Title

Stability Of Auxiliary Linear Operator And Convergence-Control Parameter In The Homotopy Analysis Method

Abstract

We consider the stability of the homotopy analysis method under the choice of both linear operator and convergence-control parameter. In particular, through several examples, we determine how changes in the linear operator can influence the convergence properties of homotopy solutions. It is seen that there is often a best way to pick the linear operator, but this can change for each problem. We consider various linear operators for some ordinary differential operators, and also discuss the method of selection for some nonlinear evolution PDEs. Throughout this chapter, we consider the optimal homotopy analysis method, which permits us to select a convergence-control parameter that minimizes residual errors. It is natural to ask whether the optimal value of the convergence-control parameter varies much as we change the number of iterations taken. For computational efficiency, we would like to take as few terms as possible in order to guarantee a low error of approximation, so learning when the optimal convergence-control parameter stabilizes could help us in knowing when to truncate our approximation. We then turn our attention to other properties of the homotopy analysis method. Through applications, we study the effect of homotopies which are nonlinear in the embedding parameter, q. In another application, we show that the auxiliary function H(x), which is often taken to unity, can be useful in a more general form. Finally, we present an application of the homotopy analysis method to a highly singular problem, and we demonstrate how to get accurate approximate solutions for such problems.

Publication Date

1-1-2013

Publication Title

Advances in the Homotopy Analysis Method

Number of Pages

123-180

Document Type

Article; Book Chapter

Personal Identifier

scopus

DOI Link

https://doi.org/10.1142/9789814551250_0004

Socpus ID

84967361657 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84967361657

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