On the conditions of the solvability of boundary value problems for a single high-order equation with variable coefficients

TL;DR

This paper investigates boundary value problems for high-order equations with variable coefficients, providing criteria for solution uniqueness, conditions for Fourier series convergence, and factors affecting solvability.

Contribution

It introduces new conditions ensuring the solvability and uniqueness of solutions for high-order variable coefficient equations with boundary conditions.

Findings

Criteria for solution uniqueness are established.

Conditions for Fourier series convergence are derived.

Solvability depends on domain size and boundary derivative orders.

Abstract

A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of the series, the problem of small denominators arises. Sufficient conditions for the separability of the denominator from zero are obtained. It is shown that the solvability of the problem is influenced not only by the dimension of the rectangle, but also by the orders of the specified derivatives at the lower boundary of the rectangle.