T. D. Omurov, K. R. Dzhumagulov

REGULARIZATION OF THE INVERSE PROBLEM WITH THE D’ALEMBERT OPERATOR IN AN UNBOUNDED DOMAIN DEGENERATING INTO A SYSTEM OF INTEGRAL EQUATIONS OF VOLTERRA TYPE

Introduction

Regularization of the inverse problem with the d’alembert operator in an unbounded domain degenerating into a system of integral equations of volterra type. Regularization of the inverse problem for the d'Alembert operator in unbounded domains, reducing to Volterra integral equations. Crucial for wave processes and mathematical physics.

Abstract

In this paper, we study the inverse problem for the wave equation with the second-order d’Alembert operator in an unbounded domain in a space with a non-uniform metric. For physical applications, inverse problems for second-order partial differential equations are of particular interest. Such inverse problems are encountered in the study of wave processes, processes of electromagnetic interactions, as well as in various reduction processes. Moreover, if there are external acting forces with respect to the indicated equations that allow additional information about the solution of the original equations, then we obtain classes of inverse problems of a coefficient nature with the d’Alembert operator, which are of particular interest to scientists in this field, in which the results of this article are relevant. Also, the relevance of the problem under study is due to the fact that it is an inverse problem, where the sought quantities are the causes of some known consequences of a particular process. Whereas for direct problems, the methods for solving are well known. Thus, this paper provides a solution to the inverse problem of mathematical physics with a hyperbolic operator and generalizes existing results. Received: June 2, 2024Accepted: July 16, 2024

Review

This paper tackles a complex and highly relevant inverse problem for the wave equation, specifically employing the second-order d’Alembert operator in an unbounded domain with a non-uniform metric. The authors correctly highlight the significant interest in such problems across various fields, including wave processes and electromagnetic interactions, particularly when dealing with coefficient identification stemming from external forces. The emphasis on inverse problems, where the aim is to deduce causes from known effects, underscores the inherent difficulty and importance of the study compared to their direct counterparts. The paper purports to offer a generalization of existing results, suggesting a substantial theoretical contribution. The core methodological innovation, as indicated by the title, lies in the regularization of this inverse problem by transforming it into a system of integral equations of Volterra type. This approach is critical for inverse problems, which are typically ill-posed and require sophisticated regularization techniques to ensure stable and unique solutions. By mapping the problem to a system of Volterra equations, the authors likely aim to leverage the well-established theory and solution methods available for this class of integral equations, potentially simplifying the analysis and computation. This degeneration strategy appears to be the central mechanism through which the challenging aspects of the unbounded domain and non-uniform metric are addressed. Should the proposed regularization and transformation prove successful, this work offers a valuable theoretical framework for a challenging class of inverse problems in mathematical physics. The stated relevance to diverse physical applications suggests potential utility beyond pure theory. To fully appreciate its impact, future considerations might include detailed analytical solutions for specific cases, numerical demonstrations of the Volterra system's tractability, and a thorough analysis of the uniqueness and stability of the obtained solutions. Overall, this paper presents a promising new theoretical pathway for analyzing and solving complex hyperbolic inverse problems, contributing to both mathematical theory and its application in physical sciences.

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