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.
| 週次 | 課程內容 | 課程影音 |
|---|---|---|
| 第一週 | PDE導論. Fundamental differences between PDE and ODE. | 線上觀看 |
| 第二週 | First and second order linear wave equations; Transport equations Characteristic lines; Travelling wave solutions. Wave equations with dispersion, dissipation, and nonlinearity. | 線上觀看 |
| 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 p | 線上觀看 |
| 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 fo | 線上觀看 |
| 第四週 | Initial-boundary value problem for a finite-line linear wave equation (II). | 線上觀看 |
| 第五週 | Linear superposition and sub-problems. Method of Separation of Variables. Fourier series representations of solutions. | 線上觀看 |
| 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. | 線上觀看 |
| 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. | 線上觀看 |
| 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. | 線上觀看 |
| 第九週 | Boundary value problem for a Poisson’s equation in a circle. | 線上觀看 |
| 第十週 | Well-posed problems for linear PDE systems (I). | 線上觀看 |
| 第十一週 | Well-posed problems for linear PDE systems (II). | 線上觀看 |
| 第十二週 | Well-posed problems for linear PDE systems (III). | 線上觀看 |
| 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 solution | 線上觀看 |
| 第十四週 | Nonlinear problems (II) - kdV equation and the solitary solutions. | 線上觀看 |
| 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. | 線上觀看 |
| 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. | 線上觀看 |
| 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. | 線上觀看 |
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