Methods for Studying Asymptotics of Solutions to Singularly Perturbed Equations in Complex Domains

Abstract

When studying the asymptotic behavior of solutions to singularly perturbed equations with analytic functions, various methods are used. The main goal of this work is to conduct a comprehensive analysis of the methods used and divide them into groups. To achieve this goal, a scalar, nonlinear first-order equation with an initial condition is considered. The equation is considered in a certain circle, the complex plane of the independent variable, and the real part of the function — the coefficient, for a linear unknown function, changes its sign from negative to positive on part of the segment of the real axis contained in the circle. The concept of the domain of attraction of the solution of a singularly perturbed equation to the solution of a nonlinear equation (degenerate equation) is introduced. The task is set to prove the existence of a region of attraction. In the process of solving the problem, the methods used are divided into three groups.