Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

Abstract

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small  parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is  to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where  additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial  perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed  homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions  and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic  estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a  small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem  under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A  modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.