^{1}Chuiko, SM^{2}Nesmelova, OV^{1}Donbass State Pedagogical University, Slovyansk^{2}Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk |

Dopov. Nac. akad. nauk Ukr. 2019, 12:3-12 |

https://doi.org/10.15407/dopovidi2019.12.003 |

Section: Mathematics |

Language: Russian |

Abstract: We have found constructive conditions of solvability and a convergent iterative scheme for constructive solutions of a nonlinear autonomous Noetherian boundary-value problem in the case of parametric resonance. As an example of applying the scheme, some approximations to the solution of a periodic boundary-value problem for an autonomous equation of the Duffing type with a parametric perturbation are determined. To control the accuracy of the approximations, residuals in the original equation are applied. |

Keywords: case of parametric resonance, Duffing-type equation, nonlinear autonomous boundary-value problem |

1. Boichuk, A. A. & Samoilenko, A. M. (2016). Generalized inverse operators and Fredholm boundary-value problems, 2-th edition. Berlin; Boston: De Gruyter. Doi: https://doi.org/10.1515/9783110378443

2. Malkin, I. G. (1956). Some problems of the theory of nonlinear oscillations. Moscow: Gostekhizdat (in Russian).

3. Boichuk, A. & Chuiko, S. (1992). Autonomous Weakly Nonlinear Boundary Value Problems in Critical Cases. Diff. Equations, 10, pp. 1353-1358.

4. Mandelstam, L. I. & Papaleksi, N. D. (1934). On the parametric excitation of electrical vibrations. Zhurn. tech. Phys., 3, pp. 5-29 (in Russian).

5. Chuiko, S. M. (2015). Nonlinear Noetherian boundary-value problem in the case of parametric resonance. J. Math. Sci. (N.Y.), 205, No. 6, pp. 859-870. Doi: https://doi.org/10.1007/s10958-015-2289-5

6. Yakubovich, V. A. & Starzhinsky, V. M. (1987). Parametric resonance in linear systems. Moscow: Nauka (in Russian).

7. Grebenikov, E. A. & Ryabov, Yu. A. (1972). Constructive methods of analysis of nonlinear systems. Moscow: Nauka (in Russian).

8. Chuiko, S. M.& Kulish, P. V. (2012). Linear Noetherian boundary value problem in the case of parametric resonance. Trudy IPMM NAN Ukrainy. 24, pp. 243-252 (in Russian).

9. Kantorovich, L. V. & Akilov, G. P. (1977). Functional analysis. Moscow: Nauka (in Russian).

10. Chuiko, S. M. (2017). To the generalization of the Newton—Kantorovich theorem. Visnyk of V.N. Ka razin Kharkiv Nat. Univ. Ser. math., appl. mathematics and mechanics. 85, No. 1, pp. 62-68.

11. Boichuk, A. A. & Krivosheya, S. A. (2001). A Critical Periodic Boundary Value Problem for a Matrix Riccati Equation. Diff. Equations, 37, No. 4, pp. 464-471. Doi: https://doi.org/10.1023/A:1019267220924

12. Chuiko, S. M. (2015). The Green’s operator of a generalized matrix linear differential-algebraic boundary value problem. Siber. Math. J., 56, No. 4, pp. 752-760. Doi: https://doi.org/10.1134/S0037446615040175

13. Gutlyanskii, V. & Ryazanov, V. & Yefimushkin, A. (2016). On the boundary-value problems for quasiconformal functions in the plane. J. Math. Sci., 214, pp. 200-219. Doi: https://doi.org/10.1007/s10958-016-2769-2

14. Skrypnik, I. I. (2016). Removability of isolated singularities for anisotropic elliptic equations with gradient absorption. Israel J. Math., 215, No. 1, pp. 163-179. Doi: https://doi.org/10.1007/s11856-016-1377-7