## Analysis of Highly Nonlinear Oscillation System Using He's Max-Min Method and Comparison with Homotopy Analysis Method and Energy Balance Methods

### Abstract

Nonlinear functions are crucial points and terms in engineering problems and the solutions of many important physical problems are centered on finding accurate solutions to these functions. In this paper, a new method called max–min method has been presented for deriving accurate/approximate analytical solution to strong nonlinear oscillators. Furthermore, it is shown that a large class of linear or nonlinear differential equations can be solved without the tangible restriction of sensitivity to the degree of the nonlinear term, adding that the method is quite convenient due to reduction in size of calculations. Results obtained by max–min are compared with Homotopy Analysis Method (HAM), energy balance and numerical solution and it is shown that, simply one term is enough to obtain a highly accurate result in contrast to HAM with just one term in series solution. Finally, the phase plane to show the stability of systems is plotted and discussed.Nonlinear functions are crucial points and terms in engineering problems and the solutions of many important physical problems are centered on finding accurate solutions to these functions. In this paper, a new method called max–min method has been presented for deriving accurate/approximate analytical solution to strong nonlinear oscillators. Furthermore, it is shown that a large class of linear or nonlinear differential equations can be solved without the tangible restriction of sensitivity to the degree of the nonlinear term, adding that the method is quite convenient due to reduction in size of calculations. Results obtained by max–min are compared with Homotopy Analysis Method (HAM), energy balance and numerical solution and it is shown that, simply one term is enough to obtain a highly accurate result in contrast to HAM with just one term in series solution. Finally, the phase plane to show the stability of systems is plotted and discussed

Topics: Min-Max Method, Nonlinear Oscillation, Duffing Equation, Homotopy Analysis Method, He's Energy Balance Method, Min-Max Method, Nonlinear Oscillation, Duffing Equation, Homotopy Analysis Method, He's Energy Balance Method
Publisher: MBD Offshore Power A/S and Aalborg University
Year: 2010
OAI identifier: oai:pure.atira.dk:publications/191a9d43-932d-4138-ac2e-f5b811c814f7
Provided by: VBN