1. National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Degenerate kernel for ellipse Huan-Zhen Liao (M9452056) Reporter: Huan-Zhen Liao (M9452056) Jeng-Tzong Chen Advisor: Jeng-Tzong Chen 2007/01/04 Date: 2007/01/04 HR2 307 Place: HR2 307

2. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 2 Outline 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

3. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 3 Outline MotivationMotivation 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

4. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 4 Motivation Straight boundary Degenerate boundary Circular inclusion Elliptic hole [Mathieu function] [Legendre polynomial] [Chebyshev polynomial] [Fourier series]

5. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 5 Present approach Fundamental solution No principal value Advantages of degenerate kernel 1.No principal value 2.Well-posed 3.Exponential convergence 4.Free of boundary-layer effect Degenerate kernel CPV and HPV

6. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 6 Outline Motivation Degenerate kernel for ellipseDegenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel A numerical example Conclusions Further studies

7. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 7 Conformal mapping technique z planew plane

8. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 8 Some properties of ellipse When R>1 z planew plane

9. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 9 Some properties of ellipse When R<1 z planew plane

10. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 10 Some properties of ellipse

11. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 11 Derivation of degenerate kernel

12. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 12 Convergence of degenerate kernel InteriorExterior

13. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 13 Outline Motivation Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel A numerical exampleA numerical example Conclusions Further studies

14. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 14 Numerical examples w planez plane

15. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15 Numerical examples

16. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 16 Numerical examples w planez plane

17. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 17 Numerical examples

18. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 18 Outline Motivation Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel A numerical example ConclusionsConclusions Further studies

19. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 19 Conclusions In the future, we will choose the correct mapping points to calculate degenerate kernels for ellipse.In the future, we will choose the correct mapping points to calculate degenerate kernels for ellipse. According to numerical results, only few terms of Fourier series can achieve accurate solutions.According to numerical results, only few terms of Fourier series can achieve accurate solutions.

20. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 20 Outline 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 studiesFurther studies

21. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 21 Further studies Application on many engineering problems using degenerate kernel for ellipse. Extension to general boundaries.Extension to general boundaries.

22. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 22 The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to visit the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab