## On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay

2014
Khusainov, Denys
Azizbayov, Elvin
##### Project
Partielle Differentialgleichungen
Preprint
##### Abstract
In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. Finally, the solution asymptotics as the delay parameter goes to zero is studied. In contrast to earlier works, no positivity conditions are imposed.
510 Mathematics
##### Keywords
functional-differential equations; harmonic oscillator; pure delay; well-posedness; solution representation
##### Cite This
ISO 690KHUSAINOV, Denys, Michael POKOJOVY, Elvin AZIZBAYOV, 2014. On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay
BibTex
@unpublished{Khusainov2014Cauch-29408,
year={2014},
title={On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay},
author={Khusainov, Denys and Pokojovy, Michael and Azizbayov, Elvin}
}

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Yes