大學課程

中文名: 國立交通大學偏微分方程(一)
英文名: Partial Differential Equations
別名: 數學物理方程
版本: 97學年度應用數學系林琦焜老師
發行時間: 2008年
地區: 台灣
對白語言: 普通話,英語
文字語言: 繁體中文
簡介:


課程介紹：

開拓視野新學習 數學交流新氣象

本課程是由交通大學應用數學系提供。

本課程屬研究所程度的微分方程課程，授課偏重於數學與物理間的連結，並且讓學生藉由此課程了解直觀地PDE概念。

授課教師 應用數學系 林琦焜老師

授課時數 每週3小時

授課學分 3學分

授課學期 97學年度

授課對象 研究所學生

預備知識Calculus, Advanced Calculus, Linear Algebra,Ordinary differential equation,Complex Analysis and Real analysis
課程綱要



課程目標/概述

本課程屬研究所程度的微分方程課程，授課偏重於數學與物理間的連結，並且讓學生藉由此課程了解直觀地PDE概念。

課程章節
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

課程書目

* Partial Differential Equations (4th Edition), Fritz John Applied Mathematical Sciences Vol.1, Springer-Verlag 1982
課程綱要
單元主題

內容綱要
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