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課程名稱 |
偏微分方程式
PARTIAL DIFFERENTIAL EQUATIONS |
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開課學期 |
98-2 |
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授課對象 |
工學院 機械工程學研究所 |
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授課教師 |
賴君亮 |
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課號 |
ME5132 |
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課程識別碼 |
522 U1540 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期二5,6,7(12:20~15:10) |
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上課地點 |
工綜213 |
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備註 |
總人數上限：40人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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課程概述 |
以緊湊的課程講授偏微分方程式的形式、分類、特性、求解方法以及相關應用。 |
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課程目標 |
Introduction : Definitions and Terminologies, Mathematical Modeling, Classification of Second-Order Partial Differential Equations
˙First Order Partial Differential Equations : The Constant Coefficient Equation, The Linear Equation, The Quasi-Linear Equation, Generalized Solutions, Systems of First-Order Equations
˙The Cauchy Problem and Hyperbolic Equations : Cauchy-Kowalewskei Theorem, The Cauchy Problem for Homogeneous Wave Equation, Initial-Boundary Value Problem, The Cauchy Problem for Nonhomogeneous Wave Equation, Riemann’s Method
˙Convolution and Parabolic Equations : Heat Equation, The Dirac-Delta Function, Scaling Method and Heat Kernel, General Initial Value Problem (Cauchy Problem), Self-Similarity Solutions, The Weak Maximum Principle, Comparison of Waves and Diffusions
˙Potential Theory and Elliptic Equations : Complex-Variable Methods, Green’s Identities, Properties of Harmonic Functions in Bounded Regions, The Mean-Value Property of Harmonic Functions, Solution of the Potential Equation for a Circle, Unbounded Domains
˙The Method of Separation of Variables Applicable to Initial-Boundary Value Problems : Separation of Variables, Nonhomogeneous Problems, Higher Dimensional Problems, Illustrations
˙Problems on Unbounded Domains and Integral Transforms : Brief Reviews of Fourier, Laplace, and Hankel Transforms, Problems on Unbounded Domains, Problems on Semi-bounded Domains
˙Green’s Functions for Partial Differential Equations : The Adjoint Operator, The Delta Function, The Green’s Function Method, Principle Solutions, Green’s Function Method For The Laplace Operator, Green’s Function Method for The Helmholtz Operator, Green’s Function Method for The Diffusion Operator, Green’s Function Method for The Wave Operator, The Eigenfunction Method
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課程要求 |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
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參考書目 |
Beginning Partial Differential Equations, by Peter V. O’Neil, John Wiley & Sons, Inc. |
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評量方式
(僅供參考) |
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週次 |
日期 |
單元主題 |
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第1週 |
2/23 |
1 Introduction
1-1 Definitions and Terminologies
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第2週 |
3/02 |
1-2 Mathematical Modeling
1-3 Classification of Second-Order Partial Differential Equations |
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第3週 |
3/09 |
2 First Order Partial Differential Equations
2-1 The Constant Coefficient Equation
2-2 The Linear Equation |
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第4週 |
3/16 |
2-3 The Quasi-Linear Equation
2-4 Generalized Solutions |
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第5週 |
3/23 |
2-5 Systems of First-Order Equations |
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第6週 |
3/30 |
3 The Cauchy Problem and Hyperbolic Equations
3-1 Cauchy-Kowalewskei Theorem
3-2 The Cauchy Problem for Homogeneous Wave Equation
3-3 Initial-Boundary Value Problem |
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第7週 |
4/06 |
3-4 The Cauchy Problem for Nonhomogeneous Wave Equation
3-5 Riemann’s Method |
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第8週 |
4/13 |
4 Convolution and Parabolic Equations
4-1 Heat Equation
4-2 The Dirac-Delta Function
4-3 Scaling Method and Heat Kernel
4-4 General Initial Value Problem (Cauchy Problem) |
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第9週 |
4/20 |
4-5 Self-Similarity Solutions
4-6 The Weak Maximum Principle
4-7 Comparison of Waves and Diffusions |
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第10週 |
4/27 |
5 Potential Theory and Elliptic Equations
5-1 Complex-Variable Methods
5-2 Green’s Identities
5-3 Properties of Harmonic Functions in Bounded Regions |
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第11週 |
5/04 |
5-4 The Mean-Value Property of Harmonic Functions
5-5 Solution of the Potential Equation for a Circle
5-6 Unbounded Domains |
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第12週 |
5/11 |
6 The Method of Separation of Variables Applicable to Initial-Boundary Value Problems
6-1 Separation of Variables
6-2 Nonhomogeneous Problems |
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第13週 |
5/18 |
6-3 Higher Dimensional Problems
6-4 Illustrations |
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第14週 |
5/25 |
7 Problems on Unbounded Domains and Integral Transforms
7-1 Brief Reviews of Fourier |
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第15週 |
6/01 |
7-2 Problems on Unbounded Domains
7-3 Problems on Semi-bounded Domains |
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第16週 |
6/08 |
8 Green’s Functions for Partial Differential Equations
8-1 The Adjoint Operator
8-2 The Delta Function
8-3 The Green’s Function Method
8-4 Principle Solutions |
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第17週 |
6/15 |
8-5 Green’s Function Method For The Laplace Operator
8-6 Green’s Function Method for The Helmholtz Operator |
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第18週 |
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8-7 Green’s Function Method for The Diffusion Operator
8-8 Green’s Function Method for The Wave Operator
8-9 The Eigenfunction Method |
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