Abstract
The averaging method is justified for normal systems of differential equations with rapidly oscillating summands proportional to the square root of the oscillation frequency in the case of the boundary-value problem on a finite interval and for the problem of bounded solutions on the positive semiaxis with boundary condition at its left endpoint.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. N. Bogolyubov, On Statistical Methods in Mathematical Physics (Izd. AN UkrSSR, Lvov, 1945) [in Russian].
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations (Nauka, Moscow, 1974).
S. K. Godunov, Ordinary Differential Equations with Constant Coefficients (Izd. Novosibirsk. Univ., Novosibirsk, 1994), Vol. 1 [in Russian].
V. B. Levenshtam, “Asymptotic integration of differential equations with oscillatory terms of large amplitudes. I,” Differ. Uravn. 41 (6), 761–770 (2005) [Differ. Equations 41 (6), 797–807 (2005)].
V. B. Levenshtam, “Asymptotic integration of differential equations with rapidly oscillating terms of large amplitude. II,” Differ. Uravn. 41 (8), 1084–1091 (2005) [Differ. Equations 41 (8), 1137–1145 (2005)].
V. B. Levenshtam, “Asymptotic integration of differential equations with large high-frequency terms,” Dokl. Ross. Akad. Nauk 405 (2), 169–172 (2005) [Dokl. Math. 72 (3), 872–875 (2005)].
V. B. Levenshtam and G. L. Khatlamadzhiyan, “Extension of the averaging theory to differential equations with large-amplitude rapidly oscillating terms. The problem of periodic solutions,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 6, 35–47 (2006).[Russian Math. (Iz. VUZ) 50 (6), 33–45 (2006).]
V. I. Yudovich, “Vibration dynamics of systems with constraints,” Dokl. Ross. Akad. Nauk 354 (2), 622–624 (1997) [Phys. Dokl. 42 (6), 322–325 (1997)].
V. I. Yudovich, “Vibration dynamics and vibration geometry of mechanical systems with constraints,” UspekhiMekhaniki 4 (3), 26–158 (2006).
V. B. Levenshtam and P. E. Shubin, “Averaging evolution system with boundary conditions,” Sci. Publ. of the State Univ. of Novi Pazar Ser. A: Appl. Math., Inform. and Mech. 5 (2), 61–67 (2013).
V. B. Levenshtam and P. E. Shubin, “Justification of the averaging method for the boundary value problems on a finite or semi-infinite interval,” Sci. Publ. of the State Univ. of Novi Pazar Ser. A: Appl. Math., Inform. and Mech. 7 (2), 81–89 (2015).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, in Nonlinear Analysis and Its Applications (Nauka, Moscow, 1970) [in Russian].
I. B. Simonenko, “A justification of the averaging method for abstract parabolic equations,” Mat. Sb. 81 (123) (1), 53–61 (1970) [Math. USSR-Sb. 10 (1), 51–59 (1970)].
M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost-Periodic Oscillations (Nauka, Moscow, 1970) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. B. Levenshtam, P. E. Shubin, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 1, pp. 94–108.
Rights and permissions
About this article
Cite this article
Levenshtam, V.B., Shubin, P.E. Justification of the averaging method for differential equations with large rapidly oscillating summands and boundary conditions. Math Notes 100, 80–92 (2016). https://doi.org/10.1134/S0001434616070075
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616070075