Exact solutions of initial boundary value problems for Sobolev type equations of elastic oscillations

Abstract

The article is devoted to the problem of the unique solvability of initial boundary value problems for partial differential equations. These initial boundary value problems are dissipative dynamic mathematical models from the field of linear elasticity. The considered equations are not resolved with respect to the highest time derivative. Therefore, these equations are referred to the so-called Sobolev type partial differential equations. The research of initial boundary value problems is carried out from general position in the form of the complete second order differential equation in abstract spaces. The theory of the distributions with values in Banach spaces and the concept of a fundamental solution of differential operator are used. Theorems of existence and uniqueness of the solutions of considered initial boundary value problems in the class of functions of finite smoothness with respect to time are proved. These solutions are obtained as an functional series of the Kummer confluent hypergeometric function.