Determining an optimal value for the convergence control parameter in the HAM. (2018)

• Record Type: Journal Article
• Authors: Hermann, Martin; Kaiser, Dieter
• Is Part Of: International journal of applied nonlinear science. Volume 3:Number 1(2018)
• Page Start: 41
• Page End: 65
• Publication Date: 2018
• Subjects: nonlinear ODE -- homotopy analysis method -- HAM -- auxiliary parameter; Science -- Mathematical models -- Periodicals; Engineering -- Mathematical models -- Periodicals; Nonlinear theories -- Periodicals; 501.5118
• Languages: English
• ISSNs: 1752-2862

Abstract

In the applications, the homotopy analysis method (HAM) is an often used method to determine an analytical approximate solution of lower-dimensional nonlinear ordinary differential equations. This approximation consists of an infinite series which depends on an auxiliary real parameter h . This parameter must be adjusted such that the series converges towards the exact solution of the given problem. In this paper we propose a computational approach, which is based on the residual of the truncated series, to determine an optimal value h opt or an optimal region for h . Using the numerical computing environment MATLAB, we describe several possibilities how this approach can be realised. Finally, by means of three examples (an IVP, a two-point BVP, as well as a BVP on an infinite interval) we show how this mathematically sophisticated strategy can be applied and we present the optimal parameter h opt for each example.