Abstract
In this paper, an optimal q-homotopy analysis method (Oq-HAM) is proposed. We present some examples to show the reliability and efficiency of the method. It is compared with the one-step optimal homotopy analysis method. The results reveal that the Oq-HAM has more accuracy to determine the convergence-control parameter than the one-step optimal HAM.
Highlights
The search for a better and easy to use tool for the solution of nonlinear equations illuminating the nonlinear phenomena of our life keeps continuing
The results reveal that the Oq-homotopy analysis method (HAM) has more accuracy to determine the convergence-control parameter than the one-step optimal HAM
How to find a proper convergence-control parameter so as to gain a convergent series solution? A straight-forward way to check the convergence of a homotopy-series solution is to substitute it into original governing equations and boundary/initial conditions and to check the corresponding squared residual integrated in the whole region
Summary
The search for a better and easy to use tool for the solution of nonlinear equations illuminating the nonlinear phenomena of our life keeps continuing. Liao [10,11,12,13,14] employed the basic ideas of the homotopy in topology to propose a general analytic method for linear and nonlinear problems, namely homotopy analysis method (HAM) In recent years, this method has been successfully applied to solve many types of nonlinear problems in science and engineering [6, 9, 17, 20].The HAM contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. The use of the convergence-control parameter is a great progress in the frame of the HAM It seems that more “artificial” degrees of freedom imply larger possibility to gain better approximations by means of the homotopy analysis method. This optimal method contains only one convergencecontrol parameter and is computationally rather efficient
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