Solvability of the Dirichlet problem

Abstract

In this paper, we consider a partial differential equation with mixed derivatives — first-order derivatives with respect to time and second-order derivatives with respect to the spatial variable. Such equations are usually referred to as one-dimensional pseudo-parabolic equations. We prove the existence and uniqueness of a classical solution to problems for a pseudo-parabolic equation involving a secondorder differential operator with a pure involution, under certain conditions imposed on the initial data of the problem. The applicability of the Fourier method is based on the Riesz basis property of the eigenfunctions of the considered second-order differential operator with a pure involution. The presence of the Bessel inequality for the Fourier coefficients facilitates the proof of the uniform convergence of the differentiated Fourier series. The solutions are obtained explicitly in the form of Fourier series. Such representations can be used for numerical computations.