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

• Date: 2023
• Main Authors: Chuiko , S., Nesmelova, O., Чуйко, Сергій, Нєсмєлова, Ольга
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2023
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7362
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.