Skip to main content
SpringerLink
Log in

On Analytic Periodic Solutions to Nonlinear Differential Equations With Delay (Advance)

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study a system of the reaction–diffusion type, where diffusion coefficients depend in an arbitrary way on spatial variables and concentrations, while reactions are expressed as homogeneous functions whose coefficients depend in a special way on spatial variables. We prove that the system has a family of exact solutions that are expressed through solutions to a system of ordinary differential equations (ODE) with homogeneous functions in right-hand sides. For a special case of theODE systemwe construct a general solution represented by Jacobi higher transcendental functions. We also prove that these periodic solutions are analytic functions that can be expressed near each point on the period by convergent power series.

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

Similar content being viewed by others

References

  1. Cherniha, R., King, J. R. “Nonlinear Reaction–Diffusion Systems With Variable Diffusivities: Lie Symmetries, Ansätze and Exact Solutions”, J.Math. Anal. Appl. 308, No. 1, 11–35 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  2. Polyanin, A. D. and Zaitsev, V. F. Handbook of Nonlinear Equations of Mathematical Physics: Exact Solutions (Fizmatlit,Moscow, 2002) [in Russian].

    Google Scholar 

  3. Slin’ko, M. G., Zelenyak, T. I., Akramov, T. A., Lavrent’ev, M. M. and Shcheplev, V. S. “Nonlinear Dynamics of Catalytic Reactions and Processes (review)”, Mat. Model. 9, No. 12, 87–109 (1997) [in Russian].

    MathSciNet  MATH  Google Scholar 

  4. Nefedov, N. N., Nikulin, E. I. “Existence and Stability of Periodic Solutions for Reaction–Diffusion Equations in the Two-Dimensional Case”, Model. Anal. Information Syst. 23, No. 3, 342–348 (2016).

    Article  MathSciNet  Google Scholar 

  5. Erugin, N. P. Book for Reading on General Course of Differential Equations (Nauka i Tekhnika, Minsk, 1972) [in Russian].

    Google Scholar 

  6. Cherepennikov, V. B. “Analytic Solutions of the Cauchy Problem for Some Linear Systems of Functional-Differential Equations of Neutral Type”, RussianMathematics 38, No. 6, 88–96 (1994).

    MathSciNet  Google Scholar 

  7. Cherepennikov, V. B. “On the Solvability in the Class of Analytic Functions of Some Linear Systems of Functional-Differential Equations in a Neighborhood of a Regular Singular Point”, Russian Mathematics 40, No. 5, 71–76 (1996).

    MathSciNet  MATH  Google Scholar 

  8. Cherepennikov, V. B. and Ermolaeva, P. G. “Numerical Experiment in the Investigation of Polynomial Quasisolutions of Linear Differential-Difference Equations”, RussianMathematics 52, No. 7, 48–60 (2008).

    MathSciNet  MATH  Google Scholar 

  9. van Brunt, B., Kim, Hong Oh, Derfel, G. “Holomorphic Solutions to Functional Differential Equations”, J. Math. Anal. Appl. 365, No. 1, 350–357 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. Akhiezer, N. I. Elements of the Theory of Elliptic Functions (OGIZ, Gostekhizdat, 1948) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, ul. Lermontova 134, Irkutsk, 664033, Russia

    A. A. Kosov & E. I. Semenov

Authors
  1. A. A. Kosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. I. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Kosov.

Additional information

Original Russian Text © A.A. Kosov, E.I. Semenov, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 10, pp. 34–42.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kosov, A.A., Semenov, E.I. On Analytic Periodic Solutions to Nonlinear Differential Equations With Delay (Advance). Russ Math. 62, 30–36 (2018). https://doi.org/10.3103/S1066369X18100043

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X18100043

Keywords

Search

Navigation