1. Chapter 11 偏微分方程式(Partial Differential Equations)
1. 基本觀念
2. 振動弦波(一維波動方程式)
3. 變數分離 : 利用傅立葉級數
4. D’Alembert’s Solution of the Wave Equation
5. Heat Equation: Solution by Fourier Series
6. Heat Equation: Solution by Fourier Integrals and Transforms
7. Modeling: Membrane, Two-Dimensional Wave Equation
8. Rectangular Membrane: Use of Double Fourier Series
9. Laplacian in Polar Coordinate
10. Circular Membrane: Use of Fourier-Bessel Series
11. Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
12. Solution by Laplace Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

2. 超過一個以上的獨立變數(Exist more than one independent variables)
Basic Concepts
Partial Differential Equation : (PDE)
超過一個以上的獨立變數(Exist more than one independent variables)
偏微分方程式之通式
u 與兩個變數 x, y 有關
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

3. Basic Concepts
Consider a function of two or more variables e.g. f(x,y). We can talk about derivatives of such a function with respect to each of its variables:
The higher order partial derivatives are defined recursively and include the mixed x,y derivatives:
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

4. Basic Concepts
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

5. Basic Concepts
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

6. General Forms of second-order P.D.E. (2 variables)橢圓 拋物 雙曲
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

7. Hyperbolic (propagation)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

8. Parabolic (Time- or space- marching)
時間或空間步推
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

9. Elliptic (Diffusion, Equilibrium Problems)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

10. System of Coupled P.D.E.s
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

11. Boundary and Initial Conditions
Dirichlet condition : specify
Neumann condition : specify
Robin condition : specify
At Boundary
or or both are prescribed at t = 0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

12. Modeling: Vibrating String, Wave Equation
一維波動方程式
Assumptions:
Homogeneous and perfectly elastic string.
Neglect the action of gravitational force.
Small vertical displacements
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

13. Modeling: Vibrating String, Wave Equation
一維波動方程式
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

14. Separation of Variables : Use of Fourier Series
一維波動方程式
Dirichlet boundary conditions :
For all t
Initial deflection
Initial conditions :
Initial velocity
Method of separating variables
(Product method)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

15. Separation of Variables : Use of Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

16. Separation of Variables : Use of Fourier Series
Dirichlet boundary conditions :
For all t
For all t
For k = 0 X
For positive k = μ2 X
For negative k = -p2
令 B = 1
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

17. Separation of Variables : Use of Fourier Series通解
Eigenfunction (Characteristic function)
Eigenvalues (Characteristic values)
Spectrum)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

18. Separation of Variables : Use of Fourier Series
Initial deflection
Initial conditions :
Initial velocity
一個某特定n的 un (x,t) 解通常並不會剛好滿足初始條件,而且下式為線性且齊次
因此,為滿足初始條件,我們考慮無窮級數
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

19. Separation of Variables : Use of Fourier Series
可見 Bn 為 f(x)之傅立葉正弦級數的係數
可見 為 g(x)之傅立葉正弦級數的係數
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

20. Separation of Variables : Use of Fourier Series
Suppose g(x) = 0 (初速度為零)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

21. Separation of Variables : Use of Fourier Series
f* 為 f 以週期 2L的奇函數展開
f(x) : initial deflection
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

22. Separation of Variables : Use of Fourier Series駐波
向右行進的波
向左行進的波
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

23. D’Alembert’s Solution of the Wave Equation
Introduce the new independent variables
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

24. D’Alembert’s Solution of the Wave Equation同理
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

25. D’Alembert’s Solution of the Wave Equation
Initial deflection
Initial conditions :
Initial velocity
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

26. D’Alembert’s Solution of the Wave Equation
Initial conditions :
對 x 積分
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

27. D’Alembert’s Solution of the Wave Equation
If g(x) = 0 (初速度為零)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

28. Heat Equation: Solution by Fourier Series
熱傳導方程式
u(x,y,z,t) 為一均勻物質中某處某一時間的溫度
c2 為材料的熱擴散率(thermal diffusivity)
K 為材料的熱傳導率(thermal conductivity)
σ 為材料的比熱(specific heat)
ρ 為材料的密度(density)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

29. Heat Equation: Solution by Fourier Series
一維熱傳導方程式
Dirichlet boundary conditions :
For all t
Initial conditions :
Initial temperature
Method of separating variables
(Product method)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

