On the uniqueness of solutions of two inverse problems for the subdiffusion equation

TL;DR

This paper investigates the uniqueness and existence of solutions for inverse problems involving a subdiffusion equation with Caputo derivatives, revealing that inverse problems can have unique solutions even when forward problems do not.

Contribution

It establishes new existence and uniqueness theorems for inverse problems of subdiffusion equations with non-local boundary conditions, highlighting the impact of the parameter lpha on solution properties.

Findings

Proved existence and uniqueness theorems for inverse problems.

Discovered that inverse problems can have unique solutions even when forward problems do not.

Analyzed the influence of the parameter lpha on solution existence and uniqueness.

Abstract

Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $\phi$ in the non-local boundary value problem $D_t^{\rho} u(t) + Au(t) = f(t)$ ($0 < \rho < 1, 0 < t \leq T$), $u(\xi) = \alpha u(0) + \phi$, ($\alpha$ is a constant and $0 < \xi \leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(\xi_1) = V$ is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant $\alpha$ on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution $u(t)$ was violated, while when solving the inverse problem for the same values of $\alpha$, the solution $u(t)$ became unique.