Skip to main content
Log in

The theory of the numerical-analytic method: Achievements and new trends of development. II

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We analyze results concerning the application of the numerical-analytic method suggested by Samoilenko in 1965 to second-order differential equations.

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

Access this article

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. M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: Achievements and new trends of development. I,” Ukr. Mat. Zh., 50, No. 1, 102–117 (1998).

    Article  Google Scholar 

  2. N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  3. A. M. Samoilenko and M. I. Rontó, Numerical-Analytic Methods for the Investigation of Periodic Solutions [in Russian], Vyshcha Shkola, Kiev (1976).

    Google Scholar 

  4. A. M. Samoilenko, “On periodic solutions of nonlinear second-order equations,” Differents. Uravn., 3, No. 11, 1903–1912 (1967).

    Google Scholar 

  5. N. N. Bautin and E. A. Leontovich, Methods and Techniques for the Qualitative Investigation of Dynamical Systems in the Plane [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  6. V. A. Pliss, Nonlocal Problems in the Theory of Oscillations [in Russian], Nauka, Moscow-Leningrad (1964).

    Google Scholar 

  7. J. H. Heinbockel and R. A. Struble, “Periodic solutions for differential systems with symmetries,” J. Soc. Indust. Appl. Math., 13, No. 2, 425–440 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Z. Chornyi, Periodic Solutions of Second-Order Integro-Differential and Difference Equations of the Wave Type [in Russian], Preprint No. 91-46, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991).

    Google Scholar 

  9. V. Z. Chornyi, Investigation of Periodic Solutions of Some Classes of Second-Order Differential Equations [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992).

    Google Scholar 

  10. V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).

    MATH  Google Scholar 

  11. L. Cesari, “Functional analysis and periodic solutions of nonlinear differential equations,” in: Contributions to Differential Equations, Vol. 1, Wiley, New York (1963), pp. 149–187.

    Google Scholar 

  12. F. Tricomi, Integral Equations [Russian translation], Nauka, Moscow (1960).

    MATH  Google Scholar 

  13. E. P. Trofimchuk, “Integral operators of the method of periodic successive approximations,” Mat. Fiz. Nelin. Mekh., 13, 31–36 (1990).

    MathSciNet  Google Scholar 

  14. M. I. Rontó and S. I. Trofimchuk, Numerical-Analytic Method for Nonlinear Differential Equations, Preprint No. 96-01, Institute of Mathematics, University of Miskolc, Miskolc (1996).

    Google Scholar 

  15. M. I. Rontó, A. N. Rontó, and S. I. Trofimchuk, Numerical-Analytic Method for Differential and Difference Equations in Partially Ordered Banach Spaces and Some Applications, Preprint No. 96-02, Institute of Mathematics, University of Miskolc, Miskolc (1996).

    Google Scholar 

  16. T. G. Strizhak, “On periodic solutions of systems of nonlinear equations of the second order,” in: Asymptotic and Qualitative Methods in the Theory of Nonlinear Oscillations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1971), pp. 35–46.

    Google Scholar 

  17. A. M. Samoilenko and M. I. Rontó, Numerical-Analytic Methods of Investigating Periodic Solutions, Mir, Moscow (1980).

    Google Scholar 

  18. Yu. A. Mitropol’skii, G. P. Khoma, and M. I. Gromyak, Asymptotic Methods for Investigation of Quasilinear Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  19. N. A. Perestyuk and A. N. Rontó, “Numerical-analytic method for the equation of a nonlinear oscillator,” Publ. Univ. Miskolc, Ser. D. Natur. Sci. (Math.), 36, No. 2, 115–124 (1996).

    MATH  Google Scholar 

  20. Yu. D. Shlapak, “On periodic solutions of implicit nonlinear differential equations of the second order,” Ukr. Mat. Zh., 26, No. 6, 850–854 (1974).

    Google Scholar 

  21. Yu. D. Shlapak, Investigation of Oscillations of Systems of Certain Types [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992).

    Google Scholar 

  22. A. M. Samoilenko and N. A. Perestyuk, “On the justification of the averaging method for second-order differential equations with pulses,” Ukr. Mat. Zh., 26, No. 3, 441–448 (1974).

    Google Scholar 

  23. R. I. Sobkovich, Numerical-Analytic Method for the Investigation of Boundary-Value Problems with Control [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1983).

    Google Scholar 

  24. R. I. Sobkovich, “On the periodic problem of control for second-order differential systems,” in: Analytic Methods in Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1981), pp. 125–133.

    Google Scholar 

  25. V. N. Shovkoplyas, “Periodic solutions of nonlinear second-order differential equations with pulses,” Visn. Kyiv. Univ., Ser. Mat. Mekh., No. 20, 131–140 (1978).

  26. V. N. Shovkoplyas, Periodic Solutions of Nonlinear Differential Equations with Pulse Influence [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev University, Kiev (1979).

    Google Scholar 

  27. M. I. Rontó and O. M. Martynyuk, “Investigation of periodic solutions of second-order countable systems,” Ukr. Mat. Zh., 44, No. 16, 83–93 (1991).

    Google Scholar 

  28. O. M. Martynyuk, “On the numerical-analytic method for countable periodic systems of the second order,” in: Nonlinear Problems in the Theory of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991), pp. 49–59.

    Google Scholar 

  29. O. M. Martynyuk, Investigation of Solutions of Boundary-Value Problems for Countable Systems of Nonlinear Differential Equations [in Ukrainian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).

    Google Scholar 

  30. O. M. Martynyuk and S. V. Martynyuk, “Investigation of periodic solutions of countable systems of the second order,” in: Nonlinear Evolution Equations in Applied Problems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991), pp. 88–90.

    Google Scholar 

  31. I. I. Korol’, “On investigation of two-point boundary-value problems for systems of ordinary differential equations of the second order with parameters,” Dop. Akad. Nauk. Ukr., Ser. A., No. 9, 6–12 (1995).

  32. I. I. Korol’, Numerical-Analytic Methods for the Investigation of Solutions of Two-Point Boundary-Value Problems with Parameters [in Ukrainian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev University, Kiev (1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 225–243, February, 1998.

This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology (project No. 1.4/269) and OTKA (grant No. T 019095).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rontó, M.I., Samoilenko, A.M. & Trofimchuk, S.I. The theory of the numerical-analytic method: Achievements and new trends of development. II. Ukr Math J 50, 255–277 (1998). https://doi.org/10.1007/BF02513450

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02513450

Keywords

Navigation