The Dirichlet problem for the Laplace equation in supershaped annuli
© Caratelli et al.; licensee Springer. 2013
Received: 20 December 2012
Accepted: 17 April 2013
Published: 3 May 2013
The Dirichlet problem for the Laplace equation in normal-polar annuli is addressed by using a suitable Fourier-like technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by 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.
Many problems of mathematical physics and electromagnetics are related to the Laplacian . In recent papers [2–9], the classical Fourier projection method [10, 11] for solving boundary-value problems (BVP s) for the Laplace and Helmholtz equations in canonical domains has been extended in order to address similar differential problems in simply connected starlike domains, whose boundaries may be regarded as an anisotropically stretched unit circle centered at the origin.
In this contribution, a suitable technique useful to compute the coefficients of the Fourier-like expansion representing the solution of the Dirichlet boundary-value problem for the Laplace equation in complex annular domains is presented. In particular, the boundaries of the considered domains are supposed to be defined by the so-called Gielis formula . Regular functions are assumed to describe the boundary values, but the proposed approach can be easily generalized in the case of weakened hypotheses. In order to verify and validate the developed methodology, a suitable numerical procedure based on the computer algebra system Mathematica© has been adopted. By using such a procedure, a point-wise convergence of the Fourier-like series representation of the solution has been observed in the regular points of the boundaries, with Gibbs-like phenomena potentially occurring in the quasi-cusped points. The obtained numerical results are in good agreement with theoretical findings by Carleson .
The Laplacian in stretched polar coordinates
with , the considered annular domain can be readily obtained by assuming and .
Remark Note that in the stretched coordinate system , the original domain is transformed into the circular annulus of radii a and b, respectively. Hence, in this system one can use classical techniques to solve the Laplace equation, including the eigenfunction method .
As it can be easily noticed, upon setting and , the classical expression of the Laplacian in polar coordinates is recovered.
The Dirichlet problem for the Laplace equation
Under the mentioned assumptions, one can prove the following theorem.
with , and .
It is worth noting that the derived expressions still hold under the assumption that are piecewise continuous functions, and the boundary values are described by square integrable, not necessarily continuous, functions so that the relevant Fourier coefficients , in equation (14) are finite quantities.
as introduced by Gielis in . Very different characteristic geometries, including ellipses, Lamé curves, ovals, and m-fold symmetric figures are obtained by assuming suitable values of the parameters , , , , , , in (27). It is emphasized that almost all two-dimensional normal-polar annular domains can be described, or closely approximated, by (27).
Remark It is to be noticed that where the boundary values exhibit a rapidly oscillating behavior, the order N of the expansion (29) approximating the solution of the problem should be increased accordingly in order to achieve the desired numerical accuracy.
Remark It has been observed that an norm of the difference between the exact solution and the relevant approximation is generally negligible. Point-wise convergence seems to be verified in the considered domains, with the only exception of a set of measure zero consisting of quasi-cusped points. In the neighborhood of these points, oscillations of the truncated order solution, recalling the classical Gibbs phenomenon, usually take place.
A Fourier-like projection method, in combination with the adoption of a suitable stretched coordinate system, has been developed for solving the Dirichlet problem for the Laplace equation in supershaped annuli. In this way, analytically based expressions of the solution of the considered class of BVP s can be derived by using classical quadrature rules, thus overcoming the need for cumbersome numerical techniques such as finite-difference or finite-element methods. The proposed approach has been successfully validated by means of a dedicated numerical procedure based on the computer-aided algebra tool Mathematica©. A point-wise convergence of the expansion series representing the solution seems to be verified with the only exception of a set of measure zero consisting of the quasi-cusped points along the boundary of the problem domain. In these points, Gibbs-like oscillations may occur. The computed results are found to be in good agreement with the theoretical findings on Fourier series.
Dedicated to Professor Hari M Srivastava.
This research has been carried out under the grant PRIN/2006 Cap. 7320.
- Gakhov FD: Boundary Value Problems. Dover, New York; 1990.Google Scholar
- Natalini P, Patrizi R, Ricci PE: The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface. Numer. Algorithms 2001, 28: 215-227. 10.1023/A:1014059219005MathSciNetView ArticleGoogle Scholar
- Caratelli D, Ricci PE: The Dirichlet problem for the Laplace equation in a starlike domain. Proceedings of the International Conference on Scientific Computing, Las Vegas, 14-17 July 2008 2008, 160-166.Google Scholar
- Caratelli D, Germano B, Gielis J, He MX, Natalini P, Ricci PE Lecture Notes of Tbilisi International Centre of Mathematics and Informatics. In Fourier Solution of the Dirichlet Problem for the Laplace and Helmholtz Equations in Starlike Domains. Tbilisi University Press, Tbilisi; 2010.Google Scholar
- Caratelli D, Natalini P, Ricci PE, Yarovoy A: The Neumann problem for the Helmholtz equation in a starlike planar domain. Appl. Math. Comput. 2010, 216: 556-564. 10.1016/j.amc.2010.01.077MathSciNetView ArticleGoogle Scholar
- Caratelli D, Gielis J, Natalini P, Ricci PE, Tavkelidze I: The Robin problem for the Helmholtz equation in a starlike planar domain. Georgian Math. J. 2011, 18: 465-480.MathSciNetGoogle Scholar
- Caratelli D, Gielis J, Ricci PE: Fourier-like solution of the Dirichlet problem for the Laplace equation in k -type Gielis domains. J. Pure Appl. Math., Adv. Appl. 2011, 5: 99-111.MathSciNetGoogle Scholar
- Caratelli D, Ricci PE, Gielis J: The Robin problem for the Laplace equation in a three-dimensional starlike domain. Appl. Math. Comput. 2011, 218: 713-719.MathSciNetGoogle Scholar
- Gielis J, Caratelli D, Fougerolle Y, Ricci PE, Gerats T: Universal natural shapes from unifying shape description to simple methods for shape analysis and boundary value problems. PLoS ONE 2012. doi:10.1371/journal.pone.0029324Google Scholar
- Tolstov GP: Fourier Series. Dover, New York; 1962.Google Scholar
- Lebedev NN: Special Functions and Their Applications. Dover, New York; 1972.Google Scholar
- Gielis J: A generic geometric transformation that unifies a wide range of natural and abstract shapes. Am. J. Bot. 2003, 90: 333-338. 10.3732/ajb.90.3.333View ArticleGoogle Scholar
- Carleson L: On convergence and growth of partial sums of Fourier series. Acta Math. 1966, 116: 135-157. 10.1007/BF02392815MathSciNetView ArticleGoogle Scholar
- Lenjou, K: Krommen en oppervlakken van Lamé and Gielis. MSc thesis, Catholic University of Leuven (2005)