Control of error in the homotopy analysis of semi-linear elliptic boundary value problems

    Authors

    R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Numer. Algorithms

    Keywords

    Elliptic boundary value problem; Yamabe equation; Brinkman-Forchheimer; equation; Liouville's equation; Homotopy analysis method; Error analysis; and control; LANE-EMDEN EQUATION; NONLINEAR DIFFERENTIAL-EQUATIONS; VISCOUS-FLOW; PROBLEMS; NON-NEWTONIAN FLUIDS; ANALYTIC SOLUTION; SERIES SOLUTIONS; YAMABE-EQUATION; 2ND KIND; PERTURBATION SOLUTION; FORCED-CONVECTION; Mathematics, Applied

    Abstract

    In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman-Forchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville's equation.

    Journal Title

    Numerical Algorithms

    Volume

    61

    Issue/Number

    4

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    613

    Last Page

    629

    WOS Identifier

    WOS:000310999600005

    ISSN

    1017-1398

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