The Dirichlet problem for the Laplace equation in supershaped annuli: [Doc 369]

Abstract

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^sup ©^ 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.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

The Dirichlet problem for the Laplace equation in supershaped annuli

Diego Caratelli1, Johan Gielis2*, Ilia Tavkhelidze3 and Paolo E Ricci4

*Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Bioscience Engineering, University of Antwerp, Antwerp, BelgiumFull list of author information is available at the end of the article

Abstract

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 dened 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.

Introduction

Many problems of mathematical physics and electromagnetics are related to the Laplacian []. In recent papers [], the classical Fourier projection method [, ] for solving boundary-value problems (BVPs) for the Laplace and Helmholtz equations in canonical domains has been extended in order to address similar dierential 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 coecients 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 dened 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 ndings by Carleson [].

The Laplacian in stretched polar coordinates

Let us introduce in the real plane the usual polar coordinate system

x = r cos ,y = r sin , ()

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and the polar equations

r = R(), ()

relevant to the boundaries of the supershaped annulus A which is described by the fol

lowing chain of inequalities:

R() r R+(), ()

with . In () R() are assumed to be piece-wise C functions satisfying the

condition

R+() > R() > , . ()

In this way, upon introducing the stretched radius such that

r = (b

)R() (a )R+()b a , ()

with b > a > , the considered annular domain A can be readily obtained by assuming

and a b.

Remark Note that in the stretched coordinate system , , the original domain A is trans

formed 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 [].

Let us consider a piece-wise C(

A) function v(x, y) = v(r cos , r sin ) = u(r, ) and the

Laplace operator in polar coordinates

u =

u

r +

r

u

r +

r

u . ()

In the considered stretched coordinate system can be represented by setting

U( , ) = u

(b )R() (a )R+()b a ,

. ()

In this way, by denoting R() as R for the sake of shortness, one can readily ndu

r =

b a R+ R

U

, ()

u

r =

b a R+ R

U , ()

u =

(b )[ R( R+ R) R(R+ R)] (R+ R)

(a

)[ R+( R+ R) R+(R+ R)]

(R+ R)

U

+

(b ) R (a ) R+ R+ R

U (b ) R (a ) R+R+ RU +U , ()

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where the dot superscript denotes the dierentiation with respect to the angle . Substituting equations ()-() into equation () nally yields

u =

b a R+ R

+

(b ) R (a ) R+ (b )R (a )R+

U

(b )[(R R)(R+ R) + R( R+ R)] [(b )R (a )R+]

(a

+

)[(R+ R+)(R+ R) + R+( R+ R)][(b )R (a )R+]

U

(R+ R) (b

) R (a ) R+ [(b )R (a )R+]

U

R+ R

+ (b )R (a )R+

U

. ()

As it can be easily noticed, upon setting R() = a = and R+() = b = , 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

A, whose boundaries A are described by the polar equations r = R() respectively

v(x, y) = , (x, y) A,

v(x, y) = f(x, y), (x, y) A.

()

Under the mentioned assumptions, one can prove the following theorem.

Theorem Let

f R() cos , R() sin

= F() =

+

m=

()m cos m + ()m sin m

, ()

where

()m ()m

cos msin m d, ()

m being the usual Neumann symbol. Then the boundary-value problem () for the Laplace equation admits a classical solution v(x, y) L(A) such that the following Fourier-

like series expansion holds true:

v

(b )R() (a )R+() b a

=

m

F(

)

cos , (b )R() (a )R+() b a

sin

= U( , )

=

+

m=

(b )R() (a )R+() b a

m(Am cos m + Bm sin m)

+ ln

(b )R() (a )R+() b a

. ()

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For each index m, dene

()m() ()m()

= R(

, ()

and set, for shortness, ()() = ln R(). In this way, the coecients and Am, Bm appearing in () can be determined by solving the innite linear system

+

m=

)

cos m sin m

X()Cn,m Y()Cn,m

X()Sn,m Y()Sn,m

X(+)Cn,m Y(+)Cn,m

X(+)Sn,m Y(+)Sn,m

Am Bm

+

Z()Cn

Z()Sn

Z(+)Cn

Z(+)Sn

=

()n ()n (+)n (+)n

, ()

where

X()

C S

n,m =

n

()m()

cos nsin n d, ()

Y()

C S

n,m =

n

()m()

cos nsin n d, ()

Z()

C S

n =

n

()()

cos nsin n d, ()

with m

Z, and n

N := N {}.

