Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse
MetadataShow full item record
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.
Showing items related by title, author, creator and subject.
Wang, Kan ; El-Mowafy, Ahmed (2021)In road transport, continuous high-accuracy positioning is required in real time. To ensure the proper functioning and safety of vehicular applications, integrity monitoring (IM) is needed to protect from the positioning ...
Characterizing the optical variability of bright blazars: Variability-based selection of fermi active galactic nucleiRuan, J.; Anderson, S.; MacLeod, C.; Becker, A.; Burnett, T.; Davenport, J.; Ivezic, Z.; Kochanek, C.; Plotkin, Richard; Sesar, B.; Scott Stuart, J. (2012)We investigate the use of optical photometric variability to select and identify blazars in large-scale time-domain surveys, in part to aid in the identification of blazar counterparts to the ~ 30% of ?-ray sources in the ...
Prakhova, Sofya (2014)In this thesis we present a new approach for modelling insolation from the space perspective. The incoming radiation was modelled as a flux passing through a cross-section (latitudinal belt), and the changes of light ...