Asymptotics of Solutions of Boundary Value Problems for the Equation εy'' + xp(x)y' - q(x)y = f

Abstract

Uniform asymptotic expansions of solutions of two-point boundary value problems of Dirichlet, Neumann and Robin for a linear inhomogeneous ordinary differential equation of the second order with a small parameter at the highest derivative are constructed. A feature of the considered two-point boundary value problems is that the corresponding unperturbed boundary value problems for an ordinary differential equation of the first order has a regularly singular point at the left end of the segment. Asymptotic solutions of boundary value problems are constructed by the modified Vishik-Lyusternik-Vasilyeva method of boundary functions. Asymptotic expansions of solutions of two-point boundary value problems are substantiated. We propose a simpler algorithm for constructing an asymptotic solution of bisingular boundary value problems with regular singular points, and our boundary functions constructed in a neighborhood of a regular singular point have the property of "boundary layer", that is, they disappear outside the boundary layer.