中文名: 国立交通大学 偏微分方程(一)
英文名: Partial Differential Equations
别名: 数学物理方程
版本: 97学年度 应用数学系 林琦焜老师
发行时间: 2008年
地区: 台湾
对白语言: 普通话,英语
文字语言: 繁体中文
简介:
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。
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。
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