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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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