Asymptotics of the Solution of a Two-Point Bundary-Value Problem with Single Points

Abstract

In many areas of science, complex problems are described by differential equations with a small parameter. One famous physicist is credited with the phrase: A phenomenon is not physical if it lacks a small parameter. A differential equation (ordinary or partial differential) with a small parameter at the highest derivative is called a singularly perturbed differential equation. Such equations arise in electrical and radio engineering, mechanics, hydraulic and aerodynamics, etc. The article is devoted to the construction of a complete expansion of the solution to a singularly perturbed two-point boundary value problem with two singular points on the boundaries of the segment under consideration. The solution is sought in the form of a sum of three functions that can be represented by asymptotic series. It is impossible to construct a uniform asymptotic expansion on a straight line, so an auxiliary function is introduced, with the help of which it is possible to construct an asymptotic expansion on the entire segment, including singular points.