Asymptotic Solutions of the System of Integro-Differential Equations with Retarded Argument
by
Domnytsky Vladimir

Many physical problems reduce to the determination of a solution of partial integro-differential equations with deviating argument. As is well known, finding exact solutions to equations of this type is extraordinarily difficult, therefore, approximation methods are employed. In the present article, making use the methods N.Bogolyubov-Yu.Mitropolskii, we study the problem of constracting an asymptotic solution of the systeme of partial integro-differential equations with delayed argument, with initial conditions and boundary conditions. We distinguish two types of solution, which we call the "Resonance" solution and "Nonresonance" solution. We show that the formal solutions have an asymptotic representation. For an applied mathematician the computation of an asymptotic expansion is often just the initial step in evaluating the wanted solution. We also investigate the computation of the solutions by numerical integration of the systems of integro-differetial equations.

Date received: March 14, 1999