Asymptotics of the Solution of a Two-Point Boundary Value Problem With An Interior Layer

Abstract

This article examines a two-point boundary value problem for a singularly perturbed, linear, inhomogeneous ordinary differential equation. Singularly perturbed problems are frequently encountered in various fields of science – for example, in the study of vibrations of thin or flexible structures (such as beams and plates), in the modeling of fast–slow dynamic processes, and in optics and quantum physics, where the small parameter ε characterizes the wavelength or the energy scale. The distinguishing features of the singularly perturbed two-point boundary value problem considered in this work are the presence of a perturbation term – that is, a small parameter multiplying the derivative of the unknown function – and the existence of singular points of both equations of the system within the interval. These singularities give rise to two types of layers: the classical boundary layer and an internal layer. The solution of the system is represented as the sum of three components: a regular outer solution, a solution defined in the neighborhoods of the boundary points describing the classical boundary layers, and a solution valid in the neighborhoods of the singular points inside the interval [0,1], which describes the internal layers. The aim of the article is to construct a uniform asymptotic expansion of the solution to the singularly perturbed two-point boundary value problem on the interval [0,1].