On the solvability of the duo-periodic problem for the hyperbolic equation system with a mixed derivative

Abstract

One of the main and most studied problems in the theory of second order hyperbolic equations is a periodic boundary value problem. To solve such problems apply Fourier method, method of successive approximations, methods of functional analysis, variational method, etc. The development of information technologies imposes new requirements on the developed methods, paying special attention to their constructibility. One of such constructive methods is the method of a parametrization proposed in the works of D. S.Dzhumabaev for solving two-point boundary-value problems for ordinary differential equations. In this paper, we consider a boundary value problem for both variables for a system of hyperbolic equations with a mixed derivative. To solve this problem, the notation is introduced and the periodic boundary value problem is reduced to an equivalent problem consisting of a family of periodic boundary value problems for an ordinary differential equation and an integral relation. To solve the problem obtained, the method of a parametrization is used. The application of this method allowed us to construct an algorithm for finding an approximate solution of the periodic boundary value problem for a system of hyperbolic equations with a mixed derivative. In addition, the coefficient conditions of convergence and feasibility of the proposed algorithm are obtained.