Resolution of the boundary value problem for a mixed equation of the fourth order

The study of boundary value problems for high-order partial differential equations plays an important role, because many scientific and practical studies lead to boudary value problems for fourth-order partial differential equtions. In this work in a rectangular region for a fourth-order equation, a boundary value problem is considered. A criterion for the uniqueness and existence of a solution to a boundary value problem for a fourth-order equation is established. The solution is constructed as the sum of a series in terms of eigenfunctions of the corresponding spectral problem. The stability of the solution of this problem is proved.