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

    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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