Asymptotic behavior of the solution of a singularly perturbed three-point boundary value problem with boundary jumps

Abstract

In this paper, the three-point boundary value problem is considered for the third-order linear differential equation with a small parameter with at the two highest derivatives when the roots of additional characteristic equation have negative signs. The aim of this paper is to bring asymptotic estimation of the solution of a singularly perturbed three-point boundary value problem with boundary jumps and the asymptotic convergence of the solution of a singularly perturbed initial value problem to the solution of an unperturbed initial value problem. In this paper the fundamental system of solutions, initial functions of a singularly perturbed homogeneous di�erential equation are constructed and their asymptotic estimates are obtained. An asymptotic behavior of the solution of the three-point boundary value problem at the points of initial jumps is established. A degenerate boundary-value problem is constructed. It is proved that the solution of the original singularly perturbed boundary value problem tends to the solution of the degenerate boundary value problem.