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    Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse

    Access Status
    Fulltext not available
    Authors
    Keady, Grant
    Wiwatanapataphee, Benchawan
    Type
    Journal Article
    
    Metadata
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    Citation
    Keady, G. and Wiwatanapataphee, B. Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse.
    Faculty
    Faculty of Science and Engineering
    School
    Engineering Support Services
    School of Electrical Engineering, Computing and Mathematical Sciences (EECMS)
    URI
    http://hdl.handle.net/20.500.11937/79264
    Collection
    • Curtin Research Publications
    Abstract

    We study the problem of finding functions, defined within and on an ellipse, whose Laplacian is -1 and which satisfy a homogeneous Robin boundary condition on the ellipse. The parameter in the Robin condition is denoted by beta. The general solution and various asymptotic approximations are obtained. To find the general solution, the boundary value problem is formulated in elliptic cylindrical coordinates. A Fourier series solution is then derived. The Fourier coefficients satisfy a 3-term recurrence relation which can be solved. The integral of the solution over the ellipse, denoted by Q, is a quantity of interest in some physical applications. The dependence of Q on beta and the ellipse geometry is found. Finding asymptotics directly from the pde formulations is easier than from our series solution. We use the asymptotic approximations to Q as checks on the series solution. Several other inequalities are also used to check the solution. It is intended that this arXiv preprint will be referenced by the journal version, which will be submitted soon, as the arXiv contains material, e.g. codes for calculating Q, not in the much shorter journal version. Maple codes used in deriving or checking results in this paper are in the process of being tidied prior to being made available via links given at the URL given in the pdf version.

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