Trigonometric and hyperbolic type solutions to a generalized Drinfel'd-Sokolov equation

    Authors

    E. Sweet;R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Appl. Math. Comput.

    Keywords

    Generalized Drinfel'd-Sokolov equations; Exact solution; Analytical; solution; Nonlinear partial differential equation; HOMOTOPY ANALYSIS METHOD; NONLINEAR DIFFERENTIAL-EQUATIONS; TANH METHOD; W-ALGEBRAS; PERIODIC-SOLUTIONS; WILSON EQUATION; WAVE-EQUATIONS; REDUCTION; OPERATORS; KDV; Mathematics, Applied

    Abstract

    A class of trigonometric and hyperbolic type solutions to the generalized Drinfel'd-Sokolov (GDS) equations u(t) + alpha(1)uu(x) + beta(1)u(xxx) + gamma(v(delta))(x) - 0 and v(t) + alpha(2)uv(x) + beta(2)v(xxx) - 0 is obtained for the case in which alpha(2) = 0, for various values of the other model parameters. The method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter alpha(2), in effect extending the exact solutions. We do not assume traveling wave solution forms for the analytical solutions; that is, we solve the generalized Drinfel'd-Sokolov equations as PDEs without resorting to transforming the system to ODEs. An error analysis of the obtained approximate local analytical solutions is provided. Then, we outline a general framework by which one many construct solutions in either sine/cosine or sinh/cosh basis. We provide the general perturbation expansion via homotopy analysis, and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors. Travelling solutions with time-dependent amplitude are then discussed. (C) 2010 Elsevier Inc. All rights reserved.

    Journal Title

    Applied Mathematics and Computation

    Volume

    217

    Issue/Number

    8

    Publication Date

    1-1-2010

    Document Type

    Article

    Language

    English

    First Page

    4147

    Last Page

    4166

    WOS Identifier

    WOS:000284600700051

    ISSN

    0096-3003

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