Halley's accelerated convergence method for a nonlinear periodic boundary value problem for the Rayleigh equation
DOI:
https://doi.org/10.37069/3154-8229-2026-40-7Keywords:
weakly nonlinear periodic boundary value problem, periodic problem for the Rayleigh equation, Halley method, pendulum oscillation equationAbstract
The study of weakly nonlinear boundary value problems for systems of ordinary differential equations, differential-algebraic equations and functional-differential equations is a traditional focus of the Kyiv School of Nonlinear Oscillations. In this article, we studied a weakly nonlinear periodic boundary value problem for an ordinary differential equation, in particular the nonlinear periodic problem for the Rayleigh equation. We propose two approaches to constructing approximations to the periodic solution of the considered problem, based on the use of the traditional Newton method and the Halley accelerated convergence method. We demonstrated the advantages of using Halley accelerated convergence method compared with the classical Newton method. In particular, the Newton and Newton–-Kantorovich methods have quadratic convergence, whilst the Halley accelerated convergence method, under certain conditions, has cubic convergence. We established the conditions for the existence of a unique periodic solution to the problem for the Rayleigh equation, and also obtained the convergence conditions for the constructed iterative scheme using the Halley method. We used the construction of the generalised Green’s operator for a periodic boundary value problem, developed by A.M. Samoilenko and O.A. Boichuk, to obtain the conditions for the existence of a unique periodic solution to the problem for the Rayleigh equation. Also we used the condition of compression for the operator that corresponds to an operator system which is equivalent to a periodic boundary value problem for an ordinary differential equation to obtain convergence conditions for the constructed iterative scheme for the nonlinear periodic Rayleigh equation. As opposed to the traditional technique of linearising boundary value problems, the obtained approximations to the solution of the periodic Rayleigh equation are periodic. The effectiveness of the developed approach is demonstrated in the example of a periodic boundary value problem for the equation of a pendulum oscillation with perturbation, which, contrary to traditional weakly nonlinear boundary value problems, is nonlinear.
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