# On a boundary problem for the fourth order equation with the third derivative with respect to time

## DOI:

https://doi.org/10.31489/2023m4/30-40## Keywords:

Initial boundary problem, Fourier method, uniqueness, existence, eigenvalue, eigenfunction, functional series, absolute and uniform convergence## Abstract

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions *X *(*x*) and *Y *(*y*). To determine *X *(*x*),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0*,p*], and for a *Y *(*y*) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0*,q*]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.