AN INVERSE HYPERBOLIC NONLOCAL PROBLEM FOR SOURCE RECONSTRUCTION WITH AN INTEGRAL CONDITION

This paper investigates a linear inverse boundary value problem for a second-order hyperbolic equation with an unknown right-hand side depending on time. The problem is posed with initial conditions, a Neumann boundary condition, a nonlocal integral condition, and an additional condition. The aim of the study is to establish conditions for the existence and uniqueness of a classical solution to the inverse problem. Under the imposed zero-mean and compatibility conditions, the original problem with a nonlocal integral condition is reduced to an equivalent auxiliary problem with homogeneous Neumann boundary conditions at both endpoints and an additional relation, for which the uniqueness of the solution is established. The Fourier method is applied to investigate solvability. The solution of the hyperbolic equation is represented as a series in the corresponding system of eigenfunctions, after which the expansion coefficients are determined. Based on the obtained representations and estimates, the necessary properties of the unknown functions are established. It is proved that the obtained functions satisfy the differential equation, the initial conditions, the Neumann boundary condition, the nonlocal integral condition, and the additional condition in the classical sense. As a result, under the formulated requirements on the given functions and the necessary compatibility conditions, the original linear inverse boundary value problem for the considered second-order hyperbolic equation is shown to admit exactly one classical solution.