NEW STATEMENT OF BOUNDARY-VALUE 2Π-PERIODIC PROB-LEM FOR THE HYPERBOLIC SECOND ORDER EQUATION IN THE ASYMPTOTIC THEORY OF NONLINEAR OSCILLATIONS - Khoma, N. H.; Khoma-Mohylska, S. H.; Khokhlova, L. H. (Computer Science and Applied Mathematics, 2020)
Until now the hyperbolic second order equations with small parameter in right side have been researched by asymptotic methods of Krylov–Boholiubov–Mytropolskyi–Moseienkov provided that undisturbed ( ) equation had solution as trigonometric Fourier series. With the assumption of the parameter is very small these methods allow to build an approximate solution of the boundary-value periodic problem for hyperbolic second order equation in which right side has small parameter and left side is formed by the operator of D’Alembert. In the process of research a logical question arises under which conditions the undisturbed ( ) equation has solution as trigonometric Fourier series. Our work is devoted to establishment of such conditions. In the first part of the article we consider the undisturbed equation in the left part of which there is the D’Alembert operator and on the right side there is an arbitrary function . Using the operator method, a formal solution of this equation is constructed. The theorems and lemmas that establish the conditions for the existence of a classical solution to the boundary-value periodic in x problem for an undisturbed equation are proved. The specific class of functions in which the above problem has a classical solution is defined. The subclass of functions in which the classical solution to the problem is an odd function and hence taking into account -periodicity decomposes into a trigonometric sine series is selected. The obtained results make it possible to construct an approximate solution to a quasilinear boundary-value -periodic in problem for a hyperbolic second order equation right side of which is a function eF (x, t, u) with small parameter . In the second part of the article the scheme of constructing an approximate solution is given. As a zero approximation, we take the representation of the classical solution to the boundary-value -periodic in problem for the undisturbed equation established in the first part of the article.