ON THE SOLVABILITY OF CERTAIN INITIAL-BOUNDARY VALUE PROBLEMS FOR A NONLOCAL HYPERBOLIC EQUATION
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Keywords:
involution, nonlocal operator, hyperbolic equation, initial-boundary value problem, existence, uniqueness, Dirichlet condition, Neumann conditionAbstract
This paper investigates the well-posedness of initial-boundary value problems for a nonlocal analogue of the hyperbolic equation. The elliptic part of the considered equations involves a nonlocal analogue of the Laplace operator. We find the eigenfunctions and eigenvalues of boundary value problems for a nonlocal analogue of the Laplace operator. The eigenfunctions of the problems are represented as even and odd parts with respect to the considered mapping. The symmetry properties of the eigenfunctions of boundary value problems with Dirichlet and Neumann conditions are investigated. These properties are further applied in obtaining solutions to the main problems. The equations we consider fall into the class of differential equations with transformed arguments. The problems are considered in an -dimensional cylindrical domain with a spherical base. The boundary conditions include Dirichlet and Neumann conditions. The problems are solved by reducing them to equivalent problems for the classical hyperbolic equation. Using known results for initial-boundary value problems for the classical hyperbolic equation, theorems on the existence and uniqueness of solutions are proved. It is shown that the well-posedness of the problems under consideration significantly depends on the coefficients involved in defining the nonlocal Laplace operator. The solutions of the problems are constructed as a series.
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