Asymptotic behavior of the solution of the boundary value problem for a singularly perturbed system of the integro-differential equations

Abstract

In this paper, we study the asymptotic behavior of solutions to the boundary value problem for singularly perturbed systems of integro-differential equations. The aim of the work is to obtain an analytical formula, an asymptotic estimate of the solution of a boundary value problem, and to determine the asymptotic behavior of the solution by a smaller parameter at the starting point. The boundary value problem given in the paper is reduced to a boundary value problem posed in a singularly perturbed integral-differential equation of mixed type with respect to a fast variable. The Cauchy function, boundary functions and Green’s function of a singularly perturbed homogeneous differential equation are obtained, and their asymptotic estimates are also determined. With the help of these constructed functions, an analytical formula and an asymptotic estimate of this solution of the boundary value problem are obtained. The asymptotic behavior of the solution with respect to a small parameter is determined and the order of growth of its derivatives at the left point of a given segment is shown. It is established that the solution of the boundary value problem under consideration has an initial jump of zero order at the initial point.