《國立交通大學 偏微分方程(一)》(Partial Differential Equations)97學年度 應用數學系 林琦焜老師

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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: [url=http://tc.wangchao.net.cn/bt/detail_90137.html][img]http://image.wangchao.net.cn/bt/1271791659091.jpg[/img][/url] 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