# 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.
Published

25.07.2005

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 57, no. 7, July 2005, pp. 912–921, https://umj.imath.kiev.ua/index.php/umj/article/view/3653.

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Section

Research articles