Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I - Mitropolskiy, Yu. A.; Khoma-Mohyl's'ka, S. H. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
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.