Numerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION

Transcription

1 alaysia Joural of athematical Scieces 3(1): (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty of Sciece, Uiversiti Tekologi alaysia, 8131 Skudai, Johor. 1 alihassa@utm.my, tehyuayig@yahoo.com.sg ABSTRACT A iterpolatio formula based o Fourier method for the umerical solutio of a Fredholm itegral equatio related to coformal mappig of a simply coected regio oto a uit disc is preseted. The itegral equatio ivolves the euma kerel. The umerical results obtaied from the iterpolatio formula based o Fourier method are the compared with the umerical results obtaied from the iterpolatio formula based o yström s method. umerical compariso shows that the iterpolatio formula based o yström s method gives better performace. umerical implemetatios o some test regios are preseted. ITRODUCTIO It has bee established that Fourier method is equivalet to yström s method for the umerical solutio of Fredholm itegral equatio (Berrut ad Trummer, 1987). This implies that both methods will produce the same approximatio to the solutio of a Fredholm itegral equatio at the collocatio poits. However, Berrut ad Trummer (1987) did ot give ay umerical examples to support their fidigs. Furthermore, o umerical compariso has bee give for the performace of the iterpolatio formulas based o Fourier method ad yström s method. The aim of this paper is to provide relevat umerical examples to fill up this gap through umerical coformal mappig. Coformal mappig has bee a familiar tool of sciece ad egieerig for geeratios. The practical limitatio has always bee that oly for certai special regios are exact coformal mappig kow, while for the rest, they must be computed umerically. Thus several methods have bee developed for umerical coformal mappig. The mappig of a simply coected regio oto a uit disc is kow as a Riema mappig ad the mappig fuctio is kow as Riema mappig fuctio. The most commoly used method to compute the Riema mappig fuctio is derived from itegral equatios ivolvig the boudary correspodece fuctio that relates the two regios boudaries poitwise. Typically the

2 Ali Hassa ohamed urid & Teh Yua Yig boudaries are discretized at poits, so that the itegral equatio reduces to a algebraic system. A method to compute the Riema mappig fuctio via the Bergma kerel is preseted i Razali et al. (1997) which expresses the Bergma kerel as the solutio of a secod kid itegral equatio ivolvig the euma kerel. I Razali et al. (1997), the itegral equatio via the Bergma kerel has bee solved usig the iterpolatio formula based o yström s method with trapezoidal rule. I this paper, we shall use iterpolatio formula based o Fourier method to solve the itegral equatio ad compare umerically the performace of these two types of iterpolatio formulas. The umerical solutio of the itegral equatio will the be used to approximate the boudary correspodece fuctio for several test regios. The exact boudary correspodece fuctios for several test regios will also be calculated ad compared with the approximate results. The orgaizatio of this paper is as follows. Sectio cotais a brief review of the itegral equatio for the Bergma kerel. I Sectio 3 we show how to treat the itegral equatio umerically usig Fourier method. I Sectio 4, we make some umerical comparisos betwee iterpolatio formulas based o yström s ad Fourier methods usig some test regios ad draw some coclusio. ITEGRAL EQUATIO FOR THE BERGA KEREL Let Ω be a simply coected regio i the complex plae whose boudary Γ is assumed to be aalytic C Jorda curve. Suppose z = z( t) is the parametric represetatio of Γ with T z z t = z t deotes the uit taget i the directio of icreasig parameters at the poit z. Let a Ω ad let R be the Riema map of Ω whose boudary is Γ, with the usual ormalizatio R a =, R a >, a Ω. 84 alaysia Joural of athematical Scieces

3 umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod Due to Γ beig aalytic, R ca be exteded to a fuctio that is aalytic o Ω = Ω Γ. A classical relatioship betwee the Riema map ad the Bergma kerel is give by (, ) 1 B z a R( z) = T ( z), z Γ, i B z, a where B( z, a ) is the Bergma kerel which is aalytic o Ω (Razali et al., 1997). I Razali et al. (1997), it is show that the fuctio B ˆ ( z, a) = T ( z) B( z, a) is the uique cotiuous solutio to the itegral equatio ˆ ˆ 1 T z B z, a + z, w B( w, a) dw =, z Γ, (1) Γ π ( z a) where 1 T z Im, z, w Γ, w z π z w ( z, w) = 1 Im z ( t) z ( t), z = w Γ. z ( t) 3 () The real kerel is the familiar euma kerel which arises frequetly i the itegral equatios of potetial theory ad coformal mappig (Herici, 1986). Sice Γ is aalytic, the euma kerel is z, w Γ Γ. certaily cotiuous at all poits Assume the parametric represetatio of Γ is z z( t) Let w z( s) =, s, t β. =. Cosequetly, itegral equatio (1) ad euma kerel () ca be expressed as φ ( t) λ β v( t, s) φ ( s) ds = ψ ( t), (3) where λ = 1, ad for s, t β, φ t = z t Bˆ z t, a, (4) ( ) alaysia Joural of athematical Scieces 85

