Skip to main content
SpringerLink
Log in

Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation

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

Abstract

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil’eva’s method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given.

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

Fig. 1.

Similar content being viewed by others

REFERENCES

  1. Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., Self-excited relaxation oscillations in networks of impulse neurons, Russ. Math. Surv., 2015, vol. 70, no. 3, pp. 383–452.

    Article  Google Scholar 

  2. Grigor’eva, E.V., Kashchenko, I.S., and Kashchenko, S.A., Quasinormal forms for Lang–Kobayashi equations with a large control coefficient, Model. Anal. Inf. Sist., 2013, vol. 20, no. 1, pp. 18–29.

    Article  Google Scholar 

  3. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie metody v teorii singulyarnykh vozmushchenii (Asymptotic Methods in Singular Perturbation Theory), Moscow: Vyssh. Shkola, 1990.

    Google Scholar 

  4. Butuzov, V.F., Vasil’eva, A.B., and Nefedov, N.N., Asymptotic theory of contrast structures (a survey), Avtom. Telemekh., 1997, vol. 7, pp. 4–32.

    MathSciNet  MATH  Google Scholar 

  5. Vasil’eva, A.B., Butuzov, V.F., and Nefedov, N.N., Contrast structures in singularly perturbed problems, Fundam. Prikl. Mat., 1998, vol. 4, no. 3, pp. 799–851.

    MathSciNet  MATH  Google Scholar 

  6. Nefedov, N.N. and Ni, M.K., Internal layers in the one-dimensional reaction–diffusion equation with a discontinuous reactive term, Comput. Math. Math. Phys., 2015, vol. 55, no. 12, pp. 2001–2007.

    Article  MathSciNet  Google Scholar 

  7. Vasil’eva, A.B., Step-like contrasting structures for a system of singularly perturbed equations, Comput. Math. Math. Phys., 1994, vol. 34, no. 10, pp. 1215–1223.

    MathSciNet  MATH  Google Scholar 

  8. Nefedov, N.N., Method of differential inequalities for some singularly perturbed partial derivative problems, Differ. Uravn., 1995, vol. 31, no. 4, pp. 719–722.

    Google Scholar 

  9. Vasil’eva, A.B., Butuzov, V.F., and Nefedov, N.N., Singularly perturbed problems with boundary and internal layers, Proc. Steklov Inst. Math, 2010, vol. 268, pp. 258–273.

    Article  MathSciNet  Google Scholar 

  10. Ni, M. and Lin, W., The asymptotic solutions of delay singularly perturbed differential difference equations, Acta Math. Sci., 2010, vol. 30, no. 6, pp. 1–11.

    MathSciNet  MATH  Google Scholar 

  11. Hongjiong, T., Asymptotic expansion for the solutions of singularly perturbed delay differential equations, J. Math. Anal. Appl., 2003, vol. 281, pp. 678–696.

    Article  MathSciNet  Google Scholar 

  12. Ren Jingli and Ge Weigao, Singularly perturbed boundary value problems for Volterra retarded differential equations with nonlinear boundary conditions, Acta Math. Sci., 2003, vol. 23A, no. 4, pp. 504–512.

    MathSciNet  MATH  Google Scholar 

  13. Lange, C.G. and Miura, R.M., Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations, SIAM J. Appl. Math., 1994, vol. 54, pp. 273–283.

    Article  MathSciNet  Google Scholar 

  14. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of Solutions to Singularly Perturbed Equations), Moscow: Nauka, 1973.

    Google Scholar 

  15. Fife, P.C. and McLeod, J.B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 1977, vol. 65, no. 4, pp. 335–361.

    Article  MathSciNet  Google Scholar 

  16. Pao, C.V., Nonlinear Parabolic and Elliptic Equations, New York: Plenum, 1992.

    MATH  Google Scholar 

  17. Nefedov, N., Comparison principle for reaction–diffusion–advection problems with boundary and internal layers. Numerical analysis and its applications, Lect. Notes in Comput. Sci., 2013, vol. 8236, pp. 62–72.

    Article  MathSciNet  Google Scholar 

Download references

Funding

The research by N.N. Nefedov and N.T. Levashova was supported by the Russian Foundation for Basic Research, project no. 19-01-00327. The research by M.K. Ni was supported by the National Natural Science Foundation of China, project no. 11871217 and the Science and Technology Commission of Shanghai Municipality, project no. 18dz2271000, School of Mathematical Sciences & Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice.

Author information

Authors and Affiliations

  1. East China Normal University, Shanghai, 200241, China

    M. K. Ni

  2. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. N. Nefedov & N. T. Levashova

Authors
  1. M. K. Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Nefedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. T. Levashova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. K. Ni, N. N. Nefedov or N. T. Levashova.

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

Ni, M.K., Nefedov, N.N. & Levashova, N.T. Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation. Diff Equat 56, 290–303 (2020). https://doi.org/10.1134/S0012266120030027

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266120030027

Search

Navigation