A comparative study of some analytic approximate methods for the anharmonic oscillator described by cubic quartic potential

TL;DR

This study compares various analytical approximate methods for solving the anharmonic oscillator with cubic and quartic potentials, introducing new rational functions and assessing their accuracy against exact solutions.

Contribution

It introduces two new rational functions for the harmonic balance method and applies rational functions as trial functions in the energy balance method for the first time.

Findings

Harmonic balance method with variants is most successful.

Fourier series approach yields accurate results.

Lindstedt Poincare perturbation methods are less reliable for asymmetric potentials.

Abstract

The improved formulations of the Lindstedt Poincare perturbation method, the harmonic balance method and its modifications using the rational functions, the energy balance method and the two terms Fourier series expansion have been employed to find approximate solution to the equation of motion for the cubic quartic potential AHO and its special cases (the cubic, the quartic, and the purely cubic quartic potential AHOs), to assess the relative merits and demerits of these techniques. Two new rational functions as approximate solution for the harmonic balance method have been proposed and the rational functions have been used as trial functions in the energy balance method for the first time. The time periods T+ (for q(t) > 0), T- (for q(t) < 0) and the total period T so determined have been compared with the corresponding exact values for an extensive range of the AHO parameter values. The results have been thoroughly analysed in terms of their percentage difference from the relevant exact time period. It is concluded that the methods based on the Lindstedt Poincare perturbation approach do not lead to reliable results for T+ and T-, which makes their applicability for the asymmetric potential AHOs doubtful. The harmonic balance technique together with its different variants is the most successful, while the Fourier series approach also yields commendably accurate results. The outcome of the third-order energy balance method is reasonably good though the usage of the rational functions for this has not been encouraging.