Proof Upon noting that in the stretched coordinate system , introduced in the x, y plane, the considered domain A turns into the circular annulus of radii a and b, one can

readily adopt the usual eigenfunction method [] in combination with the separation of variables (with respect to r and ). As a consequence, elementary solutions of the problem can be searched in the form

u(r, ) = U

b[r R()] a[r R+()] R+() R() ,

= P(r) (). ()

Substituting into the Laplace equation, one easily nds that the functions P(), () must

satisfy the ordinary dierential equations

d ()

d +

() = , ()

r dP(r) dr + r

P(r) = , ()

respectively. The parameter is a separation constant whose choice is governed by the physical requirement that at any xed point in the real plane the scalar eld u(r, ) must be single-valued. So, by setting = m

N, one can easily nd

() = am cos m + bm sin m, ()

dP(r) dr

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C denote arbitrary constants. The radial function P() satisfying () can

be readily expressed as follows:

P(r) =

where am, bm

cmrm + dmrm, m = ,c + d ln r, m = , ()

with cm, dm

C. Therefore, the general solution of the Dirichlet problem () can be searched in the form

u(r, ) =

+

m=rm(Am cos m + Bm sin m) + ln r. ()

Enforcing the Dirichlet boundary condition readily yields F() = u(R(), ). Hence, using the classical Fourier projection method, equations ()-() follow after some trivial manipulations.

It is worth noting that the derived expressions still hold under the assumption that R() are piecewise continuous functions, and the boundary values are described by square integrable, not necessarily continuous, functions so that the relevant Fourier coecients ()m, ()m in equation () are nite quantities.

Numerical procedure

In the following numerical examples, let us assume, for the boundaries A of the con

sidered annulus, general polar equations of the type

R() =

dx

cos kx

x

+

dy

sin ky

y

/, ()

as introduced by Gielis in []. Very dierent characteristic geometries, including ellipses, Lam curves, ovals, and m-fold symmetric gures are obtained by assuming suitable values of the parameters kx, ky, dx, dy, x, y, in (). It is emphasized that almost

all two-dimensional normal-polar annular domains can be described, or closely approximated, by ().

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:

eN = UN(a,

) F()

+ UN(b,

) F+()

, ()

with being the usual L norm, and where UN( , ) denotes the partial sum of order

N relevant to the Fourier-like series expansion representing the solution of the boundary-value problem for the Laplace equation, namely

UN( , ) =

N

m=N

F()

F+()

(b )R() (a )R+() b a

m(Am cos m + Bm sin m)

+ ln

(b )R() (a )R+() b a

. ()

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Figure 1 Relative boundary error eN as a function of the order N of the truncated Fourier-like series expansion representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A

described by the Gielis formula with parameters kx = ky = 3, dx = dy = 3/4, d+x = d+y = 5/2, x = y = 12, 0 = 21.

Figure 2 Boundary behavior along A (a) and +A (b) of the partial sum UN( , ) of order N = 22

representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis formula with parameters kx = ky = 3, dx = dy = 3/4, d+x = d+y = 5/2,

x = y = 12, 0 = 21.

Remark It is to be noticed that where the boundary values exhibit a rapidly oscillating behavior, the order N of the expansion () approximating the solution of the problem should be increased accordingly in order to achieve the desired numerical accuracy.

First example

By assuming in () kx = ky = , dx = dy = /, d+x = d+y = /, x = y = , = , the annulus A features a triangular strip-like shape. Let f(x, y) = ex+y + iy and f+(x, y) =

cos x+y + i sin xy be the functions describing the boundary values. Under these assumptions, the relative boundary error eN as a function of the number N of terms in the truncated series expansion () exhibits the behavior shown in Figure . As it appears from Figure , the selection of the expansion order N = leads to a very accurate Fourier-like representation vN(x, y) of the solution (featuring boundary error eN < .%). The spatial distribution of vN(x, y) is shown in Figure , whereas the magnitude and phase of the relevant Fourier expansion coecients Am and Bm (|m| N) are plotted in Figure .

