On the solvability of the initial-boundary value problem for a nonlocal hyperbolic equation

Abstract

In this paper, we consider a partial differential equation with involutively transformed arguments in a rectangular domain. The considered equation is a non-local analog of the second-order hyperbolic type equation. This equation is subject to initial-boundary conditions, and the order of the boundary operators exceeds the order of the equation. Questions of correctness of the considered problem are investigated. To solve the problem, the Fourier method is used, i.e. separation of variables method. The properties of eigenfunctions and eigenvalues of the corresponding spectral problem are studied. For the main problem under consideration, theorems on the uniqueness and existence of a solution are proved. When proving the theorem on the uniqueness of the solution, the problem under study is reduced to two auxiliary, homogeneous initial-boundary value problems for a classical equation of hyperbolic type. The resulting equations depend on the coefficients of the main equation and certain conditions are imposed on them. Further, using the completeness of the eigenfunctions of the auxiliary spectral problem, the solution of the main problem is sought in the form of a series in this system. For the unknown coefficients of the series, a system of ordinary differential equations with high-order boundary conditions is obtained. Solving these problems, we find an explicit form of the solution of the main problem under study.