Inverse problem for the abstract diffusion-wave equation with Caputo fractional derivative

TL;DR

This paper investigates the inverse problem of identifying a potential in a fractional wave equation with Caputo derivative, establishing conditions for its solvability, uniqueness, and stability using Fourier methods and fixed point theory.

Contribution

It introduces a novel approach to solve the inverse potential problem for fractional wave equations with Caputo derivatives, including existence, uniqueness, and stability results.

Findings

Proved local existence of solutions

Established uniqueness of the inverse problem solution

Demonstrated stability under certain conditions

Abstract

In this work, we study the inverse problem of determining a potential coefficient in an abstract wave equation that includes a lower-order term. The equation incorporates a time-fractional derivative in the Caputo sense, as well as a fractional power of an abstract operator defined on a Hilbert space. Using the Fourier decomposition method, we analyze the solvability of the direct problem. Leveraging the properties of the solution to the direct problem, we conduct an examination of the inverse problem. By applying the fixed point theorem within a suitable Banach space, we derive results concerning the local existence, uniqueness, and stability of the solution.