He’s Max-Min Approach for Coupled Cubic Nonlinear Equations Arising in Packaging System

Jun Wang

doi:10.1155/2013/382509

Abstract

He’s inequalities and the Max-Min approach are briefly introduced, and their application to a coupled cubic nonlinear packaging system is elucidated. The approximate solution is obtained and compared with the numerical solution solved by the Runge-Kutta algorithm yielded by computer simulation. The result shows a great high accuracy of this method. The research extends the application of He’s Max-Min approach for coupled nonlinear equations and provides a novel method to solve some essential problems in packaging engineering.

Highlights

Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically
In order to avoid some restrictions of Perturbation Method, some other methods are developed, including the homotopy perturbation method (HPM), the variational interation method (VIM), many well-established asymptotic methods [2], a novel Max-Min method [3]
The Max-Min method, which has been widely applied to many kinds of strong nonlinear equations such as pendulum and Duffing equations, is applied to study the nonlinear response of coupled cubic nonlinear packaging system in this study for the first time

Summary

Introduction

Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically. Among the methods for analytical solution, the Perturbation method [1] is one of the most wellknown approaches and is based on the existence of small or large parameters which is not commonly contained in many nonlinear problems. Among current researches about He’s Max-Min approach and its applications [4,5,6,7,8,9,10], few involve coupled nonlinear problems such in packaging engineering, especially the higher-dimensional coupled nonlinear problems. He’s Max-Min approach is applied to the second order coupled cubic nonlinear packaging system to get its frequencies and periods under different situations. According to He’s Max-Min approach, in order to obtain the exact solution of certain variable x, its minimum of Max values and maximum of Min values should firstly gained as follows:. The method in [3] is used to determine the value of k

Modelling and Equations

Application of He’s Max-Min Approach

Conclusion