First and second order linear wave equations;
Transport equations
Characteristic lines;
Travelling wave solutions.
Wave equations with dispersion, dissipation, and nonlinearity.
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
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
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.
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.
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.
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.
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.
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.
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.
