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S. Liang and D. J. Jeffrey, “Comparison of Homotopy Analysis Method and Homotopy Perturbation Method through an Evolution Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4057-4064. doi:10.1016/j.cnsns.2009.02.016

has been cited by the following article:

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

  AUTHORS: Mustafa Turkyilmazoglu

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

  JOURNAL NAME: Applied Mathematics, Vol.2 No.6, June 22, 2011

  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.