Математичне моделювання коливних процесів у смузі

Abstract

The boundary-value periodic problem for differential equations in partial derivatives, including hyperbolic equations, are complicated and controversial subject to study. Boundary-value problems with data throughout the border region as well as the problem of non-local (including integrated) conditions for hyperbolic equations in limited areas are, generally speaking, relatively correct. Many authors link the solvability of such problems with the problem of small denominators and use the methods of nonlinear functional analysis, the theory of implicit functions, variation methods. Another authors use the analytical methods in the research of periodic boundary-value problems for the second order hyperbolic equations. They construct integral operators and search the solutions in specially defined spaces of continuously differentiated functions for specific cases of periodicity.In this paper we find an analytic formula of the function v(x, t), which is a solution of the boundary-value 2π-periodic time-varying problem in the class of odd functions for which f (t) = – f(π – t). The properties of this function are established and the estimates of the solution of the boundary-value 2π-periodic problem are given.The results of the study are used for mathematical modeling of oscillating processes described by the second order hyperbolic equations.On the basis of the found function v(x, t) we can draw conclusions about the behavior of the solution of the undisturbed equation (ε = 0, ε is a small parameter) in the study of the general nonlinear the second order hyperbolic equation by the asymptotic methods.