大學課程

中文名: 國立交通大學偏微分方程(二)
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:

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。

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
章節

主題
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

課程書目

* 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
單元主題

內容綱要
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