The Dirichlet problem for the Laplace equation in supershaped annuli
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RESEARCH
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The Dirichlet problem for the Laplace equation in supershaped annuli Diego Caratelli1 , Johan Gielis2* , Ilia Tavkhelidze3 and Paolo E Ricci4 *
Correspondence: [email protected] Department of Bioscience Engineering, University of Antwerp, Antwerp, Belgium Full list of author information is available at the end of the article 2
Abstract The Dirichlet problem for the Laplace equation in normal-polar annuli is addressed by using a suitable Fourier-like technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called ‘superformula’ introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Introduction Many problems of mathematical physics and electromagnetics are related to the Laplacian []. In recent papers [–], the classical Fourier projection method [, ] for solving boundary-value problems (BVPs) for the Laplace and Helmholtz equations in canonical domains has been extended in order to address similar differential problems in simply connected starlike domains, whose boundaries may be regarded as an anisotropically stretched unit circle centered at the origin. In this contribution, a suitable technique useful to compute the coefficients of the Fourier-like expansion representing the solution of the Dirichlet boundary-value problem for the Laplace equation in complex annular domains is presented. In particular, the boundaries of the considered domains are supposed to be defined by the so-called Gielis formula []. Regular functions are assumed to describe the boundary values, but the proposed approach can be easily generalized in the case of weakened hypotheses. In order to verify and validate the developed methodology, a suitable numerical procedure based on the computer algebra system Mathematica© has been adopted. By using such a procedure, a point-wise convergence of the Fourier-like series representation of the solution has been observed in the regular points of the boundaries, with Gibbs-like phenomena potentially occurring in the quasi-cusped points. The obtained numerical results are in good agreement with theoretical findings by Carleson []. The Laplacian in stretched polar coordinates Let us introduce in the real plane the usual polar coordinate system
x = r cos ϑ, y = r sin ϑ,
()
© 2013 Caratelli et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Caratelli et al. Boundary Value Problems 2013, 2013:113 http://www.boundaryvalueproblems.com/content/2013/1/113
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and the polar equations r = R± (ϑ),
()
relevant to the boundaries
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