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

  • Authors:

  • S. Abbasbandy;M. Jalili

  • Affiliations:

  • Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran;Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

  • Venue:
  • Numerical Algorithms
  • Year:
  • 2013

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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.