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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    Mathew Baxter, Robert A. Van Gorder & Kuppalapalle Vajravelu

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  1. Mathew Baxter
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  2. Robert A. Van Gorder
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  3. Kuppalapalle Vajravelu
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Correspondence to Robert A. Van Gorder.

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Baxter, M., Van Gorder, R.A. & Vajravelu, K. On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem. Numer Algor 66, 269–298 (2014). https://doi.org/10.1007/s11075-013-9733-8

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  • DOI: https://doi.org/10.1007/s11075-013-9733-8

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