偏微分方程導論

Abstract

課程首頁

本課程是由交通大學應用數學系提供。

Mathematical models as PDE ─ qualitative and quantative analysis.

Three classical types of linear PDEs and the corresponding theory.

A short topic on nonlinear PDE.

課程目標/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.

課程章節

授課週次

老師授課主題

課本參考章節

第一週

PDE導論

Fundamental differences between PDE and ODE.

1.1* What is a Partial Differential Equation?

1.2* First-Order Linear Equations

第二週

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

第二週

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

第三週

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

第四週

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

3.2 Reflections of Waves

Supplement to lecture notes

第五週

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

第六週

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

第七週

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

第八週

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

第九週

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

6.3* Poisson’s Formula

Chapter 5 Fourier Series

第十週

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

1.5 Well-Posed Problems

6.1* Laplace’s Equation

第十一週

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

1.5 Well-Posed Problems

6.3* Poisson’s Formula

第十二週

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

2.1* The Wave Equation

第十三週

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

第十四週

Nonlinear problems (II) -

kdV equation and the solitary solutions

14.1 Shock Waves

14.2 Solitary waves and Solitons

Supplement to lecture notes

第十五週

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

第十六週

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)

第十七週

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)

課程書目

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

評分標準

項目

百分比

四次考試(最佳3次每次30%，剩餘1次10%)

100%

授課對象：大學二年級學生

預備知識：常微分方程 (Differential Equations)