The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation

Mustafa Turkyilmazoglu
DOI: 10.4236/am.2011.26105                

Abstract

The present paper is concerned with two novel approximate analytic solutions of the undamped Duffing equation. Instead of the traditional perturbation or asymptotic methods, a homotopy technique is employed, which does not require a small perturbation parameter or a large parameter for an asymptotic expansion. It is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method can yield uniformly valid and accurate solutions. The obtained explicit analytical expressions for the solution predict the displacement, frequency and period of the oscillations much more accurate than the previously known asymptotic or perturbation formulas.

Keywords

Homotopy Analysis Method, Auxiliary Linear Operator, Initial Approximation, Duffing Oscillator, Uniform Solution

Conflicts of Interest

The authors declare no conflicts of interest.

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