Періодична задача для рівняння типу Релея, не розв'язаного відносно похідної

S.  Chuiko; O.  Nesmelova

doi:10.3842/umzh.v75i10.7362

S. Chuiko Donbas State Pedagogical University, Slovyansk, Donetsk region; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region. https://orcid.org/0000-0001-7186-0129
O. Nesmelova Donbas State Pedagogical University, Slovyansk, Donetsk region; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region.

DOI: https://doi.org/10.3842/umzh.v75i10.7362

Keywords: Periodic boundary-value problem, an equation unsolved with respect to the derivative, Rayleigh-type equation, an equation of motion of a satellite in an elliptical orbit, method of least squares

Abstract

UDC 517.9

We establish constructive necessary and sufficient conditions for the solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative. The urgency of studying nonautonomous boundary-value problems, unsolved with respect to the derivative is explained by the fact that the investigation of traditional problems resolved with respect to the derivative is sometimes complicated, e.g., in the case of nonlinearities that are not integrable in elementary functions. We consider the critical case in which the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation does not turn into the identity. The least-squares method is used to find constructive conditions for the solvability and obtain convergent iterative schemes for constructing approximate solutions to a nonautonomous nonlinear boundary-value problem unsolved with respect to the derivative. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems unsolved with respect to the derivative for the case of periodic problem for the equation used to describer the motion of a satellite on the elliptic orbit. We obtain an estimate for the range of values of a small parameter for which the iterative procedure of construction of the solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent. To check the accuracy of the presented approximations, we evaluate the discrepancies in the equation used to model the motion of satellites along the elliptic orbit.

References

В. Ф. Зайцев, А. Д. Полянин, Справочник по нелинейным обыкновенным дифференциальным уравнениям, Факториал, Москва (1997).

Ю. Д. Шлапак, О периодических решениях нелинейных уравнений второго порядка, не разрешенных относительно старшей производной, Укр. мат. журн., 26, № 6, 850–854 (1974).

A. M. Samoilenko, S. M. Chuiko, O. V. Nesmelova, Nonlinear boundary-value problems unsolved with respect to the derivative, Ukr. Math. J., 72, № 8, 1280–1293 (2021).

A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, 2nd ed., De Gruyter, Berlin, Boston (2016).

А. П. Торжевский, Периодические решения уравнения плоских колебаний спутника на эллиптической орбите, Космические исслед., 2, № 5, 667–678 (1964).

S. M. Chuiko, O. V. Nesmelova (Starkova), Autonomous Noether boundary-value problems not solved with respect to the derivative, J. Math. Sci., 230, № 5, 799–803 (2018).

D. Nuсez, P. J. Torres, Stable odd solutions of some periodic equations modeling satellite motion, J. Math. Anal., 279, № 2, 700–709 (2003).

A. Cabada, A. J. A. Cid, On a class of singular Sturm–Liouville periodic boundary value problems, Nonlinear Anal. Real World Appl., 12, № 4, 2378–2384 (2011).

Л. В. Канторович, Г. П. Акилов, Функциональный анализ, Наука, Москва (1977).

S. M. Chuiko, To the generalization of the Newton–Kantorovich theorem, Visnyk V. N. Karazin Kharkiv Nat. Univ. Ser. Math., Appl. Math. and Mech., 85, № 1, 62–68 (2017).

Н. И. Ахиезер, Лекции по теории аппроксимации, Наука, Москва (1965).

S. M. Chuiko, On approximate solution of boundary value problems by the least square method, Nonlinear Oscillations, 11, № 4, 585–604 (2008).

И. Г. Малкин, Некоторые задачи теории нелинейных колебаний, Гостехиздат, Москва (1956).

G. T. Gilbert, Positive definite matrices and Sylvester's criterion, Amer. Math. Monthly, 98, № 1, 44–46 (1991).

S. M. Chuiko, Domain of convergence of an iterative procedure for an autonomous boundary value problem, Nonlinear Oscillations, 9, № 3, 405–422 (2006).

S. Chuiko, Weakly nonlinear boundary value problem for a matrix differential equation, Miskolc Math. Notes, 17, № 1, 139–150 (2016).

S. M. Chuiko, A. S. Chuiko, On the approximate solution of periodic boundary value problems with delay by the least-squares method in the critical case, Nonlinear Oscillations, 14, № 3, 445–460 (2012).

Г. В. Демиденко, С. В. Успенский, Уравнения и системы, не разрешенные относительно производной, Научная книга, Новосибирск (1998).

A. F. Filippov, Uniqueness of the solution of a system of differential equations unsolved for the derivatives, Different. Equat., 41, № 1, 90–95 (2005).

A. V. Arutyunov, E. S. Zhukovskii, S. E. Zhukovskii, On the well-posedness of differential equations unsolved for the derivative, Different. Equat., 47, № 11, 1541–1555 (2011).

С. М. Чуйко, О. В. Старкова, О. Е. Пирус, Нелинейные нетеровы краевые задачи, не разрешенные относительно производной, Динам. системы, 2(30), № 1-2, 169–186 (2012).

A. A. Boichuk, A. A. Pokutnyi, Perturbation theory of operator equations in the Frechet and Hilbert spaces, Ukr. Math. J., 67, № 6, 1327–1335 (2016).

О. А. Бойчук, О. А. Покутний, Нормально-розв’язнi крайовi задачi для операторно-диференцiальних рiвнянь, Наук. думка, Київ (2022).

Periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative

Abstract

References