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Model. Anal. Inform. Sist., 2016, Volume 23, Number 1, Pages 61–85 (Mi mais483)  

Asymptotic expansions of eigenvalues of periodic and antiperiodic boundary problems for singularly perturbed second order differential equation with turning points

S. A. Kashchenkoab

a P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
b National Engineering Physics Institute "MEPhI", Moscow

Abstract: For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solved.

Keywords: singularly perturbed equation, turning points, asymptotic, boundary value problem, eigenvalues.

DOI: https://doi.org/10.18255/1818-1015-2016-1-61-85

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UDC: 517.9
Received: 20.12.2015

Citation: S. A. Kashchenko, “Asymptotic expansions of eigenvalues of periodic and antiperiodic boundary problems for singularly perturbed second order differential equation with turning points”, Model. Anal. Inform. Sist., 23:1 (2016), 61–85

Citation in format AMSBIB
\Bibitem{Kas16}
\by S.~A.~Kashchenko
\paper Asymptotic expansions of eigenvalues of periodic and antiperiodic boundary problems for singularly perturbed second order differential equation with turning points
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 1
\pages 61--85
\mathnet{http://mi.mathnet.ru/mais483}
\crossref{https://doi.org/10.18255/1818-1015-2016-1-61-85}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3502275}
\elib{http://elibrary.ru/item.asp?id=25475542}


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