Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 592–605 | Cite as

Asymptotic Expansions of Eigenvalues of the First Boundary-Value Problem for Singularly Perturbed Second-Order Differential Equation with Turning Points

  • S. A. Kashchenko


For singularly perturbed second-order equations, the dependence of the eigenvalues of the first boundary-value problem on a small parameter at the highest derivative is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable. This leads to the emergence of so-called turning points. Asymptotic expansions with respect to a small parameter are obtained for all eigenvalues of the considered boundary-value problem. It turns out that the expansions are determined only by the behavior of the coefficients in the neighborhood of the turning points.


singularly perturbed equation turning points asymptotic boundary-value problem eigenvalues 


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  1. 1.
    Kashchenko, S.A., Ustoychivost uravneniy vtorogo poryadka s periodicheskimi koeffitsientami (Stability of Second-Order Equations with Periodic Coefficients), Yaroslavl, 2006.Google Scholar
  2. 2.
    Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya resheniy singulyarno vozmushchennykh uravneniy (Asymptotic Expansions of Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.MATHGoogle Scholar
  3. 3.
    Dorodnitsyn, A.A., An asymptotic solution of the van der Pol equation, Prikl. Mat. Mekh., 1947, vol. 11, pp. 313–328.MathSciNetMATHGoogle Scholar
  4. 4.
    Cole, J., Perturbation Methods in Applied Mathematics, London: Blaisdell Publishing Company, 1968.MATHGoogle Scholar
  5. 5.
    Vishik, M.I. and Lyusternik, L.A., Regular degeneracy and the boundary layer for linear differential equations with a small parameter, Usp. Mat. Nauk, 1957, vol. 12, no. 5, pp. 3–122.MATHGoogle Scholar
  6. 6.
    Kolesov, Yu.S. and Chaplygin, V.F., On the non-oscillation of solutions of singularly perturbed second-order equations, Dokl. Akad. Nauk SSSR, 1971, vol. 199, no. 6, pp. 1240–1242.Google Scholar
  7. 7.
    Ch. J. de la Vallie-Poussin, Sur l'equation differentielle lineaire du second ordre. Determination d'une integrale par deux valeurs assignees. Extension aux equations d’ordre, J. Math. Pure et Appl., 1929, vol. 8, no. 1, pp. 125–144.MATHGoogle Scholar
  8. 8.
    Kashchenko, S.A., Limit values of eigenvalues of the first boundary value problem for a singularly perturbed second-order differential equation with turning points, Vestn. Yarosl. Univ., 1974, vol. 10, pp. 3–39.MathSciNetGoogle Scholar
  9. 9.
    Kashchenko, S.A., Asymptotics of eigenvalues of periodic and antiperiodic boundary value problems for singularly perturbed second-order differential equations with turning points, Vestn. Yarosl. Univ., 1975, vol. 13, pp. 20–83.MathSciNetGoogle Scholar
  10. 10.
    Kashchenko, S.A., Asymptotics of eigenvalues of first boundary value problem for singularly perturbed secondorder differential equation with turning points, Model. Anal. Inf. Sist., 2015, vol. 22, no. 5, pp. 682–710.MathSciNetCrossRefGoogle Scholar

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© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia

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