On the stability of stationary solutions of a linear integro-differential equation. (2001)

• Record Type: Journal Article
• Authors: Dorogovtsev, A. Ya.; Trofimchuk, O. Yu.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 14:Number 2(2001)
• Page Start: 139
• Page End: 150
• Publication Date: 2001
• Subjects: stationary solutions -- singular perturbations -- telegraph equation -- time-stationary solutions -- asymptotic expansions; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953301000107
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation ϵ x ″ ( t ) + x ′ ( t ) = A x ( t ) + ∫ − ∞ t E ( t − s ) x ( s ) d s + ξ ( t ), t ∈ R containing a small parameter ϵ in Banach space B is considered. Here A ∈ ℒ ( B ) is a fixed operator, E ∈ C ( [ 0, + ∞ ), ℒ ( B ) ) and ξ is a stationary process. The asymptotic expansion of the stationary solution for equation (1) in the series on degrees of e is given. We have proved also the existence of a stationary with respect to time solution of the boundary value problem in B for a telegraph equation (6) containing the small parameter ϵ . The asymptotic expansion of this solution is also obtained.