偏微分方程
Partial Differential Equations
| 節 | 週一 | 週五 |
|---|---|---|
3 10:10–11:00 | 偏微分方程 SA215（光復） | |
5 13:20–14:10 | 偏微分方程 SA215（光復） 2 節連堂 | |
6 14:20–15:10 |
* 根據陽明交大上課時間表所列
This course aims to teach some fundamentals of Partial differential equations (PDEs), a subject related to many branches of pure/applied mathematics and applied science. This one-semester course aims to introduce some analytical tools to analyze the existence, uniqueness, regularity of weak/classical solutions to first-order and second-order PDEs. In addition, students will learn to write clear, correct, rigorous proofs.
Calculus (along with multivariable calculus ), Advanced Calculus, Linear Algebra, ODE (not a prerequisite)
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Homework 40% Report 30% Final Exam 30%
Four important PDEs
Transport equation, Laplace equation, Heat/Diffusion equation and Wave equation
Non-linear first-order PDEs
Characteristics, Hamilton-Jacob equation
Some ways to represent solutions
Separation of variables, Self-similar solutions, Traveling wave solutions, Transform methods
Function spaces
Sobolev space, Holder space
| 週次 | 主題 |
|---|---|
| 第 1 週 | Preliminary : Miltiindex, gradient, divergence, Gauss-Green Theorem, Green's formula, Polar coordinates formulas. 9/13 |
| 第 1 週 | Ch 1 Introduction: linear/nonlinear, some examples, well-posed problems 9/17 |
| 第 2 週 | The Mid-Autumn Festival 9/20 |
| 第 2 週 | 2-1 Transport Equations 9/24 |
| 第 3 週 | 2-2 Maximum Principle, Mollifier, Regularity, Harnack's Inequality 10/1 |
| 第 3 週 | 2-2 Laplace Equation, Fundamental Solutions, Poisson Equation, Mean-Value Property 9/27 |
| 第 4 週 | 2-2 Green's functions 10/4 |
| 第 5 週 | 國慶連假 10/11 |
| 第 5 週 | 2-3 Fundamental solution for heat equations 10/15 |
| 第 6 週 | 2-3 Initial value problems 10/18 |
| 第 6 週 | 2-3 MVP for heat equation 10/22 |
| 第 7 週 | Online 習題討論 10/25 |
| 第 7 週 | 2-3 Mean Value Principle for the heat equation 10/29 |
| 第 8 週 | 2-3 Maximum Principle for the heat equation 11/1 |
| 第 8 週 | 2-3 Maximum principle for the Cauchy problem 11/5 |
| 第 9 週 | 2-4 Wave equation in 1D 11/12 |
| 第 9 週 | 2-3 Regularity of solutions for the heat equation 11/8 |
| 第 10 週 | Online 習題研討 11/15 |
| 第 10 週 | 2-4 Wave equation in 1D (cont.) 11/19 |
| 第 11 週 | Midterm Presentation 11/22 |
| 第 11 週 | Midterm Presentation 11/26 |
| 第 12 週 | Midterm Presentation 11/29 |
| 第 12 週 | 實體習題研討 12/3 |
Lawrence C. Evans. Partial Differential Equations. Second edition. Graduate Studies in Mathematics. David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Springer, 2001. Walter A, Strauss. Partial Differential Equations: An Introduction. New York, NY: Wiley.
- 地點
- SA236
- 時間
- By appointment.
- 聯絡方式
- email: changhong@math.nctu.edu.tw