4 Ali Hassa ohamed urid & Teh Yua Yig ( s) z ( s) B( z ( s) a) φ = ˆ,, (5) ψ ( t) (, ) v t s 1 = π z ( t ) ( z( t) a), z ( t ) z ( s) 1 Im, if t s, π z t = 1 z Im ( t), if t = s. z ( t) (6) (7) ote that β is fiite, v is a give fuctio of two variables, ψ is a give fuctio, ad φ is to be determied. A importat fact is that λ = 1 is ot a eigevalue of v. The homogeous equatio correspodig to equatio (3) thus has the trivial solutio. By the Fredholm alterative (Kreyszig, 1978), the o-homogeeous equatio has exactly oe cotiuous solutio φ for ay cotiuous fuctio ψ. UERICAL IPLEETATIO USIG FOURIER ETHOD If we set β =, equatio (3) becomes (8) ( t) + v( t, s) ( s) ds = ( t) φ φ ψ ote that the fuctios φ, ψ ad v are -periodic. To implemet Fourier method, we choose equidistat collocatio poits s = t =, ad iterpolate the kerel v ad the ihomogeeity ψ by trigoometric polyomials, i.e., it ims v ( t, s) = ame e (9) = m= 86 alaysia Joural of athematical Scieces

5 umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod ad where ψˆ i ( t) d e = t = (1), if is odd, = + 1; = 1, if is eve, =. The coefficiets a m ad d are the elemets of the discrete Fourier trasforms give by (Berrut ad Trummer, 1987) 1 1 km a = v t s w w (11) (, ) m k, k = 1 1 = where w exp ( i ) ( ) d = ψ t w (1) =, v k = v( t, sk ) ad ψ ψ ( t ) = are the iterpolated values. We seek a solutio at the collocatio poits of the form ˆ φ it ( t ) be ( t ) = (13) = with ukow coefficiet b. Replacig the fuctios φ, ψ, ad v i equatio (8) by their respective approximatios ˆ φ, ψ ˆ, ad v, we obtai it it ims iks it be + ame e bk e ds = de = = m= k = = or. it it ims iks it be + ambk e e e ds = de = = m= k = =. (14) alaysia Joural of athematical Scieces 87

6 Ali Hassa ohamed urid & Teh Yua Yig Observe that equatio (14) ca be simplified by usig the orthogoality relatios for the periodic complex expoetials. These orthogoality relatios have the followig property (Kammler, ): p ikx p ilx p p, if k = l e e dx k l =, =, ± 1, ±,..., otherwise. Applyig the above orthogoality relatios with p =, equatio (14) becomes it it it be + π amb me = de = = m= =. O multiplyig both sides by to, we get ikt e ad the itegrate each term from π it ikt π it ikt π it ikt + π m m = = = m= = b e e dt a b e e dt d e e dt. Applyig the orthogoality properties, we obtai bk ( ) + akmb m ( ) = dk ( π ), k =,...,. which implies m= b + π amb m = d, =,...,. (15) m= Hece, equatio (15) is a system of liear equatios for the b. Sice v ad ψ are kow fuctios, equatios (11) ad (1) ca be used readily to compute the coefficiets a m ad d. Substitute these values ito the system (15), we ca the solve the values for b. Fially, substitute b ito equatio (13) will give us the approximate solutio for equatio (8). 88 alaysia Joural of athematical Scieces

7 umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod Equatio (15) ca be rewritte as a by system ( I + A) b = d, (16) where I is the idetity matrix, A is a matrix defied by = { d } matrix ( I + ) π, = { } am b b ad d, for =,...,, m =,...,. ote that the coefficiet A, b ad d are complex. Oce the discretized solutios ˆ( φ t ) are kow at the collocatio poits, equatio (13) provides a iterpolatio formula for φ based o Fourier method: ( t) φ it = be. (17) = ' Aother appealig procedure for solvig equatio (8) umerically is usig the yström s method with the trapezoidal rule. Choosig equidistat collocatio poits t = ad the trapezoidal rule for yström s method to discretize equatio (8), we obtai (Razali et al., 1997) 1 φ ( t ) + v( t, tk ) φ ( tk ) = ψ ( t ), 1. (18) k = y Defiig the matrix Q by Q k = πv( t, tk ), ad x φ ( t ) ( t ) = ψ, equatio (18) ca be rewritte as a by system =, ( I + Q) x = y. (19) Sice equatio (8) has a uique solutio, the for a wide class of quadrature formula, the system (19) also has a uique solutio, as log as is sufficietly large (Atkiso, 1986). Similarly, oce the discretized solutios φ ( t ) are kow at the collocatio poits, the iterpolatio formula based o yström s method is give by 1 φ ( t) = ψ ( t) v( t, t ) φ ( t ). () = alaysia Joural of athematical Scieces 89

