Convergence Accelerating in the Homotopy Analysis Method: A New Approach (Advances in Applied Mathematics and Mechanics, 2018)
One of the flash analytic approximate methods of nowadays for the solutionof highly nonlinear algebraic or differential equations arising from the mathematicalmodelling of industrial and technological applications is the homotopy analysismethod (HAM). The success of the HAM is mainly due to a so-called convergence controlparameter, $h$, plugged into the system externally which is missing in other competingmethods. A simple algorithm to determine this parameter is introduced in this paper,besides the well-known approaches of constant $h$−level curves, the squared residualerror and the recent ratio technique. Comparison of the four approaches yieldsnearly the same convergence control parameters with the advantage of the newly proposedapproach in terms of its simplicity and less CPU time requirement. Moreover,a convergence accelerating method is suggested here based on updating the initialguess of the solution at some low-order homotopy series approximation of the solution.It appears to extend the interval of convergence control parameter. The providedexamples of real life phenomena in combination with this technique demonstrate asuccessful improvement over the classical HAM method.