Skip to main content
Log in

Inner turning point in the theory of singular Perturbations

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We construct the uniform asymptotics of a solution of a singularly perturbed differential equation of Liouville type with an interior turning point.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R. E. Langer, “The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point,” Trans. Amer. Math. Soc, 67, 461–490 (1949).

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Abramowitz and Y. Stegun (editors), Handbook of Mathematical Functions, National Bureau of Standards, New York 1964.

    MATH  Google Scholar 

  3. F. Olver, Asymptotics and Special Functions [Russian translation] Nauka, Moscow 1990.

    MATH  Google Scholar 

  4. A. A. Dorodnitsyn, “Asymptotic distribution laws for eigenvalues of second-order differential equations of certain special types,” Usp. Mat. Nauk, 7, Issue 6 (52), 3–96 (1952).

    Google Scholar 

  5. V. K. Dzyadyk, “Some special functions and their role in the solution of inhomogeneous differential equations with turning point,” in: Theory of Functions and Its Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1979), pp. 61–81.

    Google Scholar 

  6. S. Yu. Dzyadyk, “On the construction of solutions of inhomogeneous differential equations with turning point and a small parameter with the derivative,” Ukr. Mat. Zh., 25, No. 5, 653–659 (1973).

    MathSciNet  Google Scholar 

  7. S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian] Nauka, Moscow 1981.

    MATH  Google Scholar 

  8. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian] Nauka, Moscow 1983.

    MATH  Google Scholar 

  9. V. N. Bobochko, “Singularly perturbed Cauchy problem with turning point,” in: Mathematical Physics and Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Issue 16 (50), Kiev, (1991), pp. 68–74.

    Google Scholar 

  10. V. N. Bobochko and V. G. Kolomiets, Asymptotic Integration of Equations of the Orr-Sommerfeld Type [in Russian], Preprint No. 90.45, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990).

    Google Scholar 

  11. V. N. Bobochko, “Asymptotic integration of a system of differential equations with turning point,” Differents. Uravn., 27, No. 9, 1505–1515 (1991).

    MATH  MathSciNet  Google Scholar 

  12. V. N. Bobochko, “Equations of the Orr-Sommerfeld type with two turning points,” Differents. Uravn., 28, No. 10, 1559–1570 (1991).

    MathSciNet  Google Scholar 

  13. V. N. Bobochko and II. Markush, Asymptotic Integration of Differential Equations with Unstable Spectrum of the Limit Operator [in Russian] Vipol, Kiev 1993.

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bobochko, V.N. Inner turning point in the theory of singular Perturbations. Ukr Math J 48, 988–1005 (1996). https://doi.org/10.1007/BF02390957

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02390957

Keywords