APPROXIMATION OF BOUNDARY VALUE PROBLEMS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY

In mathematical modeling of physical and technical processes, the evolution of which depends on prehistory, we arrive at differential equations with a delay. With the help of such equations it was possible to identify and describe new effects and phenomena in physics, biology, technology. An important task for differential-functional equations is to construct and substantiate finding approximate solutions, since there are currently no universal methods for finding their precise solutions. Of particular interest are studies that allow the use of methods of the theory of ordinary differential equations for the analysis of delay differential equations. Schemes for approximating differential-difference equations by special schemes of ordinary differential equations are proposed in the works N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy in various functional spaces. The purpose of this paper is to apply approximation schemes of differential-difference equations to approximation of solutions of boundary-value problems for integro-differential equations with a delay. The paper presents sufficient conditions for the existence of a solution of the boundary value problem for integro-differential equations with many delays. The scheme of its approximation by a sequence of boundary value problems for ordinary integro-differential equations is proposed and the conditions of its convergence are investigated. A model example is considered to demonstrate the given approximation scheme.