Advertisement

Journal of Mathematical Sciences

, Volume 223, Issue 3, pp 337–350 | Cite as

Weakly Nonlinear Matrix Boundary-Value Problem in the Case of Parametric Resonance

  • S. M. Chuiko
  • A. S. Chuiko
  • D. V. Sysoev
Article
  • 25 Downloads

We establish necessary and sufficient conditions for the existence of solutions of a nonlinear matrix boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. We construct a convergent iterative scheme for finding approximate solutions of the problem. As an example of application of the proposed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Riccati-type equation with parametric perturbation. To check the accuracy of the obtained approximations, we introduce residuals in the original equation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).CrossRefMATHGoogle Scholar
  2. 2.
    E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods for the Analysis of Nonlinear Systems [in Russian], Nauka, Moscow (1979).MATHGoogle Scholar
  3. 3.
    L. I. Mandel’shtam and N. D. Papaleksi, “On the parametric excitation of electric oscillations,” Zh. Tekh. Fiz., No. 3, 5–29 (1934).Google Scholar
  4. 4.
    G. Schmidt, Parameterregte Schwingungen, VEB Deutscher Verlag der Wissenschaften, Berlin (1975).Google Scholar
  5. 5.
    V. A. Yakubovich and V. M. Starzhinskii, Parametric Resonance in Linear Systems [in Russian], Nauka, Moscow (1987).Google Scholar
  6. 6.
    V. P. Silin, Parametric Action of High-Power Radiation on Plasmas [in Russian], Nauka, Moscow (1973).Google Scholar
  7. 7.
    V. V. Bolotin, Dynamic Stability of Elastic Systems [in Russian], Gostekhizdat, Moscow (1956).Google Scholar
  8. 8.
    Yu. F. Kopelev, “Parametric vibrations of machines,” in: Republican Interdepartmental Scientific and Engineering Collection, Metal-Cutting Machines [in Russian], Kiev, Issue 12 (1984), pp. 3–8.Google Scholar
  9. 9.
    A. A. Boichuk and S. A. Krivosheya, “A critical periodic boundary-value problem for matrix Riccati equations,” Different. Equat., 37, No. 4, 464–471 (2001).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).MATHGoogle Scholar
  11. 11.
    R. Bellman, Introduction to Matrix Analysis [Russian translation], Nauka, Moscow (1969).Google Scholar
  12. 12.
    A. A. Boichuk and S. A. Krivosheya, “Criterion of the solvability of matrix equations of the Lyapunov type,” Ukr. Math. J., 50, No. 8, 1162–1169 (1998).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    V. P. Derevenskii, “Matrix Bernoulli equations. I,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 14–23 (2008).Google Scholar
  14. 14.
    S. M. Chuiko, “On the solution of the matrix Sylvester equation,” Vestn. Odes. Nats. Univ., Ser. Mat., 19, Issue 1(21), 49–57 (2014).Google Scholar
  15. 15.
    S. M. Chuiko, “On the solution of the matrix Lyapunov equation,” Vestn. Kharkov. Nats. Univ., Ser. Mat., Prikl. Mat., Mekh., No. 1120, 85–94 (2014).Google Scholar
  16. 16.
    S. M. Chuiko, “On the solution of the generalized matrix Sylvester equation,” in: Chebyshev Collection [in Russian], 16, Issue 1 (2015), pp. 52–66.Google Scholar
  17. 17.
    S. M. Chuiko, “Green operator of the linear Noetherian boundary-value problem for the matrix differential equation,” Dinam. Sist., 4(32), No. 1, 2, 101–107 (2014).MATHGoogle Scholar
  18. 18.
    V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Calculations [in Russian], Nauka, Moscow (1984).MATHGoogle Scholar
  19. 19.
    S. M. Chuiko, “Domain of convergence of an iterative procedure for an autonomous boundary-value problem,” Nelin. Kolyv., 9, No. 3, 416–432 (2006); English translation: Nonlin. Oscillat., 9, No. 3, 405–422 (2006).Google Scholar
  20. 20.
    A. S. Chuiko, “Domain of convergence of an iteration procedure for a weakly nonlinear boundary-value problem,” Nelin. Kolyv., 8, No. 2, 278–288 (2005); English translation: Nonlin. Oscillat., 8, No. 2, 277–287 (2005).Google Scholar
  21. 21.
    S. M. Chuiko and A. S. Chuiko, “On the approximate solution of periodic boundary value problems with delay by the least-squares method in the critical case,” Nonlin. Oscillat.s, 14, No. 3, 445–460 (2012).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    O. Vejvoda, “On perturbed nonlinear boundary-value problems,” Czech. Math. J., No. 11, 323–364 (1961).Google Scholar
  23. 23.
    A. Boichuk and S. Chuiko, “Autonomous weakly nonlinear boundary value problems in critical cases,” Different. Equat., No. 10, 1353–1358 (1992).Google Scholar
  24. 24.
    S. M. Chuiko and I. A. Boichuk, “An autonomous Noetherian boundary-value problem in the critical case,” Nonlin. Oscillat., 12, No. 3, 405–416 (2009).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    V. N. Laptinskii and I. I. Makovetskii, “On the constructive analysis of a two-point boundary-value problem for the nonlinear Lyapunov equation,” Differents. Uravn., 41, No. 7, 994–996 (2005).MathSciNetMATHGoogle Scholar
  26. 26.
    S. M. Chuiko, “Nonlinear Noetherian boundary-value problem in the case of parametric resonance,” Nelin. Kolyv., 17, No. 1, 137–148 (2014); English translation: J. Math. Sci., 205, No. 6, 859–870 (2015).Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • S. M. Chuiko
    • 1
  • A. S. Chuiko
    • 1
  • D. V. Sysoev
    • 1
  1. 1.Donbass State Pedagogic UniversitySlavyanskUkraine

Personalised recommendations