Exact and analytical solutions for a nonlinear sigma model

    Authors

    M. Baxter;R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Math. Meth. Appl. Sci.

    Keywords

    nonlinear sigma model; exact solution; approximate solution; control of; error; auxiliary linear operator; HOMOTOPY ANALYSIS METHOD; VISCOUS-FLOW PROBLEMS; NON-NEWTONIAN FLUIDS; EMDEN-FOWLER TYPE; SERIES SOLUTIONS; DIFFERENTIAL-EQUATIONS; GENERAL-APPROACH; WAVES; Mathematics, Applied

    Abstract

    We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error-minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright (c) 2013 John Wiley & Sons, Ltd.

    Journal Title

    Mathematical Methods in the Applied Sciences

    Volume

    37

    Issue/Number

    11

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    1642

    Last Page

    1651

    WOS Identifier

    WOS:000338011300007

    ISSN

    0170-4214

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