Partial Differential Equations I
本課程是由 國立陽明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。
Textbook：Fritz John, Partial Differential Equations (4th Edition), Applied Mathematical Sciences Vol.1 Springer-Verlag 1982.
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| Instructor(s) | Department of Applied Mathematics Prof. C. K. Lin |
|---|---|
| Course Credits | 3 Credits |
| Academic Year | 97 Academic Year |
| Level | Master's Student |
| Prior Knowledge | Calculus, Advanced Calculus, Linear Algebra, Ordinary differential equation, Complex Analysis and Real analysis |
| Related Resources | Course Video Course Syllabus Course Calendar |
| Week | Course Content | Course Video |
|---|---|---|
| 第一章 The Single First-Order Equation 1-1 Introduction | Watch Online | |
| 第一章 The Single First-Order Equation 1-2 Examples | Watch Online | |
| 第一章 The Single First-Order Equation 1-3 Analytic Solution and Approximation methods in a simple example | Watch Online | |
| 第一章 The Single First-Order Equation 1-4 Quasilinear Equation | Watch Online | |
| 第一章 The Single First-Order Equation 1-5 The Cauchy Problem for the Quasilinear-linear Equations | Watch Online | |
| 第一章 The Single First-Order Equation 1-6 Examples | Watch Online | |
| 第一章 The Single First-Order Equation 1-7 The general first-order equation for a function of two variables | Watch Online | |
| 第一章 The Single First-Order Equation 1-8 The Cauchy Problem | Watch Online | |
| 第一章 The Single First-Order Equation 1-9 Solutions generated as envelopes | Watch Online | |
| 第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-1 Characteristics for Linear and Quasilinear Second-Order Equations | Watch Online | |
| 第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-2 Propagation of Singularity | Watch Online | |
| 第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-3 The Linear Second-Order Equation | Watch Online | |
| 第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-4 The One-Dimensional Wave Equation | Watch Online | |
| 第二章 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-5 System of First-Order Equations | Watch Online | |
| 第二章Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables 2-6 A Quasi-linear System and Simple Waves | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-1 Natation of Laurent Schwartz | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-2 The Cauchy Problem | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-4 The Lagrange-Green Identity | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-5 The Uniqueness Theorem of Holmgren | Watch Online | |
| 第三章 Characteristic Manifolds and Cauchy Problem 3-6 Distribution Solutions | Watch Online | |
| 第四章 The Laplace Equation 4-1 Greens Identity, Fundamental Solutions, and Poissons Equation | Watch Online | |
| 第四章 The Laplace Equation 4-2 The Maximal Principle | Watch Online | |
| 第四章 The Laplace Equation 4-3 The Dirichlet Problem, Greens Function, and Poisson Formula | Watch Online | |
| 第四章 The Laplace Equation 4-4 Perrons method | Watch Online | |
| 第四章 The Laplace Equation 4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods | Watch Online |
課程目標
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 | 主題內容 |
| 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
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.
評分標準
| 項目 | 百分比 |
| 平時作業 | 60% |
| 期中報告 | 20% |
| 期末報告 | 20% |
本課程行事曆提供課程進度與考試資訊參考。
| 章節 | 主題內容 |
| 第一章 The Single First-Order Equation | 1-1 Introduction 1-2 Examples 1-3 Analytic Solution and Approximation methods in a simple example 1-4 Quasilinear Equation 1-5 The Cauchy Problem for the Quasilinear-linear Equations 1-6 Examples 1-7 The general first-order equation for a function of two variables 1-8 The Cauchy Problem 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 2-2 Propagation of Singularity 2-3 The Linear Second-Order Equation 2-4 The One-Dimensional Wave Equation 2-5 System of First-Order Equations 2-6 A Quasi-linear System and Simple Waves |
| 第三章 Characteristic Manifolds and Cauchy Problem | 3-1 Natation of Laurent Schwartz 3-2 The Cauchy Problem 3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem 3-4 The Lagrange-Green Identity 3-5 The Uniqueness Theorem of Holmgren 3-6 Distribution Solutions |
| 第四章 The Laplace Equation | 4-1 Greens Identity, Fundamental Solutions, and Poissons Equation 4-2 The Maximal Principle 4-3 The Dirichlet Problem, Greens Function, and Poisson Formula 4-4 Perrons method 4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods |
COURSE
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