标题: 偏微分方程导论
Introduction to Partial Differential Equations
作者: 李荣耀
Open Education Office
开放教育推动中心
公开日期: 2014
摘要: 课程首页

 

本课程是由交通大学应用数学系提供。

 


Mathematical models as PDE ─ qualitative and quantative analysis.

Three classical types of linear PDEs and the corresponding theory.

A short topic on nonlinear PDE.
课程目标/概述  


Mathematical models as PDE – qualitative and quantative analysis.

Three classical types of lineat PDEs and the corresponding theory.

A short topics on nonlinear PDE.


 

课程章节  




授课周次


老师授课主题



课本参考章节




第一周



PDE导论

Fundamental differences between PDE and ODE.





1.1* What is a Partial Differential Equation?

1.2* First-Order Linear Equations





第二周



First and second order linear wave equations;

Transport equations

Characteristic lines;

Travelling wave solutions.

Wave equations with dispersion, dissipation, and nonlinearity





2.1* The Wave Equation





第二周



Classical linear wave equations with travelling wave solutions.

Dispersive linear wave equations.

Dissipative linear wave equations.

Nonlinear wave equations with shock wave solutions.

Nonlinear wave equations with solitary wave solutions.

Initial value problem for a whole-line linear wave equation and the dAlembert solution (I)





1.1* What is a Partial Differential Equation?

2.1* The Wave Equation

Supplement to lecture notes





第三周



Classification of 3 types of second order linear PDEs (I).

Initial value problem for a whole-line linear wave equation and the dAlembert solutions (II).

Initial-boundary value problem for a half-line linear wave equation.

Initial-boundary value problem for a finite-line linear wave equation (I) – method of Reflection and method of Separation of Variables.





2.1* The Wave Equation

3.2 Reflections of Waves

1.6 Types of Second-Order Equations





第四周



Initial-boundary value problem for a finite-line linear wave equation (II).





3.2 Reflections of Waves

Supplement to lecture notes





第五周



Linear superposition and sub-problems

Method of Separation of Variables

Fourier series representations of solutions





4.1* Separation of Variables, The Dirichlet Condition

Chapter 5 Fourier Series





第六周



Classification of 3 types of second order linear PDEs (II).

Initial value problem for a whole-line linear heat equation solved by the Fundamental solution





2.4* Diffusion on the Whole Line

4.1* Separation of Variables, The Dirichlet Condition





第七周



Initial-boundary value problem for a finite-line linear heat equations solved by method of Separation of Variables.

Initial value problem for an infinite-line linear heat equation solved by Fourier transform and inverse Fourier transform.





4.1* Separation of Variables, The Dirichlet Condition

Chapter 5 Fourier Series

12.3 Fourier Transform





第八周



Boundary value problem for a Laplace’s equation in a rectangle solved by method of Separation of Variables.

Boundary value problem for a Laplace’s equation in a circle solved by method of Separation of Variables.





6.1* Laplace’s Equation

6.2* Rectangles and Cubes 161

6.3* Poisson’s Formula

Chapter 5 Fourier Series





第九周



Boundary value problem for a Poisson’s equation in a circle.





6.3* Poisson’s Formula

Chapter 5 Fourier Series





第十周



Well-posed problems for linear PDE systems (I).





1.5 Well-Posed Problems

6.1* Laplace’s Equation





第十一周



Well-posed problems for linear PDE systems (II).





1.5 Well-Posed Problems

6.3* Poisson’s Formula





第十二周



Well-posed problems for linear PDE systems (III).





2.1* The Wave Equation





第十三周



Nonlinear problems (I) -

The effect of a combination of nonlinearity and dispersion;

The effect of a combination of nonlinearity and dissipation;

The effect of a combination of nonlinearity, dispersion, and dissipation.

Shock waves, steady-state solutions, travelling wave solutions, soliton solutions, N-soliton solutions, and wavetrains.





14.1 Shock Waves

14.2 Solitary waves and Solitons

Supplement to lecture notes





第十四周



Nonlinear problems (II) -

kdV equation and the solitary solutions





14.1 Shock Waves

14.2 Solitary waves and Solitons

Supplement to lecture notes





第十五周



Nonlinear Problems (III) - : Three famous universal nonlinear PDEs - kdV, s-G, and NLS equations.

Completely integrable systems

s-G equation and the travelling wave solutions.

NLS equation and the solitary wave solutions.





14.2 Solitary waves and Solitons

Supplement to lecture notes





第十六周



Nonlinear Problems (IV) - Introduction of Riemann surfaces of genus N (1) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS.





Supplement to lecture notes - Extension I of sec14.2 - the underlying theory of solutions of universal nonlinear PDEs (KdV, s-G, and NLS)





第十七周



Nonlinear Problems (V) - Introduction of Riemann surfaces of genus N (2) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS.





Supplement to lecture notes - Extension II of sec14-the underlying theory of solutions of universal nonlinear PDEs (KdV, s-G, and NLS)






 

课程书目 

PDE, An Introduction, 2nd ed. by Walter A. Strauss

 

评分标准  




项目

百分比





四次考试(最佳3次每次30%,剩余1次10%)



100%




 
授课对象:大学二年级学生
预备知识:常微分方程 (Differential Equations)
URI: http://ocw.nctu.edu.tw/course_detail.php?bgid=1&nid=503
http://hdl.handle.net/11536/132428
显示于类别:Open Course Ware