The Limiting Transition in Integro-Differential Equations Containing a Null Operator in the Differential Part and Several Rapidly Varying Kernels

Abstract

An integro-differential equation with a null operator in the differential part and several rapidly varying kernels is considered. The limiting transition of its solution with the small parameter tending to zero is studied. Earlier, a similar problem was considered for one rapidly varying kernel. In the case of several rapidly varying kernels, the investigation of this problem involves a rather subtle analysis of a regularized asymptotic solution, an algorithm for construction of which was previously developed.

An attempt to generalize directly the ideas of the study that dealt with one rapidly changing kernel is inefficient, because the presence of several spectral values of the integral operator’s kernels involves the need to essentially change Lomov’s regularization method algorithm, which becomes less observable. It is known that in studying the limit transition with a small parameter tending to zero in singularly perturbed problems, the asymptotics main term is usually constructed. For calculating this term in the case of integro-differential equations with a null operator in the differential part, it is necessary to solve the first two iteration problems arising in constructing the asymptotics, provided that there is a knowm right-hand side of the third iteration system. In the case of a single rapidly changing kernel, construction of such solution is quite straightforward, because the equivalent integro-differential system is of the second order. If there are several kernels, this order is greater than two, a circumstance that adds noticeably more difficulty to construction of solutions for the corresponding iterative problems. The basic ideas of the generalization and subtleties that arise in developing the corresponding regularization method algorithm are fully seen in the case of two rapidly varying kernels; this is why it is exactly this case that is presented in the article.

It is shown that the limit solution depends essentially on the number of integral operator spectral values, and that the exact solution of the initial problem tends to the limit solution on the entire considered time interval, a result that confirms the boundary layer lacking effect in the problem.