Optimal analytic method for the nonlinear Hasegawa-Mima equation

    Authors

    M. Baxter; R. A. Van Gorder;K. Vajravelu

    Comments

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    Abbreviated Journal Title

    Eur. Phys. J. Plus

    Keywords

    HOMOTOPY ANALYSIS METHOD; MAGNETIZED NONUNIFORM PLASMA; VISCOUS-FLOW; PROBLEMS; NON-NEWTONIAN FLUIDS; EMDEN-FOWLER TYPE; SERIES SOLUTIONS; DIFFERENTIAL-EQUATIONS; GENERAL-APPROACH; TURBULENCE; WAVES; Physics, Multidisciplinary

    Abstract

    The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10(-15) are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.

    Journal Title

    European Physical Journal Plus

    Volume

    129

    Issue/Number

    5

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    14

    WOS Identifier

    WOS:000336613900001

    ISSN

    2190-5444

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