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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
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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
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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
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Basic Concepts Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Basic Concepts Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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Hyperbolic (propagation)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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Elliptic (Diffusion, Equilibrium Problems)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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System of Coupled P.D.E.s Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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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
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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
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Separation of Variables : Use of Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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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
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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
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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
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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
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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
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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
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D’Alembert’s Solution of the Wave Equation
同理 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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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
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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
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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
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Heat Equation: Solution by Fourier Series
通解 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Heat Equation: Solution by Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Heat Equation: Solution by Fourier Integrals and Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Heat Equation: Solution by Fourier Integrals and Transforms
-1 < v < 1 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Heat Equation: Solution by Fourier Integrals and Transforms
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Modeling: Membrane, Two-Dimensional Wave Equation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Rectangular Membrane: Use of Double Fourier Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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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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
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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
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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
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Laplacian in Polar Coordinate
(x,y) (r,θ) ? Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Laplacian in Polar Coordinate
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Circular Membrane: Use of Fourier-Bessel Series
令 s = kr Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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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
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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
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Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Circular Membrane: Use of Fourier-Bessel Series
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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Circular Membrane: Use of Fourier-Bessel Series
當 時 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
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