Розв’язки крайової перiодичної задачi для неоднорiдного лiнiйного гiперболiчного рiвняння другого порядку

Abstract

We obtained the conditions of solvability of a boundary-value inhomogeneous periodic problem for particular values of the period ω on the base of a solution of the boundary-value periodic problem utt − uxx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). This solution is represented with the formula u(x, t) = u0(x, t) + u~(x, t). Here, u0(x, t) is the solution corresponding to the homogeneous periodic problem and u~(x, t) is a partial solution of the inhomogeneous equation where u~(x, t+ω) = u~(x, t). We demonstrate that the obtained solution contains the results that were found before.