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

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.

From Pythagoras to Fourier and From Geometry to Nature

“…Further extensions have been made to the case of 3𝐷 domains, but the relevant equations are much more involved. A list of such articles can be found in the References section (see [9,10,13,15,[19][20][21]45]).…”