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D. M. Young, “Iterative Methods for Solving Partial Difference Equations of Elliptic Type,” Transactions on American Mathematical Society, Vol. 76, 1954, pp. 92-111. doi:10.1090/S0002-9947-1954-0059635-7

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• TITLE: Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

  AUTHORS: Diego Caratelli, Johan Gielis, Ilia Tavkhelidze, Paolo Emilio Ricci

  KEYWORDS: Robin Problem; Helmholtz Equation; Spherical Harmonic Expansion; Gielis Formula; Supershaped Shell

  JOURNAL NAME: Applied Mathematics, Vol.4 No.1A, January 30, 2013

  ABSTRACT: The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of 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.