Differential and Integral Equations

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

Sasun Yakubov and Yakov Yakubov

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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.

Article information

Source
Differential Integral Equations, Volume 17, Number 1-2 (2004), 53-72.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060472

Mathematical Reviews number (MathSciNet)
MR2035495

Zentralblatt MATH identifier
1164.34458

Subjects
Primary: 34G10: Linear equations [See also 47D06, 47D09]
Secondary: 35L20: Initial-boundary value problems for second-order hyperbolic equations 35L90: Abstract hyperbolic equations

Citation

Yakubov, Sasun; Yakubov, Yakov. An initial-boundary-value problem for hyperbolic differential-operator equations on a finite interval. Differential Integral Equations 17 (2004), no. 1-2, 53--72. https://projecteuclid.org/euclid.die/1356060472


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