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

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The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical har-monic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are de-fined 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