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Abstract: The internal and external Robin problems for the Laplace equation in bounded starlike domains are addressed. We show how to derive the relevant solutions by using a suitable Fourier series-like method. Numerical results are specifically obtained considering three-dimensional domains whose boundary is defined by a generalization of the so-called “superformula” introduced by Gielis. By using the computer algebra code Mathematica©, truncated series approximations of the solutions are determined. Our findings are in good agreement with the theoretical results on the Fourier series due to Carleson. [Copyright &y& Elsevier]

Abstract: The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson. [Copyright &y& Elsevier]