Title

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Authors

Authors

M. Baxter; R. A. Van Gorder;K. Vajravelu

Comments

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Abbreviated Journal Title

Numer. Algorithms

Keywords

Cahn-Hilliard equation; Homotopy analysis method; Auxiliary linear; operator; Convergence control parameter; NONLINEAR DIFFERENTIAL-EQUATIONS; VISCOUS-FLOW PROBLEMS; NON-NEWTONIAN; FLUIDS; EMDEN-FOWLER TYPE; ANALYTIC SOLUTION; SERIES SOLUTIONS; GENERAL-APPROACH; FREE-ENERGY; WAVES; Mathematics, Applied

Abstract

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L-2-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.

Journal Title

Numerical Algorithms

Volume

66

Issue/Number

2

Publication Date

1-1-2014

Document Type

Article

Language

English

First Page

269

Last Page

298

WOS Identifier

WOS:000338336700004

ISSN

1017-1398

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