MATHEMATICS

On the solvability of the inverse problem for the wave equation with memory and acoustic boundary conditions

Zh. D. Totieva

Southern Mathematical Institute, Vladikavkaz Scientific Centre of the Russian Academy of Sciences, ul. Vatutina, 53, Vladikavkaz, 362025, Russia

Abstract: The one-dimensional inverse problem of determining the kernel of the integral convolution operator in the wave equation on a segment for media with dispersion is considered. The direct problem is an initial-boundary value problem of simultaneously finding the velocity potential and the displacement of the boundary points of the medium. The acoustic control conditions are used as boundary conditions. An integral overdetermination condition is specified as additional information for setting the inverse problem. The inverse problem is reduced to an equivalent problem of studying the solvability of a closed system of integro-differential equations of the Volterra type with zero boundary conditions. Using the technique of estimating integral operators and the principle of contractive mappings in Sobolev spaces, a local theorem of existence and uniqueness of a solution to the inverse problem is proved.

Keywords: wave equation, acoustic boundary condition, inverse problem, memory kernel

UDC: 517.958

MSC: 35L20, 35R30, 35Q99

Received: 21.02.2025
Accepted: 10.09.2025

DOI: 10.35634/vm250306

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