A Fastly Convergent Approximate Solution of the Duffing Equation with the Extended Rayleigh-Ritz Method

Abstract

The Duffing equation has been frequently encountered in many scientific and engineering fields for the accurate description of nonlinear phenomena associated with the temporal evolution, and the solution techniques of this highly nonlinear differential equation, even with complications, have been extensively studied for practical applications. Most of the methods for approximate solutions are in the asymptotic nature with the series expansions, and the convergence is always a challenge in expecting a better expression for a faster convergence. As the latest effort in solving the Duffing equation with a newer technique, the extended Rayleigh-Ritz method with an integration over the fundamental period of the solution of the harmonic function is used for the approximate frequency and amplitudes, and the fast convergence is achieved with just two terms, whether it is a weak nonlinear problem for small parameters or a strong nonlinear problem for large parameters. The convergent approximate solutions are closer to the exact values, demonstrating the effectiveness of the latest technique for strong nonlinear equations with a similar nature and property.