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Mathematical Problems in Engineering
Volume 2014, Article ID 697845, 15 pages
Research Article

On the Optimal Auxiliary Linear Operator for the Spectral Homotopy Analysis Method Solution of Nonlinear Ordinary Differential Equations

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg, Scottsville 3209, South Africa

Received 9 July 2014; Accepted 31 July 2014; Published 14 August 2014

Academic Editor: Mohammad Mehdi Rashidi

Copyright © 2014 S. S. Motsa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The purpose of this study is to identify the auxiliary linear operator that gives the best convergence and accuracy in the implementation of the spectral homotopy analysis method (SHAM) in the solution of nonlinear ordinary differential equations. The auxiliary linear operator is an essential element of the homotopy analysis method (HAM) algorithm that strongly influences the convergence of the method. In this work we introduce new procedures of defining the auxiliary linear operators and compare solutions generated using the new linear operators with solutions obtained using well-known linear operators. The applicability and validity of the proposed linear operators is tested on four highly nonlinear ordinary differential equations with fluid mechanics applications that have recently been reported in the literature. The results from the study reveal that the new linear operators give better results than the previously used linear operators. The identification of the optimal linear operator will direct future research on further applications of HAM-based methods in solving complicated nonlinear differential equations.