Initial-boundary-value problem for inhomogeneous mixed type fourth-order differential equation with power degeneration

In this paper an initial boundary value problem is studied for a fourth-order nonhomogeneous equation of mixed type with a degenerate parabolic part. Using the spectral analysis method, a criterion for the uniqueness of the solution to the problem was established and the solution is constructed as the sum of a Fourier series. The work also studied the spectral properties of the Samarskii - Ionkin type problem for an ordinary differential equation of the fourth order, found the eigenvalues and corresponding eigenfunctions, proved their completeness and basis property, and investigated the adjoint problem.