8 Ali Hassa ohamed urid & Teh Yua Yig Suppose θ ( t) is the boudary correspodece fuctio to a represetatio z = z( t), t, of Γ. The i ( t ) ( ) e θ R z t =, (1) where R is the Riema mappig fuctio. The boudary correspodece fuctio ca be computed (without itegratio) by the formula (Razali et al., 1997) θ t = arg i φ t. () COPUTATIO O SEVERAL TEST REGIOS I this sectio the umerical scheme discussed i Sectio 3 is applied to several test regios with the ormalizatio R ( ) =, R ( ) >. We have used the ATHEATICA 5. to carry out the etire umerical procedure. θ t θ t θ t is the exact We list the sup-orm error P where boudary correspodece fuctio for the test regios, ad ( t ) θ P is the approximatio obtaied by meas of equatio () usig P equally spaced,, most of which are ot the iterpolatio poits i the iterval [ ] origial collocatio poits. I all our experimets, we have chose P = 36. These allow exact comparisos with the umerical results obtaied from iterpolatio formula based o yström s method give i Razali et al. (1997). Example 1: Ellipse ε 1, axis ratio ( 1 ε ) ( 1 ε ) it i t z t = e + εe, < = +. k = 1 ( 1) ε θ ( t ) = t + si ( kt). k k 1+ ε k k 9 alaysia Joural of athematical Scieces

9 umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod TABLE 1: Error orm θ ( t) θ ( t ) for ellipse Axis ratio (-1) (-) (-3) 5.5(-) (-7) 1.1(-4) 1.(-) (-15) 7.4(-1) 4.8(-6) 1.4(-) (-15).9(-13) 6.4(-7) (-15) 1.(-13) 3.6(-7) (-11) P Example : Iverted Ellipse ( p 1) <. = ( ) it z t 1 1 p cos t e, TABLE : Error orm θ ( t) θ ( t ) for iverted ellipse Values of p (-1) (-) 6.1(-1) (-4) 8.7(-) (-7) 1.(-3) 9.8(-1) 64 6.(-15).9(-7) 4.(-) 18-1.(-14) 8.(-5) P Example 3: Oval of Cassii ( z α z α 1, α 1) + =. 1 = α + α 4 it z t cos t 1 si t e, 1 θ ( t ) = t arg ( w( t) ), 4 α α w t = 1 si t + i si t. alaysia Joural of athematical Scieces 91

10 Ali Hassa ohamed urid & Teh Yua Yig TABLE 3: Error orm θ ( t) θ ( t ) for oval of Cassii Values of α (-) 3.7(-1) (-4).4(-) (-7).3(-4).5(-) 1.5(-1) (-14) 3.(-8) 1.5(-4).9(-3) (-16) 1.8(-15) 4.1(-8) 1.3(-5) (-15).(-1) 1.7(-3) (-15) 1.8(-15) 8.6(-7) (-1).1(-4) (-13) 5.3(-8) P Example 4: Epitrochoid ( Apple ) ( α 1). z t e α e θ t = t it i t = +,. TABLE 4: Error orm θ ( t) θ ( t ) for epitrochoid Values of α (-5) 3.9(-4) 1.9(-3).4(-) 6.5(-1) (-8) 4.8(-7) 7.8(-6) 4.8(-4).(-) (-16) 5.7(-13) 1.5(-1) 4.(-7) 3.1(-4) 1.6(-) 3-8.9(-16) 8.9(-16) 5.8(-13) 5.6(-7) 5.3(-4) (-16) 8.(-13) 1.9(-7) (-16) 1.6(-13) P Eve though Berrut ad Trummer (1987) has established the equivalece of the methods of Fourier ad yström for solvig Fredholm itegral equatio of the secod kid; our umerical results have show that the iterpolatio formula based o yström s method i Razali et al. (1997) gives better accuracy compared with the iterpolatio formula based o Fourier method. These umerical experimets suggest that oe should prefer the iterpolatio formula based o yström s method i order to obtai approximatios of high accuracy for umerical coformal mappig of iterpolatio poits o the boudary. 9 alaysia Joural of athematical Scieces

11 umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod REFERECES Atkiso, K. E A Survey of umerical ethods for the Solutio of Fredholm Itegral Equatios. Philadephia: Society for Idustrial ad Applied athematics. Berrut, J. P. ad. R. Trummer Equivalece of yström s ethod ad Fourier ethods for the umerical Solutio of Fredholm Itegral Equatios. athematics of Computatio 48(1987): Herici, P Applied ad Computatioal Complex Aalysis Volume 3. ew York: Joh Wiley & Sos. Kammler, D. W.. A First Course i Fourier Aalysis. ew Jersey: Pretice-Hall. Kreyszig, E Itroductory Fuctioal Aalysis with Applicatios. ew York: Joh Wiley & Sos. Razali,. R..,. Z. ashed ad A. H.. urid umerical Coformal appig via the Bergma Kerel. Joural of Computatioal ad Applied athematics 8(1997): alaysia Joural of athematical Scieces 93