We solve a Dirichlet boundary value problem for the Klein?Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.
Pelloni, B., & Pinotsis, D. A. (2008). The Klein–Gordon Equation in a Domain with Time-Dependent Boundary. Studies in Applied Mathematics, 121(3), 291-312. https://doi.org/10.1111/j.1467-9590.2008.00416.x