High order analysis of nonlinear periodic differential equations

TL;DR

This paper refines a recent method to achieve highly accurate approximate solutions for nonlinear periodic differential equations, demonstrating fast convergence and applying it to various complex systems.

Contribution

It enhances a previous method to obtain higher precision solutions for nonlinear oscillators and extends its application to more general polynomial potentials and the Van Der Pol equation.

Findings

Fast convergence to exact solutions observed.

High-precision results for Duffing and sextic oscillators.

Successful application to octic oscillator and Van Der Pol equation.

Abstract

In this letter we apply a method recently devised in \cite{aapla03} to find precise approximate solutions to a certain class of nonlinear differential equations. The analysis carried out in \cite{aapla03} is refined and results of much higher precision are obtained for the problems previously considered (Duffing equation, sextic oscillator). Fast convergence to the exact results is observed both for the frequency and for the Fourier coefficients. The method is also applied with success to more general polynomial potentials (the octic oscillator) and to the Van Der Pol equation.