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Chapter 11 偏微分方程式 (Partial Differential Equations)

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

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Chapter 11 偏微分方程式 (Partial Differential Equations)

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  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. BasicConcepts 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. BasicConcepts 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. BasicConcepts Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  5. BasicConcepts 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 For all t Dirichlet boundary conditions : 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 For all t Dirichlet boundary conditions : For all t X For k = 0 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) Spectrum) Eigenvalues (Characteristic values) 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(x) : initial deflection f* 為 f 以週期 2L的奇函數展開 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  22. Separation of Variables : Use of Fourier Series 向左行進的波 向右行進的波 駐波 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 D’Alembert’s Solution 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 令 B = 1 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 傅立葉積分(Fourier Integrals) 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的奇函數 為w的偶函數 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

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