Abstract
The Dirichlet problem for the Laplace equation in normalpolar annuli is addressed by using a suitable Fourierlike 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 [1]. In recent papers [29], the classical Fourier projection method [10,11] for solving boundaryvalue problems (BVPs) 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 Fourierlike expansion representing the solution of the Dirichlet boundaryvalue 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 socalled Gielis formula [12]. 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 pointwise convergence of the Fourierlike 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 [13].
The Laplacian in stretched polar coordinates
Let us introduce in the real plane the usual polar coordinate system
and the polar equations
relevant to the boundaries of the supershaped annulus which is described by the following chain of inequalities:
with . In (2) are assumed to be piecewise functions satisfying the condition
In this way, upon introducing the stretched radius ϱ such that
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 [11].
Let us consider a piecewise function and the Laplace operator in polar coordinates
In the considered stretched coordinate system Δ can be represented by setting
In this way, by denoting as for the sake of shortness, one can readily find
where the dot superscript denotes the differentiation with respect to the angle ϑ. Substituting equations (8)(10) into equation (6) finally yields
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
Let us consider the interior Dirichlet problem for the Laplace equation in a starlike annulus , whose boundaries are described by the polar equations respectively
Under the mentioned assumptions, one can prove the following theorem.
TheoremLet
where
being the usual Neumann symbol. Then the boundaryvalue problem (12) for the Laplace equation admits a classical solutionsuch that the following Fourierlike series expansion holds true:
For each indexm, define
and set, for shortness, . In this way, the coefficientsand, appearing in (15) can be determined by solving the infinite linear system
where
Proof Upon noting that in the stretched coordinate system introduced in the x, y plane, the considered domain turns into the circular annulus of radii a and b, one can readily adopt the usual eigenfunction method [11] in combination with the separation of variables (with respect to r and ϑ). As a consequence, elementary solutions of the problem can be searched in the form
Substituting into the Laplace equation, one easily finds that the functions , must satisfy the ordinary differential equations
respectively. The parameter μ is a separation constant whose choice is governed by the physical requirement that at any fixed point in the real plane the scalar field must be singlevalued. So, by setting , one can easily find
where denote arbitrary constants. The radial function satisfying (23) can be readily expressed as follows:
with . Therefore, the general solution of the Dirichlet problem (12) can be searched in the form
Enforcing the Dirichlet boundary condition readily yields . Hence, using the classical Fourier projection method, equations (17)(20) follow after some trivial manipulations. □
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.
Numerical procedure
In the following numerical examples, let us assume, for the boundaries of the considered annulus, general polar equations of the type
as introduced by Gielis in [12]. Very different characteristic geometries, including ellipses, Lamé curves, ovals, and mfold symmetric figures are obtained by assuming suitable values of the parameters , , , , , , in (27). It is emphasized that almost all twodimensional normalpolar annular domains can be described, or closely approximated, by (27).
In order to assess the performance of the proposed methodology in terms of numerical accuracy and convergence rate, the relative boundary error has been evaluated as follows:
with being the usual norm, and where denotes the partial sum of order N relevant to the Fourierlike series expansion representing the solution of the boundaryvalue problem for the Laplace equation, namely
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.
First example
By assuming in (27) , , , , , the annulus features a triangular striplike shape. Let and be the functions describing the boundary values. Under these assumptions, the relative boundary error as a function of the number N of terms in the truncated series expansion (29) exhibits the behavior shown in Figure 1. As it appears from Figure 2, the selection of the expansion order leads to a very accurate Fourierlike representation of the solution (featuring boundary error ). The spatial distribution of is shown in Figure 3, whereas the magnitude and phase of the relevant Fourier expansion coefficients and () are plotted in Figure 4.
Figure 1. Relative boundary erroras a function of the orderNof the truncated Fourierlike series expansion representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,.
Figure 2. Boundary behavior along(a) and(b) of the partial sumof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,.
Figure 3. Spatial distribution of the Fourierlike series expansionof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,.
Figure 4. Magnitude (a), (b) and phase (c), (d) of the coefficientsandrelevant to the expansionof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,.
Second example
In the second numerical example, we turn to the consideration of the class of annuli having one or both boundaries featuring a polygonal contour. In this respect, it is not difficult to show that the general ksided convex regular polygon can be readily described by the following specialized version of Gielis’ formula [14]:
In this way, the methodology detailed in the previous section can be used straightforwardly. In particular, upon assuming in (27) , , , , , as well as , , and , with , the annulus may be regarded as the result of the Boolean subtraction of an ovaloid from a square. Let and be the functions describing the boundary values along , respectively. Under these assumptions, the relative boundary error exhibits the behavior shown in Figure 5. As it appears from Figure 6, the selection of the expansion order results in an extremely accurate Fourierlike series representation of the solution (with boundary error ). The spatial distribution of is shown in Figure 7, whereas the magnitude and phase of the relevant Fourier expansion coefficients , are plotted in Figure 8.
Figure 5. Relative boundary erroras a function of the orderNof the truncated Fourierlike series expansion representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,,,, and,.
Figure 6. Boundary behavior along(a) and(b) of the partial sumof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,,,, and,.
Figure 7. Spatial distribution of the Fourierlike series expansionof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,,,, and,.
Figure 8. Magnitude (a), (b) and phase (c), (d) of the coefficientsandrelevant to the expansionof orderrepresenting the solution of the Dirichlet problem for the Laplace equation in the supershaped annulusdescribed by the Gielis formula with parameters,,,,,,, and,.
Remark It has been observed that an 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.
Conclusion
A Fourierlike 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 BVPs can be derived by using classical quadrature rules, thus 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 Fourier series.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DC proved the main theorem regarding the solution of the Laplace equation in supershaped annuli and drafted the paper. JG carried out the verification of the methodology and its application to Gielis domains. IT performed the numerical examples. PER derived the analytical expression of the Laplacian operator in stretched coordinates and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This research has been carried out under the grant PRIN/2006 Cap. 7320.
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