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.

d’Alembert operator, inverse problem, integral equation, unbounded domain, Picard method, regularization method

Received: June 2, 2024; Accepted: July 16, 2024; Published: August 21, 2024

How to cite this article: T. D. Omurov and 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, Advances in Differential Equations and Control Processes 31(4) (2024), 473-486. https://doi.org/10.17654/0974324324025

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] J. Hadamard, Cauchy Problem for Linear Partial Differential Equations of Hyperbolic Type, Nauka, 1978, p. 352 (in Russian).
[2] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, 1976, p. 527 (in Russian).
[3] K. N. Dafermos, Quasilinear hyperbolic systems following from conservation laws, Nonlinear Waves, Seabass, Mir, Moscow, 1977, pp. 91-112 (in Russian).
[4] S. I. Kabanikhin, Inverse and ill-posed problems, Siberian Scientific Publishing House, Novosibirsk, 2009, p. 457 (in Russian).
[5] M. M. Lavrentiev, Regularization of operator equations of Volterra type, Problem of Mathematics, Physics, and Calculates, Nauka, 1977, pp. 199-205 (in Russian).
[6] A. Newell, Solitons in mathematics and physics, Transl. from English, Mir, 1989, p. 326 (in Russian).
[7] A. M. Nakhushev, Inverse problems for degenerate equations and Volterra integral equations of the third kind, Differential Equations 10(1) (1974), 100-111 (in Russian).
[8] T. D. Omurov, A. O. Ryspaev and M. T. Omurov, Inverse problems in applications of mathematical physics, Bishkek, 2014, p. 192 (in Russian).
[9] T. D. Omurov and M. M. Tuganbaev, Direct and inverse problems of single velocity transport theory, Bishkek, Ilim, 2010, p. 116 (in Russian).
[10] V. G. Romanov, Inverse Problems for Differential Equations, Novosibirsk State University, 1973, p. 252 (in Russian).
[11] I. I. Smulsky, Theory of interaction, Novosibirsk: From the Novosibirsk University, NSC OIGGM SB RAS, 1999, p. 294 (in Russian).
[12] A. N. Tikhonov and V. Y. Arsenin, Methods for Solving Ill-posed Problems, Nauka, 1986, p. 287 (in Russian).
[13] T. Tobias, On the inverse problem of determining the kernel of the hereditary environment, Izv. Academy of Sciences of the ESSR, Physics and Mathematics, 1984, pp. 182-187.
[14] V. A. Trenogin, Functional Analysis, Nauka, Moscow, 1980, p. 496 (in Russian).
[15] J. Whitham, Linear and Nonlinear Waves, Mir, Moscow, 1977, p. 622 (in Russian).