Open AccessAVERAGING IN MULTIFREQUENCY SYSTEMS WITH DELAY AND LOCAL INTEGRAL CONDITIONS
Author(s) -
Ya. J. Bihun,
I. D. Skutar
Publication year - 2020
Publication title -
bukovinsʹkij matematičnij žurnal
Language(s) - English
Resource type - Journals
ISSN - 2309-4001
DOI - 10.31861/bmj2020.02.02
Subject(s) - mathematics , commensurability (mathematics) , integral equation , method of averaging , interval (graph theory) , variable (mathematics) , mathematical analysis , physics , pure mathematics , nonlinear system , combinatorics , quantum mechanics
Multifrequency systems of dierential equations were studied with the help of averagingmethod in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenkoand many other scientists. The complexity of the study of such systems is their inherent resonantphenomena, which consist in the rational complete or almost complete commensurability offrequencies. As a result, the solution of the system of equations averaged over fast variables inthe general case may deviate from the solution of the exact problem by the quantity O (1). Theapproach to the study of such systems, which was based on the estimation of the correspondingoscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works byA.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systemswith initial , boundary and integral conditions.For multifrequency systems with an argument delay, the averaging method is substantiatedin the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors.In this paper, the averaging method is used to study the solvability of a multifrequencysystem with an arbitrary nite number of linearly transformed arguments in slow and fastvariables and integral conditions for slow and fast variables on parts of the interval [0, L] ofthe system of equations. An unimproved estimate of the error of the averaging method underthe superimposed conditions is obtained, which clearly depends on the small parameter andthe number of linearly transformed arguments in fast variables.