30. Heat Equation: Solution by Fourier Series通解
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

31. Heat Equation: Solution by Fourier Series
可見 Bn 為 f(x)之傅立葉正弦級數的係數
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

32. Heat Equation: Solution by Fourier Series
Steady-State Two-Dimensional Heat Flow
Steady-State  溫度不為時間的函數 
Dirichlet boundary conditions :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

33. Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

34. Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

35. Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

36. Heat Equation: Solution by Fourier Integrals and Transforms
若要擴展到無限長金屬棒時,則我們將採用傅立葉積分
此時則無邊界條件,僅有初始條件
Initial temperature
Method of separating variables
(Product method)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

37. Heat Equation: Solution by Fourier Integrals and Transforms
此處A與B為任意常數,可視為p的函數:
Initial conditions :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

38. Heat Equation: Solution by Fourier Integrals and Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

39. Heat Equation: Solution by Fourier Integrals and Transforms
利用公式 若取 並設 則
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

40. Heat Equation: Solution by Fourier Integrals and Transforms若取
只要知道初始條件 f(x), 帶入上式積分後即可得到 u(x,t)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

41. Heat Equation: Solution by Fourier Integrals and Transforms
-1 < v < 1
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

42. Heat Equation: Solution by Fourier Integrals and Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

43. Heat Equation: Solution by Fourier Integrals and Transforms
設 表 u 的傅立葉轉換,視 u 為 x 的函數
Initial conditions :
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

44. Heat Equation: Solution by Fourier Integrals and Transforms
為w的偶函數
為w的奇函數
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

45. Modeling: Membrane, Two-Dimensional Wave Equation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

46. Rectangular Membrane: Use of Double Fourier Series
Dirichlet boundary conditions : at boundaries
For all t
Initial displacement
Initial conditions :
Initial velocity
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

47. Rectangular Membrane: Use of Double Fourier Series
對時間函數G(t)的常微分方程式
對振幅函數F(x,y)的偏微分方程式
又稱為二維Helmholtz方程式
二維Helmholtz方程式中的變數分離
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

48. Rectangular Membrane: Use of Double Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

49. Rectangular Membrane: Use of Double Fourier Series
Dirichlet boundary conditions : at boundaries
For all t
為整數
為整數
m = n = 1,2,3,….
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

50. Rectangular Membrane: Use of Double Fourier Series
m = n = 1,2,3,….
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

51. Rectangular Membrane: Use of Double Fourier Series
Initial displacement
Initial conditions :
Initial velocity
Double Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

52. Rectangular Membrane: Use of Double Fourier Series
可見 Bmn 為 km之傅立葉正弦級數的係數 假設
可見 km 為 f(x,y)之傅立葉正弦級數的係數
廣義歐拉公式(generalized Euler formula)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

53. Rectangular Membrane: Use of Double Fourier Series
Initial velocity
Double Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

54. Laplacian in Polar Coordinate
(x,y)  (r,θ) ?
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

55. Laplacian in Polar Coordinate
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

56. Laplacian in Polar Coordinate
平面極座標的拉普拉斯運算
圓柱座標的拉普拉斯運算
球座標的拉普拉斯運算
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

57. Circular Membrane: Use of Fourier-Bessel Series
如果只考慮徑向對稱的解 u(r.t),則二維波動方程式將簡化為
邊界條件
初始條件
初始偏移
For all t  0
初始速度
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

58. Circular Membrane: Use of Fourier-Bessel Series
令 s = kr
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

59. Circular Membrane: Use of Fourier-Bessel Series
貝索微分方程式(Bessel’s differential equation)
為 = 0 的貝索微分方程式
W(r)為包含第一類與第二類的貝索函數J0和Y0,但Y0在0處為無窮大,所以只需考慮J0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

60. Circular Membrane: Use of Fourier-Bessel Series
在邊界 r = R 上,W(R) = J0(kR) = 0, J0有無限多個正零點
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

61. Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

62. Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

63. Circular Membrane: Use of Fourier-Bessel Series
初始偏移
初始條件
初始速度
am為傅立葉-貝索級數的係數
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

64. Circular Membrane: Use of Fourier-Bessel Series
Now n = 0
同理可證
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

65. Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

66. Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

67. Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

68. Circular Membrane: Use of Fourier-Bessel Series
當 時
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung