Skip to main content
SpringerLink
Log in

Asymptotic Expansions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Singularly Perturbed Second-Order Differential Equation with Turning Points

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

For a second-order equation with a small factor at the highest derivative, the asymptotic behavior of all eigenvalues of periodic and antiperiodic boundary value problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable, i.e., turning points exist. An algorithm to compute all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are determined only by the values of the coefficients of the original equation in a neighborhood of turning points. The asymptotic behavior of the length of Lyapunov stability and instability zones is obtained. In particular, the stability problem is solved for solutions of second-order equations with periodic coefficients and small parameters at the highest derivative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, London: McGraw-Hill Book Company, 1955.

    MATH  Google Scholar 

  2. Yakubovich, V.A. and Starzhinskii, V.M., Lineinye differentsial’nye uravneniya s periodicheskimi koeffitsientami (Linear Differential Equations with Periodic Coefficients), Moscow: Nauka, 1972.

  3. Kashchenko, S.A., Ustoichivost’ uravnenii vtorogo poryadka s periodicheskimi koeffitsientami (Stability of Second-Order Equations with Periodic Coefficients), Yaroslavl, 2006.

  4. Kashchenko, S.A., Asymptotics of eigenvalues of first boundary value problem for singularly perturbed second-order differential equation with turning points, Model. Anal. Inf. Syst., 2015, vol. 22, no. 5, pp. 682–710.

    Article  MathSciNet  Google Scholar 

  5. Kashchenko, S.A., Asymptotic expansions of the eigenvalues of the first boundary-value problem for a singularly perturbed second-order differential equation with turning points, Model. Anal. Inf. Syst., 2016, vol. 23, no. 1, pp. 41–60.

    Article  MathSciNet  Google Scholar 

  6. Kashchenko, S.A. and Kolesov, Yu.S., A stability criterion for solutions of singularly perturbed second-order equations with periodic coefficients, Usp. Mat. Nauk, 1974, vol. 29, no. 4, pp. 171–172.

    Google Scholar 

  7. Kolesov, Yu.S., Periodic solutions of second-order quasilinear parabolic equations, Tr. Mosk. Mat. O-va., 1970, vol. 21, pp. 103–134.

    Google Scholar 

  8. Naimark, M.A., Lineinye differentsial’nye operatory (Linear Differential Operators), Moscow: Nauka, 1969.

  9. Chaplygin, V.F., Positive periodic solutions of singularly perturbed second-order nonlinear differential equations, Tr. Nauchno-Issled. Inst. Mat. Voronezh. Gos. Univ., 1970, vol. 2, pp. 43–46.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Demidov Yaroslavl State University, 150000, Yaroslavl, Russia

    S. A. Kashchenko

  2. National Research Nuclear University Moscow Engineering Physics Institute (MEPhI), 115409, Moscow, Russia

    S. A. Kashchenko

Authors
  1. S. A. Kashchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Kashchenko.

Additional information

Translated by A. Muravnik

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kashchenko, S.A. Asymptotic Expansions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Singularly Perturbed Second-Order Differential Equation with Turning Points. Aut. Control Comp. Sci. 52, 728–744 (2018). https://doi.org/10.3103/S0146411618070131

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411618070131

Keywords:

Search

Navigation