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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.