Asymptotic expansion of the solution of a singularly perturbed boundary value problem with boundary initial jumps

Abstract

In this paper we consider a singularly perturbed general boundary value problem for a third - order differential equation in the conditionally stable case. The form of the asymptotics of the desired solution is determined with the help of the established asymptotic estimates for the solution of the singularly perturbed boundaryvalue problem under study. An algorithm is described with which all the terms of the asymptotic expansion for the problem under consideration are determined successively. Exponential estimates for boundary functions are established. An asymptotic accuracy estimate is obtained that is obtained by the partial sum of the asymptotic expansion of the solution of the singularly perturbed general boundary value problem under consideration. A theorem on the existence, uniqueness, and validity of the asymptotic expansion of the solution of the boundary value problem is proved. The questions of the limiting transition of the solution of the perturbed problem to the solution of the unperturbed problem are studied with the small parameter tending to zero, the existence of the boundary jump phenomenon. Formulas for boundary jumps, and the order of jumps are found.