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課程名稱 |
偏微分方程導論
Introduction to Partial Differential Equations |
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開課學期 |
103-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
陳俊全 |
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課號 |
MATH2206 |
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課程識別碼 |
201 25000 |
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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 |
必修 |
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上課時間 |
星期二7,8(14:20~16:20)星期五1,2(8:10~10:00) |
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上課地點 |
新102新102 |
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備註 |
教學改善計畫課程，有教學助理實施小班輔導。
總人數上限：130人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1032MATH2206_PDE |
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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 |
1. PDE problems from physics and mathematics
2. First order equations
3. Wave equation, causality and energy
4. Diffusion equation, maximum principle, heat kernel
5. Fourier series
6. Boundary problems
7. Harmonic functions, Poisson's foumula, Green's function
8. Eigenvalue problems |
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課程目標 |
Course Goal：
This course provides an introduction to the basic theory of partial differential equations. |
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課程要求 |
Course prerequisite:
Calculus, advanced calculus, linear algebra |
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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 |
待補 |
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參考書目 |
Walter A. Strauss
Partial Differential Equations-An Introduction, Second Edition |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
homework |
25% |
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2. |
midterm exam |
35% |
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3. |
final exam |
40% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/24,2/27 |
Introduction to PDE |
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第2週 |
3/03,3/06 |
Basic concepts: derivatives of integrals; divergence theorem
PDE model:transport equation |
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第3週 |
3/10,3/13 |
First order equation: characteristic method, linear PDE with constant/variable coefficients |
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第4週 |
3/17,3/20 |
First order equation: quasilinear equations, examples of nonexistence
PDE models: vibrating string, vibrating drumhead, diffusion of chemicals and heat
Types of second order equations |
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第5週 |
3/24,3/27 |
Wave equation on real line: general solution (characteristic method, coordinate method), initial value problem |
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第6週 |
3/31,4/03 |
Wave equation: domain of dependence, domain of influence, principle of causality
Diffusion equation: maximum principle |
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第7週 |
4/07,4/10 |
Applications of Max. Principle: uniqueness and stability of Dirichlet problem
Energy methods
Invariance properties of diffusion equation
Special solutions of diffusion equation |
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第8週 |
4/14,4/17 |
Self-similar solutions
Fundamental solution (diffusion kernel) |
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第9週 |
4/21,4/24 |
Comparison of wave eq. and diffusion eq.
Wave eq. and diffusion eq. on the half line. |
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第10週 |
4/28,5/01 |
Diffusion eq. and wave eq. on an interval
Diffusion eq. and wave eq. with sources
Duhamel's principle |
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第11週 |
5/05,5/08 |
Problems with more general boundary conditions
Separation of variables |
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第12週 |
5/12,5/15 |
Fourier series
Orthogonality and inner product
sup norm and L2 norm |
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第13週 |
5/19,5/22 |
Schwarz inequality
Least square approximation
Bessel's inequality, Parseval's identity
Pointwise convergence of Fourier series |
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第14週 |
5/26,5/29 |
Application to Wave and Diffusion equations
Symmetric boundary conditions
Robin boundary condition |
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第15週 |
6/02,6/05 |
General inhomogeneous boundary conditions for wave and diffusion
Laplace equation: invariance property of Laplacian, mean value property |
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第16週 |
6/09,6/12 |
Maximum principle (strong and weak)
Green's function and Green's representation theorem
Diffusion in higher dimensions
Kirchhoff's formula in 3 dimensions |
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第17週 |
6/16 |
Kirchhoff's formula in 3 dimensions
2 dimensional formula for wave equation |
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