开拓视野新学习 数学交流新气象
本课程是由交通大学应用数学系提供。
This is the continuous course of Partial Differential Equations (I).This is the GRADUATE level partial differential equation. We will focus on the relation between mathematics and physics and show the students how to understand PDEs intuitively.
授课教师 应用数学系 林琦焜老师
授课时数 每週3小时
授课学分 3学分
授课学期 97学年度
授课对象 研究所学生
预备知识 Calculus, Advanced Calculus, Linear Algebra,Ordinary differential equation,Complex Analysis and Real analysis
课程纲要
课程目标/概述
本课程属研究所程度的微分方程课程，授课偏重於数学与物理间的连结，并且让学生藉由此课程了解直观地PDE概念。
课程章节
第一章 The Single First-Order Equation
第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables
第三章 Characteristic Manifolds and Cauchy Problem
第四章 The Laplace Equation
课程书目
* Partial Differential Equations (4th Edition), Fritz John Applied Mathematical Sciences Vol.1, Springer-Verlag 1982
课程纲要
单元主题
内容纲要
第一章 The Single First-Order Equation
1-1 Introduction Partial differential equations occur throughout mathematics. In this part we will give some examples
1-2 Examples
1-3 Analytic Solution and Approximation methods in a simple example 1-st order linear example
1-4 Quasilinear Equation The concept of characteristic
1-5 The Cauchy Problem for the Quasilinear-linear Equations
1-6 Examples Solved problems
1-7 The general first-order equation for a function of two variables characteristic curves, envelope
1-8 The Cauchy Problem characteristic curves, envelope
1-9 Solutions generated as envelopes
第二章Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables
2-1 Characteristics for Linear and Quasilinear Second-Order Equations Characteristic
2-2 Propagation of Singularity Characteristic curve and singularity
2-3 The Linear Second-Order Equation classification of 2nd order equation
2-4 The One-Dimensional Wave Equation dAlembert formula, dimond law, Fourier series
2-5 System of First-Order Equations Canonical form, Characteristic polynominal
2-6 A Quasi-linear System and Simple Waves Concept of simple wave
第三章 Characteristic Manifolds and Cauchy Problem
3-1 Natation of Laurent Schwartz Multi-index notation
3-2 The Cauchy Problem Characteristic matrix, characteristic form
3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem Local existence of solutions of the non-characteristic
3-4 The Lagrange-Green Identity Gauss divergence theorem
3-5 The Uniqueness Theorem of Holmgren Uniqueness of analytic partial differential equations
3-6 Distribution Solutions Introdution of Laurent Schwartzs theory of distribution (generalized function)
第四章 The Laplace Equation
4-1 Greens Identity, Fundamental Solutions, and Poissons Equation Dirichlet problem, Neumann problem, spherical symmetry, mean value theorem, Poisson formula
4-2 The Maximal Principle harmonic and subharmonic functions
4-3 The Dirichlet Problem, Greens Function, and Poisson Formula Symmetric point, Poisson kernel
4-4 Perrons method Existence proof of the Dirichlet problem
4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods Functional analysis, Riesz representation theorem, Dirichlet integra
课程章节
章节
主题
第五章 Hyperbolic Equations in Higher Dimensions
第六章 Higher-Order Elliptic Equations with Constant Coefficients
第七章 Parabolic Equations
第八章 H. Lewys Example of a Linear Equation without Solutions
课程书目
* Fritz John, Partial Differential Equations (4th Edition), Applied Mathematical Sciences Vol.1 Springer-Verlag 1982.
* A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983
单元主题
内容纲要
第五章 Hyperbolic Equations in Higher Dimensions
5-1 The Wave Equation in n-Dimensional Space
(1) The method of sphereical means
(2) Hadmards method of descent
(3) Duhamels principle and the general Cauchy problem
(4) mixed problem
5-2 Higher-Order Hyperbolic Equations with Constant Coefficients
(1) Standard form of the initial-value problem
(2) solution by Fourier transform,
(3) solution of a mixed problem by Fourier transform
5-3 Symmetric Hyperbolic System
(1) The basic energy inequality
(2)Finite difference method
(3) Schauder method
第六章 Higher-Order Elliptic Equations with Constant Coefficients
6-1 The Fundamental Solution for Odd n Travelling wave
6-2 The Dirichlet Problem Lax-Milgram theorem, Garding inequality
6-3 Sobolev Space Weak solution and Hibert space
第七章 Parabolic Equations
7-1 The Heat Equation Self-Similarity, Heat kernel, maximum principle
7-2 The Initial-Value Problem for General Second-Order Parabolic Equations
(1) Finite difference and maximum principle
(2) Existence of Initial Value Problem
第八章 H. Lewys Example of a Linear Equation without Solutions
8-1 Brief introduction of Functional Analysis Hilbert and Banach spaces, projection theorem, Leray-Schauder theorem
8-2 Semigroups of linear operator Generation, representation and spectral properties
8-3 Perturbations and Approximations The Trotter theorem
8-4 The abstract Cauchy Problem Basic theory
8-5 Application to linear partial differential equations Parabolic equation, Wave equation and Schrodinger equation
8-6 Applications to nonlinear partial differential equations KdV equation, nonlinear heat equation, nonmlinear Schrodinger equation
