Title: Determining an optimal value for the convergence control parameter in the HAM

Authors: Martin Hermann; Dieter Kaiser

Addresses: Institute of Mathematics, Friedrich Schiller University, Ernst-Abbe-Platz 2, 07743 Jena, Germany ' Institute of Mathematics, Friedrich Schiller University, Ernst-Abbe-Platz 2, 07743 Jena, Germany

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.

Keywords: nonlinear ODE; homotopy analysis method; HAM; auxiliary parameter.

DOI: 10.1504/IJANS.2018.097325

International Journal of Applied Nonlinear Science, 2018 Vol.3 No.1, pp.41 - 65

Accepted: 11 Sep 2017
Published online: 15 Jan 2019 *

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