Construction of asymptotics of the solution for a system of singularly perturbed equations by the method of essentially singular functions

Abstract

Singularly perturbed problems with turning points arise as mathematical models for various physical phenomena. The internal turning point problem is a one-dimensional version of the steady-state convection-diffusion problem with a dominant convective term and a velocity field that changes sign in the reservoir. Boundary turning point problems, on the other hand, arise in geophysics and in the modeling of thermal boundary layers in laminar flow. The paper analyzes the results from the asymptotic analysis of singularly perturbed problems with turning points. For a homogeneous system of singularly perturbed differential equations with a small parameter at the highest derivative and a turning point, the conditions for constructing a uniform asymptotic solution are obtained. We consider the case when the spectrum of the limit operator contains multiple and identically zero elements. The asymptotics are constructed by the method of essentially singular functions, which allows using the Airy model operator in the vicinity of the turning point. The construction of asymptotic solutions contains arbitrary constants, which are determined uniquely during the solution of the iterative equations. At the same time, the conditions for the existence of a solution of a system of differentials with a small parameter for the highest derivative and for the presence of a turning point are obtained, provided that the turning point is located on the interval [0; l]. An example of constructing the asymptotic of a homogeneous system of differential equations is given.