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課程名稱 |
偏微分方程式一
Partial Differential Equations (Ⅰ) |
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開課學期 |
111-1 |
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授課對象 |
理學院 應用數學科學研究所 |
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授課教師 |
阮文先 |
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課號 |
MATH5218 |
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課程識別碼 |
221EU0330 |
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班次 |
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學分 |
3.0 |
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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 |
星期二3,4(10:20~12:10)星期四6(13:20~14:10) |
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上課地點 |
天數101天數101 |
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備註 |
本課程以英語授課。
總人數上限：40人
外系人數限制：10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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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
熟悉各類偏微分方程(PDE)的分析方法及其在弱拓樸(weak topology)意義下的理論建構及認識PDE在物理和幾何方面的意義。 |
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課程目標 |
學習下列主題：
1. Laplace and Poisson equations
2. Heat equations
3. Wave equations
4. Nonlinear first-order PDE
5. Special solutions of Burger, Incompressible Euler, KdV and Reaction-Diffusion equations
6. Holder and Sobolev spaces |
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課程要求 |
採課前預習、上課討論的上課方式。修課學生需每週在Ceiba下載PDF檔和在NTU COOL下載MP4檔預習當週課程內容，於上課時參與討論。PDF與MP4檔僅提供修課學生個人使用，請勿外傳。另外因NTU COOL提供的記憶體容量有限，可能無法同時儲存所有的MP4檔，將以每週上課有關內容為主，請大家儘早下載MP4檔。 |
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預期每週課後學習時數 |
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Office Hours |
另約時間 備註： By writing to vtnguyen@ntu.edu.tw |
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指定閱讀 |
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參考書目 |
[1] L. Evans, Partial Differential Equations, 1998 AMS.
[2] F. John, Partial Differential Equations, 1982 Springer-Verlag New York Inc.
[3] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 5,
Berkeley : Pub. or Perish, 1979
[4] R. Adams J. Fournier, Sobolev Spaces, Volume 140 of Series: Pure and Applied
Mathematics, 2nd Edition, Academic Press 2003 |
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評量方式
(僅供參考) |
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針對學生困難提供學生調整方式 |
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上課形式 |
提供學生彈性出席課程方式 |
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作業繳交方式 |
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考試形式 |
書面(口頭)報告取代考試 |
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其他 |
由師生雙方議定 |
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週次 |
日期 |
單元主題 |
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Week 1 |
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Overview of PDEs |
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Week 2 |
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Transport equation
and the method of characteristics |
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Week 3 |
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Laplace’s equation: Fundamental solution,
Mean-value formula, harmonic functions,
Hanack’s inequality
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Week 4 |
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Poisson’s equation: Green’s function,
representation formula, Dirichlet’s principle |
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Week 5 |
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Heat equation: Fundamental solution,
Mean-value formula, maximum principle.
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Week 6 |
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Nonhomogeneous heat equation:
uniqueness, regularity, and energy methods
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Week 7 |
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Wave equation: d’Alembert’s formula, solution
in odd and even dimensions
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Week 8 |
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Nonhomogeneous wave equation: d’Alembert’s
formula, uniqueness, domain of dependence,
and energy methods.
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Week 9 |
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Separation of variables, similarity solutions
and Fourier transform
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Week 10 |
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Overview of Sobolev space |
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Week 11 |
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Second-order elliptic equations: weak solutions,
Lax-Milgram theorem, energy estimates and
Fredholm alternative
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Week 12 |
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Second-order elliptic equations: regularity,
maximum principles
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Week 13 |
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Second-order parabolic equations: weak solutions
Galerkin approximations and energy estimates
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Week 14 |
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Second-order parabolic equations: existence and
uniqueness
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Week 15 |
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Second-order parabolic equations: regularity,
maximum principles
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Week 16 |
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Second-order hyperbolic equations: weak solutions
Galerkin approximations and energy estimates
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Week 17 |
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Second-order hyperbolic equations: existence and
uniqueness, regularity
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Week 18 |
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Discussion and related topics |
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