Boundary Value Problem for a Seventh Order Nonhomogeneous Partial Differential Equation

Abstract

In this paper, we consider a boundary value problem for a seventh order partial differential equation. It is used Samarskii–Ionkin type boundary value conditions on spatial variable
x. The non-self-adjoint spectral problem and adjoint spectral problem are studied. The systems
of eigenvalues and eigenfunctions are determined. By the biorthogonal systems of eigenvalues,
the Fourier series method of separation of variables is applied. Consequently, the unique solution of the boundary value problem is obtained in the form of Fourier series. Absolutely and
uniformly convergence of Fourier series is proved.