Improved Lindstedt-Poincare method for the solution of nonlinear problems

TL;DR

This paper introduces an enhanced Lindstedt-Poincare method by integrating the Linear Delta Expansion to improve the accuracy and convergence of approximate solutions for nonlinear oscillatory problems like the Duffing equation and nonlinear pendulum.

Contribution

The paper presents a novel combination of the Linear Delta Expansion with the Lindstedt-Poincare method, resulting in better convergence and more accurate solutions for nonlinear oscillators.

Findings

Improved convergence compared to traditional Lindstedt-Poincare method

More accurate approximate solutions for anharmonic oscillator and nonlinear pendulum

Method performs well across a wide range of parameters

Abstract

We apply the Linear Delta Expansion (LDE) to the Lindstedt-Poincare (``distorted time'') method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator (Duffing equation) and of the non-linear pendulum. The approximate solutions found with this method are better behaved and converge more rapidly to the exact ones than in the simple Lindstedt-Poincar\'e method.