Time-dependent identification problem for a fractional Telegraph equation with the Caputo derivative

TL;DR

This paper addresses the first known inverse problem for a fractional telegraph equation with Caputo derivatives, establishing existence, uniqueness, and stability of solutions in a Hilbert space setting.

Contribution

It introduces a novel inverse problem for a fractional telegraph equation with Caputo derivatives and proves its well-posedness.

Findings

Existence and uniqueness of solutions are proven.

Stability inequalities for the inverse problem are derived.

The problem is formulated for a general self-adjoint positive operator.

Abstract

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form $(D_{t}^{\rho})^{2}u(t)+2\alpha D_{t}^{\rho}u(t)+Au(t)=p( t)q+f(t)$, where $0<t\leq T$, $0<\rho<1$ and $D_{t}^{\rho}$ is the Caputo derivative. The equation contains a self-adjoint positive operator $A$ and a time-varying multiplier $p(t)$ in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition $B[u(t)] = \psi(t)$ is imposed, where $B$ is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator $A$ and the functional $B$ are discussed.