Boundary problem for equations of high-even order

As it is well known, Fourier method is one of the classical methods of studying boundary value problems for second-order equations. Recently, the spectral method researchers have begun to use it not only for the construction of solutions of boundary value problems for higher-order equations, but also to justify the uniqueness of the solution.
In this paper we consider the boundary value problem in a rectangular region for a high even-order equation.
Using the method of energy integrals shows the unique solvability of the problem. The solution is sought by separation of variables (Fourier method) to give two-dimensional boundary value problems for ordinary differential equations.
According to the variable $y$ we have the problem on eigenvalues and eigenfunctions for a high even-order equation. The asymptotic behavior of the eigenvalues is taken. In order to obtain some necessary estimates, the spectral problem is reduced to an integral equation by constructing the Green's function. Next, the Bessel inequality is used. The paper also shows the possibility of expansion of boundary functions in the system of eigenfunctions.
Next, the boundary value problem is solved for an ordinary differential equation of even order in the variable $x$. The general solution of the differential equation is found. To find the unknown constants an algebraic equation is solved and an estimate for the decision itself and its derivatives is otained.
The formal solution of the boundary value problem is obtained in the form of an infinite series in eigenfunctions. To prove the uniform convergence of the last series composed of the partial derivatives, first using Cauchy–Bunyakovsky inequality, the series consisting of two variables is decomposed into two one-dimensional series, and estimates for the Fourier coefficients are used.