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 convergence control parameter 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 constant h − curves ( h is the convergence control parameter) or from the classical squared residual error as frequently used in the literature, a novel description is proposed to find out an optimal value for the convergence control parameter yielding the same optimum values. 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. Examples involving solution of algebraic, highly nonlinear differentialdifference, integro-differential, and ordinary or partial differential equations or systems, all from the literature demonstrate the validity and usefulness of the introduced technique