An initial-boundary-value problem for hyperbolic differential-operator equations on a finite interval

Abstract

In this paper we give, for the first time, an abstract interpretation of initial--boundary-value problems for hyperbolic equations such that a part of the boundary-value conditions contains also a differentiation of the time $t$ of the same order as the equations. Initial--boundary-value problems for hyperbolic equations are reduced to the Cauchy problem for a system of hyperbolic differential-operator equations. A solution of this system is not a vector function but one function. At the same time, the system is not overdetermined. We prove the well-posedness of the Cauchy problem, and for some special cases we give an expansion of a solution to the series of eigenvectors. As application we show, in particular, a generalization of the classical Fourier method of separation of variables.