FourierHankel solution of the Robin problem for the Helmholtz equation in supershaped annular domains
 Diego Caratelli^{1},
 Johan Gielis^{2, 3}Email author,
 Ilia Tavkhelidze^{4} and
 Paolo E Ricci^{5}
https://doi.org/10.1186/168727702013253
© Caratelli et al.; licensee Springer. 2013
Received: 16 August 2013
Accepted: 23 October 2013
Published: 22 November 2013
Abstract
The Robin problem for the Helmholtz equation in normalpolar annuli is addressed by using a suitable FourierHankel series technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the socalled 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.
Introduction
Many problems of mathematical physics and electromagnetics are related to the Laplacian:

The wave equation ${v}_{tt}={a}^{2}\mathrm{\Delta}v$;

The heat propagation ${v}_{t}=\kappa \mathrm{\Delta}v$;

The Laplace equation $\mathrm{\Delta}v=0$;

The Poisson equation $\mathrm{\Delta}v=f$;

The Helmholtz equation $\mathrm{\Delta}v+{k}^{2}v=0$;

The Schrödinger equation $\frac{{h}^{2}}{2m}\mathrm{\Delta}\psi +V\psi =E\psi $.
In recent papers [1–8], the classical Fourier projection method [9, 10] 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 FourierHankel expansion representing the solution of the Robin boundary value problem for the Helmholtz equation in complex annular domains is presented. In particular, the boundaries of the considered domains are supposed to be defined by the socalled Gielis formula [11]. Regular functions are assumed to describe the boundary values, but the proposed approach can be easily generalized in 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 pointwise convergence of the FourierHankel series representation of the solution has been observed in the regular points of the boundaries, with Gibbslike phenomena potentially occurring in the quasicusped points. The obtained numerical results are in good agreement with theoretical findings by Carleson [12].
The Laplacian in stretched polar coordinates
with $b>a>0$, the considered annular domain can be readily obtained by taking $a\le \varrho \le b$.
Remark Note that in the stretched coordinate system ϱ, ϑ, the original domain is transformed into the circular annulus of radii a and b, respectively. Hence, in this system one can use classical techniques, including the eigenfunction method, to solve the Helmholtz equation [10].
As it can be easily noticed, upon setting ${R}_{}(\vartheta )=a=0$ and ${R}_{+}(\vartheta )=b=1$, the classical expression of the Laplacian in polar coordinates is recovered.
The Robin problem for the Helmholtz equation
where $k>0$ denotes the propagation constant, ${\stackrel{\u02c6}{\nu}}_{\pm}={\stackrel{\u02c6}{\nu}}_{\pm}(\vartheta )$ are the outwardpointing normal unit vectors to the domain boundaries ${\partial}_{\pm}\mathcal{A}$, respectively, and ${\lambda}_{\pm}$, ${\gamma}_{\pm}$ are given regular weighting coefficients.
Under the mentioned assumptions, one can prove the following theorem.
with $p=1,2$ and $m,n\in {\mathbb{N}}_{0}:=\mathbb{N}\cup \{0\}$.
Hence, combining equations above and using the classical Fourier projection method, equations (17)(20) follow after some algebraic manipulations. □
It is worth noting that the derived expressions still hold under the assumption that ${R}_{\pm}(\vartheta )$ are piecewise continuous functions, and the boundary values are described by square integrable, not necessarily continuous, functions, so that the relevant Fourier coefficients ${\alpha}_{m}^{(\pm )}$, ${\beta}_{m}^{(\pm )}$ in equation (14) are finite quantities.
Numerical procedure
as introduced by Gielis in [11]. Very different characteristic geometries, including ellipses, Lamé curves, ovals, and mfold symmetric figures, are obtained by assuming suitable values of the parameters ${k}_{x}^{\pm}$, ${k}_{y}^{\pm}$, ${d}_{x}^{\pm}$, ${d}_{y}^{\pm}$, ${\nu}_{x}^{\pm}$, ${\nu}_{y}^{\pm}$, ${\nu}_{0}^{\pm}$ in (30). It is emphasized that almost all twodimensional normalpolar annular domains can be described, or closely approximated, by the abovementioned Gielis formula.
Remark It is to be noticed that where the boundary values exhibit a rapidly oscillating behavior, the number N of terms in expansion (32) approximating the solution of the problem should be increased accordingly in order to achieve the desired numerical accuracy.
First example
Second example
Remark It has been observed that ${L}^{2}$ norm of the difference between the exact solution and the relevant approximation is generally negligible. Pointwise convergence seems to be verified in the considered domains, with the only exception of a set of measure zero consisting of quasicusped points. In the neighborhood of these points, oscillations of the truncated order solution, recalling the classical Gibbs phenomenon, usually take place (see Figure 6).
Conclusion
A Fourierlike projection method, in combination with the adoption of a suitable stretched coordinate system, has been developed for solving the Robin problem for the Helmholtz 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, so overcoming the need for cumbersome numerical techniques such as finitedifference or finiteelement methods. The proposed approach has been successfully validated by means of a dedicated numerical procedure based on the computeraided algebra tool Mathematica^{©}. A pointwise 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 quasicusped points along the boundary of the problem domain. In these points, Gibbslike oscillations may occur. The computed results are found to be in good agreement with the theoretical findings on a Fourier series.
Declarations
Acknowledgements
This research has been carried out under the grant PRIN/2006 Cap. 7320.
Authors’ Affiliations
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