国立交通大学 偏微分方程 一 Partial Differential Equations 97学年度 应用数学系 林琦焜老师

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资源信息： 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 发行日期: 2008年 地区: 台湾 对白语言: 普通话,英语 文字语言: 繁体中文 概述:

课程介绍： 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