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 Model. Anal. Inform. Sist., 2015, Volume 22, Number 5, Pages 682–710 (Mi mais467)

Asymptotics of eigenvalues of first boundary value problem for singularly pertubed second-order differential equation with turning points

S. A. Kaschenko

P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: We consider a linear differential equation of second order with a small factor at the highest derivative. We study the problem of the asymptotic behavior of the eigenvalues of the first boundary value problem (task Dirichlet) in situation when the turning points (points where the coefficient at the first derivative equals to zero) exist. It is shown that only the behavior of coefficients of the equation in a small neighborhood of the turning points is essential. The main result is a theorem on the limit values of the eigenvalues of the first boundary value problem.

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

DOI: https://doi.org/10.18255/1818-1015-2015-5-682-710

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UDC: 517.9

Citation: S. A. Kaschenko, “Asymptotics of eigenvalues of first boundary value problem for singularly pertubed second-order differential equation with turning points”, Model. Anal. Inform. Sist., 22:5 (2015), 682–710

Citation in format AMSBIB
\Bibitem{Kas15} \by S.~A.~Kaschenko \paper Asymptotics of eigenvalues of first boundary value problem for singularly pertubed second-order differential equation with turning points \jour Model. Anal. Inform. Sist. \yr 2015 \vol 22 \issue 5 \pages 682--710 \mathnet{http://mi.mathnet.ru/mais467} \crossref{https://doi.org/10.18255/1818-1015-2015-5-682-710} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3499145} \elib{http://elibrary.ru/item.asp?id=25063578} 

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This publication is cited in the following articles:
1. S. A. Kaschenko, “Asimptoticheskie razlozheniya sobstvennykh chisel pervoi kraevoi zadachi dlya singulyarno vozmuschennogo differentsialnogo uravneniya vtorogo poryadka s tochkami povorota”, Model. i analiz inform. sistem, 23:1 (2016), 41–60
2. S. A. Kaschenko, “Asimptoticheskie razlozheniya sobstvennykh znachenii periodicheskoi i antiperiodicheskoi kraevykh zadach dlya singulyarno vozmuschennykh differentsialnykh uravnenii vtorogo poryadka s tochkami povorota”, Model. i analiz inform. sistem, 23:1 (2016), 61–85
3. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Many-circuit canard trajectories and their applications”, Izv. Math., 81:4 (2017), 771–817
4. S. A. Kashchenko, “Asymptotic Expansions of Eigenvalues of the First Boundary-Value Problem For Singularly Perturbed Second-Order Differential Equation With Turning Points”, Autom. Control Comp. Sci., 51:7 (2017), 592–605
5. S. A. Kashchenko, “Asymptotic expansions of eigenvalues of periodic and antiperiodic boundary value problems for singularly perturbed second-order differential equation with turning points”, Autom. Control Comp. Sci., 52:7 (2018), 728–744
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