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

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

Department of Bioscience Engineering, University of Antwerp, Antwerp, Belgium.
Faculty of Engineering, Campus Bio-Medico University, Rome, Italy.
Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia.
Microwave Sensing, Signals and Systems, Delft University of Technology, Delft, The Netherlands.

DOI: 10.4236/am.2013.41A040                

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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. N. Lebedev, “Special Functions and Their Applications,” Dover Inc., New York, 1972.
[2]	G. Krall, “Meccanica Tecnica Delle Vibrazioni,” Vol. 2, Veschi, Roma, 1970.
[3]	A. Bondeson, T. Rylander and P. Ingelstrom, “Computational Electromagnetics,” Springer Science, New York, 2005.
[4]	D. Medková, “Solution of the Dirichlet Problem for the Laplace Equation,” Applications of Mathematics, Vol. 44, No. 2, 1999, pp. 143-168. doi:10.1023/A:1022209421576
[5]	B. N. Khoromski, “Integro-Difference Method of Solution of the Dirichlet Problem for the Laplace Equation,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, Vol. 24, No. 1, 1984, pp. 53-64.
[6]	A. P. Volkov, “An Effective Method for Solving the Dirichlet Problem for the Laplace Equation,” Differentsial’nye Uravneniya, Vol. 19, 1983, pp. 1000-1007.
[7]	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
[8]	G. P. Tolstov, “Fourier Series,” Dover Inc., New York, 1962.
[9]	P. Natalini, R. Patrizi, and P. E. Ricci, “The Dirichlet Problem for the Laplace Equation in a Starlike Domain of a Riemann Surface,” Numerical Algorithms, Vol. 28, No. 1-4, 2001, pp. 215-227. doi:10.1023/A:1014059219005
[10]	D. Caratelli and P. E. Ricci, “The Dirichlet Problem for the Laplace Equation in a Starlike Domain,” Proceedings of International Conference on Scientific Computing, Las Vegas, 14-17 July 2008, pp. 160-166.
[11]	D. Caratelli, B. Germano, J. Gielis, M. X. He, P. Natalini and P. E. Ricci, “Fourier Solution of the Dirichlet Problem for the Laplace and Helmholtz Equations in Starlike Domains,” Lecture Notes of Tbilisi International Centre of Mathematics and Informatics, Tbilisi University Press, Tbilisi, 2010.
[12]	D. Caratelli, P. Natalini, P. E. Ricci and A. Yarovoy, “The Neumann Problem for the Helmholtz Equation in a Starlike Planar Domain,” Applied Mathematics and Computation, Vol. 216, No. 2, 2010, pp. 556-564. doi:10.1016/j.amc.2010.01.077
[13]	D. Caratelli, J. Gielis, P. Natalini, P. E. Ricci and I. Tavkelidze, “The Robin Problem for the Helmholtz Equation in a Starlike Planar Domain,” Georgian Mathematical Journal, Vol. 18, No. 3, 2011, pp. 465-480.
[14]	D. Caratelli, J. Gielis and P. E. Ricci, “Fourier-Like Solution of the Dirichlet Problem for the Laplace Equation in k-Type Gielis Domains,” Journal of Pure and Applied Mathematics: Advances and Applications, Vol. 5, No. 2, 2011, pp. 99-111.
[15]	D. Caratelli, P. E. Ricci and J. Gielis, “The Robin Problem for the Laplace Equation in a Three-Dimensional Starlike Domain,” Applied Mathematics and Computation, Vol. 218, No. 3, 2011, pp. 713-719.
[16]	J. Gielis, D. Caratelli, Y. Fougerolle, P. E. Ricci and T. Gerats, “Universal Natural Shapes from Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems,” PLoS One, Vol. 7, No. 9, 2012, Article ID: e29324. doi:10.1371/journal.pone.0029324
[17]	G. Dattoli, B. Germano, M. R. Martinelli and P. E. Ricci, “A Novel Theory of Legendre Polynomials,” Mathematical and Computer Modelling, Vol. 54, No. 1-2, 2011, pp. 80-87. doi:10.1016/j.mcm.2011.01.037
[18]	J. Gielis, “A Generic Geometric Transformation that Unifies a Wide Range of Natural and Abstract Shapes,” American Journal of Botany, Vol. 90, No. 3, 2003, pp. 333-338. doi:10.3732/ajb.90.3.333
[19]	L. Carleson, “On Convergence and Growth of Partial Sums of Fourier Series,” Acta Mathematica, Vol. 116, No. 1, 1966, pp. 135-157. doi:10.1007/BF02392815