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 dicult to show that the general k-sided convex regular polygon can be readily described by the

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Figure 3 Spatial distribution of the Fourier-like series expansion vN(x, y) of order N = 22 representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis

formula with parameters kx = ky = 3, dx = dy = 3/4, d+x = d+y = 5/2, x = y = 12, 0 = 21.

Figure 4 Magnitude (a), (b) and phase (c), (d) of the coefcients Am and Bm relevant to the expansion vN(x, y) of order N = 22 representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis formula with parameters kx = ky = 3,dx = dy = 3/4, d+x = d+y = 5/2, x = y = 12, 0 = 21.

following specialized version of Gielis formula []:

Rk() = lim

+

d

cos k

( log cos

d

sin k

( log cos

k )

+

k )

/. ()

In this way, the methodology detailed in the previous section can be used straightforwardly. In particular, upon assuming in () kx = ky = , dx = /, dy = , x = , y =

= , as well as k+x = k+y = k+ = , d+x = d+y = /, and +x = +y = ( + log cos k+ ), with

+ +, the annulus A may be regarded as the result of the Boolean subtraction of an

ovaloid from a square. Let f(x, y) = sinh(x + y) + i cosh(xy) and f+(x, y) = be the functions describing the boundary values along A, respectively. Under these assumptions, the

relative boundary error eN exhibits the behavior shown in Figure . As it appears from

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Figure 5 Relative boundary error eN as a function of the order N of the truncated Fourier-like series expansion representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A

described by the Gielis formula with parameters kx = ky = 2, dx = 1/2, dy = 1, x = 2, y = 0 = 4, k+x = k+y = k+ = 4, d+x = d+y = 7/4, and +x = +y = 2(1 +0 log2 cos

k+ ), +0 +.

Figure 6 Boundary behavior along A (a) and +A (b) of the partial sum UN( , ) of order N = 16

representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis formula with parameters kx = ky = 2, dx = 1/2, dy = 1, x = 2,

y = 0 = 4, k+x = k+y = k+ = 4, d+x = d+y = 7/4, and +x = +y = 2(1 +0 log2 cos

k+ ), +0 +.

Figure 7 Spatial distribution of the Fourier-like series expansion vN(x, y) of order N = 16 representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis

formula with parameters kx = ky = 2, dx = 1/2, dy = 1, x = 2, y = 0 = 4, k+x = k+y = k+ = 4, d+x = d+y = 7/4, and +x = +y = 2(1 +0 log2 cos

k+ ),

+0 +.

Figure , the selection of the expansion order N = results in an extremely accurate Fourier-like series representation vN(x, y) of the solution (with boundary error eN < .%). The spatial distribution of vN(x, y) is shown in Figure , whereas the magnitude and phase of the relevant Fourier expansion coecients Am, Bm are plotted in Figure .

Remark It has been observed that an L norm of the dierence between the exact solution and the relevant approximation is generally negligible. Point-wise convergence seems to be veried 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.

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Figure 8 Magnitude (a), (b) and phase (c), (d) of the coefcients Am and Bm relevant to the expansion vN(x, y) of order N = 16 representing the solution of the Dirichlet problem for the Laplace equation in the supershaped annulus A described by the Gielis formula with parameters kx = ky = 2, dx = 1/2,

dy = 1, x = 2, y = 0 = 4, k+x = k+y = k+ = 4, d+x = d+y = 7/4, and +x = +y = 2(1 +0 log2 cos

k+ ),

+0 +.

Conclusion

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 BVPs can be derived by using classical quadrature rules, thus overcoming the need for cumbersome numerical techniques such as nite-dierence or nite-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 veried 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 ndings 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 verication 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 nal manuscript.

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Author details

1Microwave Sensing, Signals and Systems, Delft University of Technology, Delft, The Netherlands. 2Department of Bioscience Engineering, University of Antwerp, Antwerp, Belgium. 3Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia. 4Faculty of Engineering, Campus Bio-Medico University, Rome, Italy.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research has been carried out under the grant PRIN/2006 Cap. 7320.

Received: 20 December 2012 Accepted: 17 April 2013 Published: 3 May 2013

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doi:10.1186/1687-2770-2013-113Cite this article as: Caratelli et al.: The Dirichlet problem for the Laplace equation in supershaped annuli. Boundary Value Problems 2013 2013:113.