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Determination of optimal convergence-control parameter value in homotopy analysis method

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Abstract

In the framework of the Homotopy Analysis Method (HAM) the so-called convergence-control parameter \(c_{0}\) (Liao (Int J Non-Linear Mech 32:815–822, 1997) originally used the symbol \(\hbar \) to denote the auxiliary parameter. But, \(\hbar \) is well-known as Planck’s constant in quantum mechanics. To avoid misunderstanding, Liao (Commun Nonlinear Sci Numer Simulat 15:2003–2016, 2010) suggest to use the symbol \(c_0\) to denote the basic convergence-control parameter.) has a key role in convergence of obtained series solution of solving non-linear equations. In this paper a modified approach in the determining of the convergence-control parameter value \(c_{0}\) is proposed. The purpose of this paper is to find a proper convergence-control parameter \(c_0\) to get a convergent series solution, or a faster convergent one. This modified approach minimizes the norm of a discrete residual function, systematically, in which seeks to find an optimal value of the convergence-control parameter \(c_0\) at each order of HAM approximation, instead of the so-called \(c_0\)-curve process. The proved theorems and numerical results demonstrate the validity, efficiency, and performance of the proposed approach.

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Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    S. Abbasbandy & M. Jalili

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  1. S. Abbasbandy
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  2. M. Jalili
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Correspondence to S. Abbasbandy.

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Abbasbandy, S., Jalili, M. Determination of optimal convergence-control parameter value in homotopy analysis method. Numer Algor 64, 593–605 (2013). https://doi.org/10.1007/s11075-012-9680-9

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  • DOI: https://doi.org/10.1007/s11075-012-9680-9

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