Convergence in the Boundary Layer for Nonhomogeneous Linear Singularly Perturbed Systems

TL;DR

This paper investigates how solutions of nonhomogeneous linear singularly perturbed systems converge to the reduced system, using a distributional approach to include boundary layer effects near the initial time.

Contribution

It introduces a distributional method to analyze boundary layer convergence in singularly perturbed systems, providing explicit analytical expressions for the limits.

Findings

Established convergence of solutions to the reduced system

Derived explicit distributional limit expressions

Included boundary layer effects in the analysis

Abstract

Convergence of the solutions of nonhomogeneous linear singularly perturbed systems to that of the corresponding reduced singular system on the half-line [0, $\infty $) is considered. To include the situation on a neighborhood of initial instant, a boundary layer, a distributional approach to convergence is adopted. An explicit analytical expression for the limit as a distribution is proved.