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

S S Motsa

doi:10.1155/2014/697845

Abstract

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.

Highlights

The homotopy analysis method (HAM) is an analytic method that has been widely used for solving highly nonlinear equations with applications in computational and applied mathematics, economics and finance, engineering, and many other areas of fundamental science
The spectral homotopy analysis method (SHAM) algorithm using the proposed auxiliary linear operators was applied to the several test problems presented in the last section
This study gives a systematic way of choosing initial approximations and linear operators that can be employed in the spectral homotopy analysis method (SHAM) algorithm

Summary

Introduction

The homotopy analysis method (HAM) is an analytic method that has been widely used for solving highly nonlinear equations with applications in computational and applied mathematics, economics and finance, engineering, and many other areas of fundamental science. A distinctive characteristic of the HAM that sets it apart from all other analytical methods is the presence of a convergence-controlling parameter and the flexibility to select auxiliary functions and linear operators in order to guarantee convergence and improve accuracy of the approximate solutions. This makes the HAM suitable for solving many highly nonlinear problems including those that do not contain small or large embedded parameters. This makes the SHAM more robust and capable of solving a wider range of complicated nonlinear equations than its analytical counterpart

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Conclusion