse problem. We take the additional condition in the form $\int\limits_0^Tu(t)dt=\psi$. Here $\psi $ is a given element, of $H$.