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.
| 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 |
COURSE
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