Elimination of the boundary condition singularity. A new approach to solving a system of nonlinear two-dimensional singularly perturbed differential equations

Abstract

A new method of equation formation for boundary layer series has been suggested. It enables to eliminate boundary condition singularity, to construct a complete asymptotics for problem solving and to substantiate it for the system of singularly perturbed elliptic equations given that Dirichlet boundary condition for one function and Neyman boundary condition for the other have been provided in the case of degenerate equation multiply root. Solution expansion was carried out on fractional degrees of a small parameter. Asymptotics contains two different types of boundary series. First type coefficients of boundary series exponentially decrease, coefficients of the second type have a different structure, a special benchmark function emerges in order to evaluate them. The boundary layer is divided into a few zones with different ways of proceeding for each zone. The constructed boundary layer series describe solution for all zones of boundary layer. The problem is solved without using splicing method.