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

In the framework of the Homotopy Analysis Method (HAM) the so-called convergence-control parameter (Liao (Int J Non-Linear Mech 32:815-822, 1997) originally used the symbol to denote the auxiliary parameter. But, 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 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 is proposed. The purpose of this paper is to find a proper convergence-control parameter 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 at each order of HAM approximation, instead of the so-called -curve process. The proved theorems and numerical results demonstrate the validity, efficiency, and performance of the proposed approach.