Resolvent Estimates for Boundary Value Problems on Large Intervals Via the Theory of Discrete Approximations

In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abst...

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Bibliographic Details
Published in:Numerical functional analysis and optimization 2007-05, Vol.28 (5-6), p.603-629
Main Authors: Beyn, Wolf-Jürgen, Rottmann-Matthes, Jens
Format: Article
Language:English
Subjects:
Boundary value problems
Hyperbolic-parabolic systems
Resolvent estimates
Theory of discrete approximations
Traveling waves
Unbounded domains
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Summary:In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic-parabolic equations. Furthermore, we show that our results apply to the FitzHugh-Nagumo system.
ISSN:0163-0563
1532-2467
DOI:10.1080/01630560701348475