# Singularly perturbed linear oscillator with piecewise-constant argument

## DOI:

https://doi.org/10.26577/jmmcs-2018-1-480## Keywords:

piecewise-constant argument of generalized type, small parameter, singular perturbation## Abstract

The Cauchy problem for singularly perturbed linear differential equation the second order with

piecewise-constant argument is considered in the article. The definition of singularly perturbed

linear harmonic oscillator with piecewise-constant argument is given in the paper. The system of

fundamental solutions of homogeneous singularly perturbed differential equation with piecewiseconstant

argument are constructed according to the nonhomogeneous singularly perturbed differential

equation with piecewise-constant argument. With the help of the system of fundamental

solutions, the initial functions are constructed and their asymptotic representation are obtained.

By using the reduction method, the analytical formula of the solution of singularly perturbed the

initial value problem with piecewise-constant argument is obtained. In addition, the unperturbed

Cauchy problem is constructed according to the singularly perturbed Cauchy problem. The solution

of the unperturbed Cauchy problem is obtained. When the small parameter tends to the

zero, the solution of singularly perturbed the Cauchy problem with piecewise-constant argument

approaches the solution of the unperturbed Cauchy problem with piecewise-constant argument.

The theorem on the passage to the limit is proved.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*97*(1), 3–13. https://doi.org/10.26577/jmmcs-2018-1-480