Degenerate kernel for ellipse

1

National Taiwan Ocean University

MSVLABDepartment of Harbor and River Engineering

Degenerate kernel for ellipseDegenerate kernel for ellipse

Reporter: Sheng-Kai Gau (B92520121)Sheng-Kai Gau (B92520121)

Advisor: Jeng-Tzong ChenJeng-Tzong Chen

Date: 2007/01/182007/01/18

Place: HR2 307HR2 307

2MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering

OutlineOutline

• Motivation• Degenerate kernel for ellipse ◎ Conformal mapping technique

◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel

• A numerical example• Conclusions• Further studies


OutlineOutline

• MotivationMotivation• Degenerate kernel for ellipse ◎ Conformal mapping technique




MotivationMotivation

In the course, the discussion of the degenerate degenerate kernel for ellipse excited my interest.kernel for ellipse excited my interest.


OutlineOutline

• Motivation• Degenerate kernel for ellipseDegenerate kernel for ellipse ◎ Conformal mapping technique




2 2 2x y R 2 2 2( )cos

vu R

2 2

2 2 21

cos

u v

R R


Conformal mapping techniqueConformal mapping technique

1w z

z

a

bR

1a R

R

1b R

R

z plane w plane

( , ) ( cos( ), sin( ))x y R R ( , ) ( cos( ), sin( ))u v a b


Some properties of ellipseSome properties of ellipse

a

bR

z plane w planeWhen R>1



When R<1

a

bR

z plane w plane



a

b

cc

1L2L

1 2 2constaL L ant 1 2 2constaL L ant 1 2 2constaL L ant

1L2L

1L

2L


Derivation of degenerate kernelDerivation of degenerate kernel

1

21

ln ln(( )( )) [ln( ) ln( )]2s x s x s x s xr w w w w w w w w

1 1 1 1ln( ) ln( ) ln[( )(1 )] ln( ) ln(1 )s x s x s x s x

s x s x s x

w w z z z z z zz z z z z z

1 1 1[ln( ) ln(1 ) ln( ) ln(1 )]2 s x s x

s x s x

z z z zz z z z

1ln( ) ln( ) ln(1 )x x s x

s x

w w z zz z

1 1 1 1[ln[( )( )] ln(1 )]2 s x s x

s x s x s x s x

z z z zz z z z z z z z

21

1 1 2 1ln ( ) cos ( ) [1 cos( ) ]

2 ( )

m

m

R mm R R R

1 1

1 1 1ln ( ) cos ( ) ( ) cos ( )

m m

m m

R m mm R m R

2ln( ) ln ( ) 2 cos( ) 1R R R

1

1 1ln( ) (ln( ) ( ) cos ( ))

m

m

R R mm R

1

1 1( ) cos ( )

m

m

mm R


Derivation of degenerate kernelDerivation of degenerate kernel

s

x

2

s x

(s, x)(s, x)

n

(s, x)(s, x)

n

(s, x)(s, x)

n n

UT

UL

UM

¶º

¶

¶º

¶

¶º¶ ¶

1 1

1 1

1 1 1ln cos cos ,

s, x1 1 1

ln cos

1

cos

1, ; ,

, ; , ,1 1

i

e

mm

m m

m m

m m

R m mm R m R

UR

m mm

U R RR

U Rm RR

R

o

s

x

x

iU

eU


OutlineOutline

• Motivation• Degenerate kernel for ellipse ◎ Conformal mapping technique


• A numerical exampleA numerical example• Conclusions• Further studies


Numerical examplesNumerical examples

-2 -1 1 2

-1

-0.5

0.5

1

1.5

2

2.5

4 3,0

3s

1

50,2

x

2

10,2

x

w plane

-1.5 -1 -0.5 0.5 1 1.5

-1

1

2

3,0s

1,03

s

1

5 410,

4x

1

5 410,

4x

z plane2 4

2

w wz

1L2L

3L4L



1 1

1 3 1 1ln 2.85078 cos 0 cos 0

2.85078 2 23*2.85

1.22478

078

m m

m m

m mm m

1 1

1 1 1 3ln 2.85078 cos 0 cos 0

2 2.85078 23*2.85

1.22

078

478

mm

m m

m mm m

1 1

1 0.350781 1 1ln 3 cos 0 cos 0

2 23 3*0.350781

754.202

m m

m m

m mm m

1 1

1 1 1 3ln 3*0.350781 cos 0 cos 0

2 0.350781 23

mm

m m

m mm m

2 2

1.224 3 5

ln ln3 2

478r

1 1

1 1

1 1 1ln cos cos ,

s, x1 1 1

ln cos

1

cos

1, ; ,

, ; , ,1 1

i

e

mm

m m

m m

m m

R m mm R m R

UR

m mm

U R RR

U Rm RR

R

-1.5 -1 -0.5 0.5 1 1.5

-1

1

2

3,0s

1,03

s

1

5 410,

4x

1

5 410,

4x



-2 -1 1 2

-1

-0.5

0.5

1

1.5

2

2.5

4 3,0

3s

1

50,2

x

2

10,2

x

-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

3,0s

1,03

s

2

1 170,

4x

2

1 170,

4x

w plane z plane2 4

2

w wz



1 1

1 1.28078 1 1ln 3 cos 0 cos 0

2 23 3*1.28078

0.85985

m m

m m

m mm m

1 1

1 1 1 3ln1.28078 cos 0 cos 0

2 1.28078 21.28078* 3

mm

m m

m mm m

1 1

1 0.780776 1 1ln 3 cos 0 cos 0

2 23 3*1.28078

0.85989

m m

m m

m mm m

1 1

1 1 1 3ln 0.780776 cos 0 cos 0

2 0.780776 20.780776* 3

mm

m m

m mm m

2 24 3 1

ln ln 0.859893 2

r

-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

3,0s

1,03

s

2

1 170,

4x

2

1 170,

4x


The endThe end

Thanks for your kind attention.Thanks for your kind attention.

Your comments will be highly appreciated.Your comments will be highly appreciated.

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Degenerate kernel for ellipse

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