Skip to main content
SpringerLink
Log in

Solution of a Singularly Perturbed Mixed Problem on the Half-Line for a Parabolic Equation with a Strong Turning Point of the Limit Operator

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study singularly perturbed problems in the presence of spectral singularities of the limit operator using S.A. Lomov’s regularization method. In particular, a regularized asymptotic solution is constructed for a singularly perturbed inhomogeneous mixed problem on the half-line for a parabolic equation with a strong turning point of the limit operator. Based on the idea of asymptotic integration of problems with unstable spectrum, it is shown how regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for this type of singularity is described in detail, this algorithm is justified, and an asymptotic solution of any order in a small parameter is constructed.

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

REFERENCES

  1. Lomov, S.A., Vvedenie v obshchuyu teoriyu singulyarnykh vozmushchenii (Introduction to the General Singular Perturbation Theory), Moscow: Nauka, 1981.

    Google Scholar 

  2. Lomov, S.A. and Lomov, I.S., Osnovy matematicheskoi teorii pogranichnogo sloya (Fundamentals of the Mathematical Theory of the Boundary Layer), Moscow: Izd. Dom Mosk. Gos. Univ., 2011.

    Google Scholar 

  3. Eliseev, A.G. and Lomov, S.A., Asymptotic integration of singularly perturbed problems, Russ. Math. Surv., 1988, vol. 43, pp. 1–63.

    Article  MathSciNet  MATH  Google Scholar 

  4. Yeliseev, A., Ratnikova, T., and Shaposhnikova, D., Regularized asymptotics of the solution of the singularly perturbed first boundary value problem on the semiaxis for a parabolic equation with a rational “simple” turning point, Mathematics, 2021, no. 9, p. 405.

  5. Eliseev, A.G. and Kirichenko, P.V., Singularly perturbed Cauchy problem in which the limit operator has multiple spectrum and a weak first-order turning point, Differ. Equations, 2022, vol. 58, no. 6, pp. 727–740.

    Article  MathSciNet  MATH  Google Scholar 

  6. Eliseev, A.G., An example of solving a singularly perturbed Cauchy problem for a parabolic equation in the presence of a strong turning point, Differ. Uravn. i Protsessy Upr., 2022, no. 3, pp. 46–59.

  7. Arnold, V.I., On matrices depending on parameters, Russ. Math. Surv., 1971, vol. 26, no. 2, pp. 29–43.

    Article  MathSciNet  Google Scholar 

  8. Mehler, F.G., Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung, J. Reine Angew. Math., 1866, pp. 161–176.

Download references

Funding

The results by A.G. Eliseev (statement of the problem and the derivation of the equations for regularizing functions, the singular operators for regularizing the right-hand sides of iterative problems, and the boundary operator for describing the boundary layer in the vicinity of the point \(x=0 \)) were obtained with the support of the Ministry of Science and Higher Education of the Russian Federation as part of the state order, project no. FSWF-2023-0012.

Author information

Authors and Affiliations

  1. National Research University “Moscow Power Engineering Institute”, Moscow, 111250, Russia

    A. G. Eliseev, T. A. Ratnikova & D. A. Shaposhnikova

Authors
  1. A. G. Eliseev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. A. Ratnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. A. Shaposhnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. G. Eliseev, T. A. Ratnikova or D. A. Shaposhnikova.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eliseev, A.G., Ratnikova, T.A. & Shaposhnikova, D.A. Solution of a Singularly Perturbed Mixed Problem on the Half-Line for a Parabolic Equation with a Strong Turning Point of the Limit Operator. Diff Equat 59, 1032–1049 (2023). https://doi.org/10.1134/S0012266123080037

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266123080037

Search

Navigation