Finding of periodic solutions of the ordinary nonlinear second order differential equation with the delay

Authors

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2016.4.13

Keywords:

periodic solutions, nonlinear delayed differential equations, periodic boundary problem, the Green function, self-adjoint differential operator

Abstract

The work suggests an approach to finding of periodic solutions of the nonlinear delayed second order differential equations. There exists a numerical-analytical method that is generalized for delayed equations and whose idea is to reduce the equation to the system of the first order. The suggested approach explores the equation itself without its reduction to the system of the first order. The Green function for the self-adjoint differential operator of the second derivative is built, that is defined on functions that satisfy periodic boundary conditions. The necessary and sufficient existence conditions of the periodic equation solutions are given. The estimation for the rate of convergence of the method of approximate calculations is obtained.

Author Biography

Yuriy I. Bokhonov, ESC "Institute for Applied System Analysis" of NTUU "Kyiv Polytechnic Institute, Kyiv, Ukraine

Yuriy Bokhonov,

Candidate of Physical-Mathematical Sciences (Ph.D.), associate professor at ESC "Institute for Applied System Analysis" of NTUU "Kyiv Polytechnic Institute, Kyiv, Ukraine.

 

References

Samojlenko A.M. O periodicheskih reshenijah nelinejnyh uravnenij vtorogo porjadka / A.M. Samojlenko // Differentsial'nye uravnenija. — 1967. — 3, № 11. — S. 1903–1912.

Ronto N.I. Teorija chislenno-analiticheskogo metoda, dostizhenija i novye napravlenija razvitija / N.I. Ronto, A.M. Samojlenko, S.I. Trofimchuk // Ukr. mat. zhurn. — 1998. — 50, № 2. — S. 225–243.

Samojlenko A.M. Chislenno-analiticheskie metody v teorii kraevyh zadach obyknovennyh differentsial'nyh uravnenij / A.M. Samojlenko, N.Y. Ronto. — K.: Nauk. dumka, 1992. — 280 s.

Mitropol'skij Ju.A. Periodicheskie i kvaziperiodicheskie kolebanija sistem s zapazdyvaniem / Ju.A. Mitropol'skij, D.I. Martynjuk. — K.: Vyscha shk., 1979. — 248 s.

Mitropol'skij Ju.A. Sistemy evoljutsionnyh uravnenij s periodicheskimi i uslovno-periodicheskimi koeffitsientami / Ju.A. Mitropol'skij, A.M. Samojlenko, D.I. Martynjuk. — K.: Nauk. dumka, 1984. — 213 s.

Bokhonov Ju.Ye. Pro odyn pidkhid do znakhodzhennja periodychnykh rozv’jazkiv nelinijnoho zvychajnoho dyferentsial'noho rivnjannja druhoho porjadku. // Systemni doslidzhennja ta informatsijni tekhnolohiyi. —2012. — № 2. — S. 138–143.

Horn R. Matrichnyj analiz / R. Horn, Ch. Dzhonson. — M.: Mir, 1989. — 655 s.

Downloads

Published

2016-12-15

Issue

No. 4 (2016)

Section

Mathematical methods, models, problems and technologies for complex systems research

License

This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author. This is in accordance with the BOAI definition of open access.