Introduction to Partial Differential Equations
本課程是由 國立陽明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.
Textbook：
PDE, An Introduction, 2nd ed. by Walter A. Strauss; Publisher: Wiley
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| Instructor(s) | Department of Applied Mathematics Prof. Jong-Eao Lee |
|---|---|
| Course Credits | 3 Credits |
| Academic Year | 103 Academic Year |
| Level | Second year university student |
| Prior Knowledge | Differential Equations |
| Related Resources | Course Video Course Syllabus Course Calendar |
| Week | Course Content | Course Video |
|---|---|---|
| Week 01 | PDE導論. Fundamental differences between PDE and ODE. | Watch Online |
| Week 02 | First and second order linear wave equations; Transport equations Characteristic lines; Travelling wave solutions. Wave equations with dispersion, dissipation, and nonlinearity. | Watch Online |
| Week 02 | 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 p | Watch Online |
| Week 03 | 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 fo | Watch Online |
| Week 04 | Initial-boundary value problem for a finite-line linear wave equation (II). | Watch Online |
| Week 05 | Linear superposition and sub-problems. Method of Separation of Variables. Fourier series representations of solutions. | Watch Online |
| Week 06 | 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. | Watch Online |
| Week 07 | 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. | Watch Online |
| Week 08 | 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. | Watch Online |
| Week 09 | Boundary value problem for a Poisson’s equation in a circle. | Watch Online |
| Week 10 | Well-posed problems for linear PDE systems (I). | Watch Online |
| Week 11 | Well-posed problems for linear PDE systems (II). | Watch Online |
| Week 12 | Well-posed problems for linear PDE systems (III). | Watch Online |
| Week 13 | 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 solution | Watch Online |
| Week 14 | Nonlinear problems (II) - kdV equation and the solitary solutions. | Watch Online |
| Week 15 | 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. | Watch Online |
| Week 16 | 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. | Watch Online |
| Week 17 | 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. | Watch Online |
課程目標
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
| 章節 | 章節內容 |
| PDE導論 Fundamental differences between PDE and ODE. | 1.1* What is a Partial Differential Equation? 1.2* First-Order Linear Equations |
| 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 |
| 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 |
| 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 |
| Initial-boundary value problem for a finite-line linear wave equation (II). | 3.2 Reflections of Waves Supplement to lecture notes |
| 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 |
| 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 |
| 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 |
| Boundary value problem for a Poisson’s equation in a circle. | 6.3* Poisson’s Formula Chapter 5 Fourier Series |
| Well-posed problems for linear PDE systems (I). | 1.5 Well-Posed Problems 6.1* Laplace’s Equation |
| Well-posed problems for linear PDE systems (II). | 1.5 Well-Posed Problems 6.3* Poisson’s Formula |
| Well-posed problems for linear PDE systems (III). | 2.1* The Wave Equation |
| 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 |
| Nonlinear problems (II) - kdV equation and the solitary solutions | 14.1 Shock Waves 14.2 Solitary waves and Solitons Supplement to lecture notes |
| 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 |
| 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) |
| 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% |
本課程行事曆提供課程進度與考試資訊參考。
授課日期 | 上課日期 | 參考課程進度 |
2015/02/25 | PDE導論 Fundamental differences between PDE and ODE. | 1.1* What is a Partial Differential Equation? 1.2* First-Order Linear Equations |
| 2015/03/02 | 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 |
| 2015/03/04 | 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 |
| 2015/03/11 | 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 |
| 2015/03/18 | Initial-boundary value problem for a finite-line linear wave equation (II). | 3.2 Reflections of Waves Supplement to lecture notes |
| 2015/03/25 | 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 |
| 2015/04/01 | 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 |
| 2015/04/08 | 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 |
| 2015/04/15 | 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 |
| 2015/04/22 | Boundary value problem for a Poisson’s equation in a circle. | 6.3* Poisson’s Formula Chapter 5 Fourier Series |
| 2015/04/29 | Well-posed problems for linear PDE systems (I). | 1.5 Well-Posed Problems 6.1* Laplace’s Equation |
| 2015/05/06 | Well-posed problems for linear PDE systems (II). | 1.5 Well-Posed Problems 6.3* Poisson’s Formula |
| 2015/05/13 | Well-posed problems for linear PDE systems (III). | 2.1* The Wave Equation |
| 2015/05/20 | 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 |
| 2015/05/27 | Nonlinear problems (II) - kdV equation and the solitary solutions | 14.1 Shock Waves 14.2 Solitary waves and Solitons Supplement to lecture notes |
| 2015/06/03 | 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 |
| 2015/06/10 | 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) |
| 2015/06/17 | 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) |
COURSE
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