Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse
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 obtai...
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| Format: | Journal Article |
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| Online Access: | http://hdl.handle.net/20.500.11937/79264 |
| _version_ | 1848764025877299200 |
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| author | Keady, Grant Wiwatanapataphee, Benchawan |
| author_facet | Keady, Grant Wiwatanapataphee, Benchawan |
| author_sort | Keady, Grant |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T11:12:48Z |
| format | Journal Article |
| id | curtin-20.500.11937-79264 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T11:12:48Z |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-792642020-09-03T00:52:36Z Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse Keady, Grant Wiwatanapataphee, Benchawan math.AP math.AP 35J05 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. Journal Article http://hdl.handle.net/20.500.11937/79264 restricted |
| spellingShingle | math.AP math.AP 35J05 Keady, Grant Wiwatanapataphee, Benchawan Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse |
| title | Functions with constant Laplacian satisfying Robin boundary conditions
on an ellipse |
| title_full | Functions with constant Laplacian satisfying Robin boundary conditions
on an ellipse |
| title_fullStr | Functions with constant Laplacian satisfying Robin boundary conditions
on an ellipse |
| title_full_unstemmed | Functions with constant Laplacian satisfying Robin boundary conditions
on an ellipse |
| title_short | Functions with constant Laplacian satisfying Robin boundary conditions
on an ellipse |
| title_sort | functions with constant laplacian satisfying robin boundary conditions
on an ellipse |
| topic | math.AP math.AP 35J05 |
| url | http://hdl.handle.net/20.500.11937/79264 |