2019
Том 71
№ 11

# Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

Abstract

We consider the periodic boundary-value problem $u_{tt} − u_{xx} = g(x, t),\; u(0, t) = u(π, t) = 0,\; u(x, t + ω) = u(x, t)$. By representing a solution of this problem in the form $u(x, t) = u^0(x, t) + ũ(x, t)$, where $u^0(x, t)$ is a solution of the corresponding homogeneous problem and $ũ(x, t)$ is the exact solution of the inhomogeneous equation such that $ũ(x, t + ω) u_x = ũ(x, t)$, we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier.

English version (Springer): Ukrainian Mathematical Journal 57 (2005), no. 7, pp 1077-1088.

Citation Example: Khoma-Mohyl's'ka S. H., Mitropolskiy Yu. A. Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I // Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 912–921.

Full text