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課程名稱 |
數學物理方程一
Equations of Mathematical Physics (Ⅰ) |
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開課學期 |
112-2 |
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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 |
MATH7419 |
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課程識別碼 |
221EU5780 |
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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)星期四5(12:20~13:10) |
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上課地點 |
天數302天數302 |
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備註 |
本課程以英語授課。
總人數上限：30人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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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 |
介紹物理領域的偏微分方程數學建模及其數學分析方法 |
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課程目標 |
學習下列主題
1. Energy Law
2. Gradient Flow
3. Energetic Variational Approach
4. Poisson-Nernst-Planck equations
5. Poisson-Nernst-Planck equations with steric effects
6. Nonlinear Schrodinger equations
7. Nonlinear Schrodinger systems
8. Ground states
9. Eigenvalue estimates
10. Direct method
11. Ginzburg-Landau equations
12. Vortex dynamics of Ginsburg-Landau equations
13. Big ball approach
14. Saturated nonlinear Schrodinger equations |
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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NTU COOL下載MP4檔與PDF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 |
備註： Office hours: Thursday 2 - 3:30 pm |
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指定閱讀 |
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參考書目 |
1. A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic
problems on R^n, 2006 Birkhauser Verlag
2. J.B. Grotberg, Biofluid Mechanics, 2021 Cambridge University Press
3. J. Keener and J. Sneyd, Mathematical Physiology, 1998 Springer
4. W. Scherer, Mathematics of quantum computing: An Introduction, 2019 Springer
5. M. Struwe, Variational method, 2008 Springer
6. C. Sulem and P.L. Sulem, The nonlinear Schrodinger equation, 1999 Springer |
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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 |
學生與授課老師協議改以其他形式呈現 |
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考試形式 |
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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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First-order quasilinear equations: models, Cauchy problem, characteristics, domain of definition, blowup |
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Week 2 |
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First-order quasilinear equations: Solutions with discontinuities, generalized (weak) solutions, shock waves, energy estimates, and Riemann problem |
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Week 3 |
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Introduction to second-order scalar equations: models, Cauchy problem, characteristics, canonical forms |
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Week 4 |
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Fisher-KPP equation: models, traveling wave and stability |
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Week 5 |
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Predator-prey models, traveling wave and stability |
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Week 6 |
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Keller-Segel system: derivation, global solutions and long-time asymptotic behaviors. |
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Week 7 |
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Keller-Segel system: self-similarity, collapsing solutions and blowup behaviors. |
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Week 8 |
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Nonlinear Fokker-Planck equation. Midterm. |
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Week 9 |
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One-phase Stefan problem: derivation, existence and uniqueness of solutions, long-time asymptotic behavior. |
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Week 10 |
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One-phase Stefan problem: dynamics of melting ice balls. |
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Week 11 |
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Two-phase Stefan problem: derivation, existence and uniqueness of solutions, asymptotic behavior. |
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Week 12 |
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Semilinear wave equations: derivation, existence and uniqueness of solutions, asymptotic of global solutions |
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Week 13 |
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Semilinear wave equations: self-similarity, singularity formation |
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Week 14 |
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Nonlinear Schrödinger equation: derivation, solitary waves, stability |
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Week 15 |
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Nonlinear Schrödinger equation: self-similarity, singularity formation, stability |
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Week 16 |
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Final exam |
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