SPLITTING SOLUTIONS TO IRREGULARLY DGENOUS LINEAR SINGULARLY PERTURBATE EQUATIONS IN COMPLEX AREAS

Abstract

This article considers irregularly degenerate linear singularly perturbed equations in complex domains.As shown by previous studies, the asymptotic behavior of solutions of singularly perturbed equations in different parts of the considered domains has a different character of changes. Naturally, the problem arises of the possibility of splitting the solution into several component functions so that each of these functions is dominant in one of the considered parts. To solve this problem, in this article we consider a linear singularly perturbed first-order equation in the complex domain, the corresponding unperturbed equation, which has a fixed pole. By replacing the unknown function, the solution of the initial problem of the considered equation is presented as a sum of three functions. Using the level line of harmonic functions, the area is divided into several parts. The neighborhood of the pole is determined, and this neighborhood is excluded from the region under consideration. It was found out that a function having a pole does not allow close enough (along a small perimeter) to approach the pole, one of the functions determines boundary-layer lines and regions, and the third function determines regular regions (where the solution of the singularly perturbed equation tends to the solution of the unperturbed equation) and under the influence of the pole determines a new kind of boundary layer lines.