標題: 偏微分方程導論Introduction to Partial Differential Equations 作者: 李榮耀Open Education Office開放教育推動中心 公開日期: 2014 摘要: 課程首頁   本課程是由交通大學應用數學系提供。   Mathematical models as PDE ─ qualitative and quantative analysis. Three classical types of linear PDEs and the corresponding theory. A short topic on nonlinear PDE.課程目標/概述   Mathematical models as PDE – qualitative and quantative analysis. Three classical types of lineat PDEs and the corresponding theory. A short topics on nonlinear PDE.   課程章節   授課週次 老師授課主題 課本參考章節 第一週 PDE導論 Fundamental differences between PDE and ODE. 1.1* What is a Partial Differential Equation? 1.2* First-Order Linear Equations 第二週 First and second order linear wave equations; Transport equations Characteristic lines; Travelling wave solutions. Wave equations with dispersion, dissipation, and nonlinearity 2.1* The Wave Equation 第二週 Classical linear wave equations with travelling wave solutions. Dispersive linear wave equations. Dissipative linear wave equations. Nonlinear wave equations with shock wave solutions. Nonlinear wave equations with solitary wave solutions. Initial value problem for a whole-line linear wave equation and the dAlembert solution (I) 1.1* What is a Partial Differential Equation? 2.1* The Wave Equation Supplement to lecture notes 第三週 Classification of 3 types of second order linear PDEs (I). Initial value problem for a whole-line linear wave equation and the dAlembert solutions (II). Initial-boundary value problem for a half-line linear wave equation. Initial-boundary value problem for a finite-line linear wave equation (I) – method of Reflection and method of Separation of Variables. 2.1* The Wave Equation 3.2 Reflections of Waves 1.6 Types of Second-Order Equations 第四週 Initial-boundary value problem for a finite-line linear wave equation (II). 3.2 Reflections of Waves Supplement to lecture notes 第五週 Linear superposition and sub-problems Method of Separation of Variables Fourier series representations of solutions 4.1* Separation of Variables, The Dirichlet Condition Chapter 5 Fourier Series 第六週 Classification of 3 types of second order linear PDEs (II). Initial value problem for a whole-line linear heat equation solved by the Fundamental solution 2.4* Diffusion on the Whole Line 4.1* Separation of Variables, The Dirichlet Condition 第七週 Initial-boundary value problem for a finite-line linear heat equations solved by method of Separation of Variables. Initial value problem for an infinite-line linear heat equation solved by Fourier transform and inverse Fourier transform. 4.1* Separation of Variables, The Dirichlet Condition Chapter 5 Fourier Series 12.3 Fourier Transform 第八週 Boundary value problem for a Laplace’s equation in a rectangle solved by method of Separation of Variables. Boundary value problem for a Laplace’s equation in a circle solved by method of Separation of Variables. 6.1* Laplace’s Equation 6.2* Rectangles and Cubes 161 6.3* Poisson’s Formula Chapter 5 Fourier Series 第九週 Boundary value problem for a Poisson’s equation in a circle. 6.3* Poisson’s Formula Chapter 5 Fourier Series 第十週 Well-posed problems for linear PDE systems (I). 1.5 Well-Posed Problems 6.1* Laplace’s Equation 第十一週 Well-posed problems for linear PDE systems (II). 1.5 Well-Posed Problems 6.3* Poisson’s Formula 第十二週 Well-posed problems for linear PDE systems (III). 2.1* The Wave Equation 第十三週 Nonlinear problems (I) - The effect of a combination of nonlinearity and dispersion; The effect of a combination of nonlinearity and dissipation; The effect of a combination of nonlinearity, dispersion, and dissipation. Shock waves, steady-state solutions, travelling wave solutions, soliton solutions, N-soliton solutions, and wavetrains. 14.1 Shock Waves 14.2 Solitary waves and Solitons Supplement to lecture notes 第十四週 Nonlinear problems (II) - kdV equation and the solitary solutions 14.1 Shock Waves 14.2 Solitary waves and Solitons Supplement to lecture notes 第十五週 Nonlinear Problems (III) - : Three famous universal nonlinear PDEs - kdV, s-G, and NLS equations. Completely integrable systems s-G equation and the travelling wave solutions. NLS equation and the solitary wave solutions. 14.2 Solitary waves and Solitons Supplement to lecture notes 第十六週 Nonlinear Problems (IV) - Introduction of Riemann surfaces of genus N (1) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS. Supplement to lecture notes - Extension I of sec14.2 - the underlying theory of solutions of universal nonlinear PDEs (KdV, s-G, and NLS) 第十七週 Nonlinear Problems (V) - Introduction of Riemann surfaces of genus N (2) for the underlying theory of solutions of universal nonlinear PDEs such as kdV, s-G, and NLS. Supplement to lecture notes - Extension II of sec14-the underlying theory of solutions of universal nonlinear PDEs (KdV, s-G, and NLS)   課程書目  PDE, An Introduction, 2nd ed. by Walter A. Strauss   評分標準   項目 百分比 四次考試(最佳3次每次30%，剩餘1次10%) 100%  授課對象：大學二年級學生預備知識：常微分方程 (Differential Equations) URI: http://ocw.nctu.edu.tw/course_detail.php?bgid=1&nid=503http://hdl.handle.net/11536/132428 Appears in Collections: Open Course Ware