Inverse problem for wave equation of memory type with acoustic boundary conditions: Global solvability

TL;DR

This paper addresses the inverse problem of identifying the memory kernel in a wave equation with acoustic boundary conditions, establishing conditions for global solvability and uniqueness.

Contribution

It introduces a method to determine the memory kernel in a wave equation with acoustic boundary conditions, proving global existence and uniqueness of solutions.

Findings

Proved global existence of solutions for the inverse problem.

Established uniqueness of the memory kernel determination.

Applied contraction mappings in Sobolev spaces for the analysis.

Abstract

In this article, we study the one-dimensional inverse problem of determining the memory kernel by the integral overdetermination condition for the direct problem of finding the velocity potential and the displacement of boundary points. A wave equation with initial and acoustic boundary conditions in media with dispersion is used as a mathematical model. The inverse problem is reduced to an equivalent problem with homogeneous boundary conditions for the system of integro-differential equations. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the global existence and uniqueness theorem for the inverse problem is proved.