This article is cited in 3 papers

Differential Equations and Mathematical Physics

On the question of the correctness of inverse problems for the inhomogeneous Helmholtz equation

K. B. Sabitov^ab, N. V. Martem'yanova<sup>c

a</sup> Samara State Technical University, Samara, 443100, Russian Federation
^b Samara State University of Social Sciences and Education, Samara, 443099, Russian Federation
^c Samara National Research University, Samara, 443086, Russian Federation

Abstract: In the rectangular domain, the initial-boundary value problem for the Helmholtz equation and its non-local modifications are studied and the inverse problems for finding its right-hand side are studied. The solutions of direct problems with nonlocal boundary conditions and inverse problems are constructed in explicit form as the sums of orthogonal series in the system of eigenfunctions of the one-dimensional Sturm–Liouville spectral problem. The corresponding uniqueness theorems for the solution of all set problems are proved. Sufficient conditions for boundary functions are established, which are guaranteed by the existence and stability theorems for the solution of the proposed new problem statements.

Keywords: Helmholtz equation, initial-boundary value problem, nonlocal problems, inverse problems, uniqueness, existence, series, stability, integral equations.

UDC: 517.956.6

MSC: 35R30, 35M13

Received: January 10, 2018
Revised: April 21, 2018
Accepted: June 11, 2018
First online: June 27, 2018

DOI: 10.14498/vsgtu1600

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