Abstract

Asymptotic integration of differential systems of equations with fast oscillating coefficients has been carried out by the Feschenko-Shkil-Nikolenko splitting method and the Lomov regularization method. Equations of this type are often encountered in study of various questions related to dynamic stability, to properties of media with a periodic structure and other applied problems. In the monograph by Yu.L. Daletski and M.G. Krein an asymptotic analysis is given for one of these problems - the problem on parametric amplification. In the present paper, we generalize this problem to integro-differential equations, the differential part of which coincides with the parametric amplification problem. The main purpose of the research is to identify the influence of the integral term in the asymptotic behavior of the solution. It is considered the general case, i.e. the case of both the lack of resonance (when the integer linear combination of frequencies of the fast oscillating cosine does not coincide with the spectrum frequency of the limit operator), and its presence (when such coincidence takes place). The developed algorithm is obviously generalized to systems of equations with an arbitrary matrix of the differential part, the pure imaginary spectrum, and with an arbitrary number of fast oscillating coefficients (such as the cosine considered in the paper).

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 2
Year: 2020

DOI: 10.12732/ijam.v33i2.12

Download Section

Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

1. [1] Yu.L. Daletskiy and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Moscow, Nauka (1970).
2. [2] Yu.L. Daletskiy, Asymptotic method for some differential equations with oscillating coefficients, DAN SSSR, 143, No 5 (1962), 1026-1029.
3. [3] S.F. Feschenko, N.U. Shkill and L.D. Nikolenko, Asymptotic Methods in the Theory of Linear Differential Equations, Kiev, Naukova Dumka (1966).
4. [4] B.T. Kalimbetov and V.F. Safonov, Integro-differentiated singularly perturbed equations with fast oscillating coefficients, Bulletin of KarSU, Ser. Mathematics, 94, No 2 (2019), 33-47.
5. [5] S.A. Lomov, Introduction to General Theory of Singular Perturbations, Moscow, Nauka (1981).
6. [6] S.A. Lomov and I.S. Lomov, Foundations of Mathemathical Theory of Boundary Layer, Publishing House of Moscow State University (2011).
7. [7] A.D. Ryzhih, Asymptotic solution of linear differential equation with rapidly oscillating coefficient, Trudy MPEI, 357 (1978), 92-94.
8. [8] A.D. Ryzhih, Application of regularization method for an equation with rapidly oscillating coefficients, In: Mater. All-Union. Conf. by Asympt. Methods, Science, Alma-Ata (1979), 64-66.
9. [9] V.F. Safonov and A.A. Bobodzhanov, Course of Higher Mathematics. Singularly Perturbed Problems and Regularization Method: Textbook, Moscow, Publishing House of MPEI (2012).