Mixed Boussinesq-type differential equation

The problems of solvability and construction of the solution of a non-local mixed problem for a homogeneous mixed differential equation of Boussinesq type are considered. Used spectral method based on the separation of variables. The solution of the problem is represented as a Fourier series with separated variables. The criterion of uniqueness of the solution is established. When this criterion is fulfilled, the unique solvability of the problem is proved. When the criterion of uniqueness is violated, the solution of this problem under certain conditions is represented as the sum of Fourier series.