Closeness of Solutions for Singularly Perturbed Systems via Averaging

TL;DR

This paper investigates the behavior of singularly perturbed nonlinear differential equations with boundary-layer solutions, establishing bounds on how closely solutions of the perturbed system follow the averaged and boundary layer systems, with an error of order O(√ε).

Contribution

It provides new theoretical results on the closeness of solutions for singularly perturbed systems without requiring boundary-layer solutions to converge to an equilibrium.

Findings

Solutions stay close to averaged and boundary layer systems over finite intervals

Error bound of order O(√ε) for solution approximation

Applicable to systems with boundary layers not converging to equilibrium

Abstract

This paper studies the behavior of singularly perturbed nonlinear differential equations with boundary-layer solutions that do not necessarily converge to an equilibrium. Using the average of the fast variable and assuming the boundary layer solutions converge to a bounded set, results on the closeness of solutions of the singularly perturbed system to the solutions of the reduced average and boundary layer systems over a finite time interval are presented. The closeness of solutions error is shown to be of order O(\sqrt(\epsilon)), where \epsilon is the perturbation parameter.