A Study on Particular Solutions of Coupled PDE System

1. A Study on Particular Solutions of Coupled PDE System Chia-Cheng Tasi 蔡加正 Department of Information Technology Toko University, Chia-Yi County, Taiwan

2. Overview Motivation and Introduction Method of Particular Solutions (MPS) Particular solutions of Chebyshev polynomials Numerical example I Particular solutions of spline Numerical example II Conclusions

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4. Motivation and Introduction Boundary-type numerical method: BEM, Treffz method, MFS Advantage: Reduction of dimensionalities Disadvantage: Domain integration => the method of particular solutions (MPS) or the dual reciprocity method (DRM) Active research fields of BEM: Singularity and Domain integration

5. Motivation and Introduction

6. Motivation and Introduction

7. Motivation and Introduction BIEM Innovation Hypersingularity DBEM MPS MRM How to prove? Application

8. Motivation and Introduction RBF Golberg (1995) Chebyshev MPS with Chebyshev Polynomials exponential convergence MFS Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

9. Motivation and Introduction

10. Method of particular solutions

11. Method of particular solutions Method of particular solutions Method of fundamental solutions, Trefftz method, boundary element method, et al.

12. Method of particular solutions (basis functions)

13. Method of particular solutions Coupled PDE system Hörmander operator decomposition technique Product operator Partial fraction decomposition Polyharmonic operator Poly-Helmholtz operator ? Generating theorem Laplacian operator Helmholtz operator

14. Method of particular solutions (Hörmander Operator Decomposition technique) Particular solutions for the engineering problems

15. Example

16. Example

17. Other examples Stokes flow Thermal Stokes flow

18. Other examples Thick plate Solid deformation

19. Remark Particular solutions for product operator Particular solutions for engineering problems Hörmander operator decomposition technique

20. Method of particular solutions (Partial fraction decomposition) Particular solutions for Particular solutions for product operator Partial fraction decomposition

21. Partial fraction decomposition (Theorem)

22. Partial fraction decomposition (Proof 1)

23. Partial fraction decomposition (Proof 2)

24. Example (1)

25. Example (2)

26. Remark Partial fraction decomposition

27. Particular solutions of Chebyshev polynomials (why orthogonal polynomials) Fourier series: exponential convergence but Gibb’s phenomena Lagrange Polynomials: Runge phenomena Chebyshev Polynomials (one of the orthogonal polynomials): exponential convergence

28. Chebyshev interpolation (1)

29. Chebyshev interpolation (2)

30. Chebyshev interpolation (3)

31. Chebyshev interpolation (4)

32. Particular solutions of Chebyshev polynomials

33. Particular solutions of Chebyshev polynomials

34. Particular solutions of of Chebyshev polynomials (Generating Theorem)

35. Particular solutions of of Chebyshev polynomials (Generating Theorem)

36. Particular solutions of of Chebyshev polynomials (Generating Theorem)

37. Particular solutions of Chebyshev polynomials (poly-Helmholtz) Generating Theorem Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

38. Particular solutions of Chebyshev polynomials (polyharmonic)

39. Particular solutions of Chebyshev polynomials (polyharmonic)

40. Fig. 2: Geometry configuration of the MFS. Method of fundamental solutions

41. Method of fundamental solutions (example)

42. Numerical example I Example (2D modified Helmholtz)

43. Numerical example I Example (2D Laplace)

44. Numerical example I Example (3D modified Helmholtz)

45. Numerical example I Example (3D Laplace)

46. Numerical example I Example (2D polyharmonic)

47. Numerical example I Example (2D product operator)

48. Fig. 1: Geometric configuration of the Ressiner plate model. Numerical example I Example (Reissner plate)

49. Example (Reissner plates: particular solutions)

50. Example (Reissner plates: particular solutions)