An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method

A rapid and effective way of working out the optimum parameter of convergence control in the homotopy analysis method (HAM) is introduced in this paper. As compared with the already known ways of evaluating the convergence control parameter in HAM either through the classical h-level curves with h being the convergence control parameter or from the classical squared residual formula as adopted in the HAM society, an elegant way of calculating the convergence control parameter yielding the same optimum values is offered. In most cases, the new method is shown to perform quicker and better against the residual error method when integrations are much harder to evaluate or even by numerical means. Examples originating from real life applications selected from the literature demonstrate the validity and usefulness of the introduced technique.