IMA Journal of Applied Mathematics Advance Access originally published online on September 7, 2009
IMA Journal of Applied Mathematics 2009 74(5):668-684; doi:10.1093/imamat/hxp023
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Analytic solution of an exterior Dirichlet problem in a non-convex domain
School of Science and Technology, Hellenic Open University, 167 Riga Feraiou Street, GR-26 222 Patras, Greece
Email: hadjinicolaou{at}eap.gr
Received on August 10, 2008; Revision received March 18, 2009. Accepted on July 22, 2009
An exterior Dirichlet problem in a non-convex domain is solved analytically by combining two powerful methods: the well-known Kelvin transformation and the newly established method of generalized integral transforms introduced by Fokas (2001, Proc. R. Soc. A., 457, 371–393). In fact, our approach leads to an integral representation for the solution of Laplace's equation in the unbounded domain formed by the exterior of the Kelvin image of an equilateral triangle. First, we apply the Kelvin transformation of the given boundary with arbitrary data. Second, we use the Dirichlet-to-Neumann map (Dassios & Fokas, 2005, Proc. R. Soc. A., 461, 2721–2748). to obtain the unknown Neumann boundary values on the image boundary. Third, by inverting the Kelvin transformation, we derive the Neumann data on the original boundary. We demonstrate the 2D version of Kelvin transformation and we apply it to the equilateral triangle, which, through the Dirichlet-to-Neumann map, leads us in a natural way to a known integral representation of the solution.
Keywords: exterior Dirichlet problem; analytic solution of Laplace's equation; equilateral triangle; Kelvin inversion; general transform method; Dirichlet-to-Neumann map.