A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

Alvaro H Salas,Simeon Casanova Trujillo

doi:10.1155/2020/3985975

Alvaro H Salas, Simeon Casanova Trujillo

Open Access

https://doi.org/10.1155/2020/3985975

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Abstract

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

Highlights

It is known that most phenomena in nature have a nonlinear character, i.e., their laws of evolution are governed by either nonlinear ordinary or nonlinear partial differential equations
It is desiderable to make an analytical study of the behavior of the equation solutions by means of the stability analysis of some associated linear systems. is “linearization” is not possible in all cases. is is the reason why analytical techniques are required to analyze the behavior of these solutions. ere are analytical methods that give necessary and sufficient conditions for the existence and uniqueness of solution to nonlinear equations
We meet in literature different techniques for integrating nonlinear equations, such as parameter perturbation techniques and homotopic perturbation methods, among others

Summary

Introduction

It is known that most phenomena in nature have a nonlinear character, i.e., their laws of evolution are governed by either nonlinear ordinary or nonlinear partial differential equations. As a contribution to the literature, in this article, we present the exact solution to the cubic-quintic Duffing oscillator equation by means of the famous Weierstrass elliptic function. Mathematical Problems in Engineering quintic Duffing equation for given initial conditions. A PHP script for solving the damped cubic-quintic equation may be found at http://fizmako.com/ duffing35.php.

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