SES # | TOPICS | LECTURE SUMMARIES |
---|---|---|

1 | Mechanics of the course. Example PDE. Initial and boundary value problems. Well and ill-posed problems. | (PDF) |

2 | Conservation laws and PDE. Integral and differential forms. Closure strategies. Quasi-equillibrium. | (PDF) |

3 | Classification of PDE. Examples. Kinematic waves and characteristics. | (PDF) |

4 | First order scalar PDE. Examples of solutions by characteristics. Domain of influence. | (PDF) |

5 | Domains of influence and dependence. Causality and uniqueness. Allowed boundary conditions. Examples. | (PDF) |

6 | Graphical interpretation of solution by characteristics. Conservation. Wave steepening and breaking. Back to the physics. | (PDF) |

7 | Region of multiple values. Envelope of characteristics. | (PDF) |

8 | More on envelopes. Infinite slopes at envelope. Shocks. Conservation and entropy. Irreversibility. Examples from traffic flow. | (PDF) |

9 | Continues lecture 8. More examples. | |

10 | Shocks in the presence of source terms. Example. Riemann problems and Godunov's type methods. | (PDF) |

11 | The Riemann problem for the kinematic wave equation with convex/concave flux. Example of a conservation law with a point source term. | (PDF) |

12 | Shock structure and detailed physics. Examples: Viscosity solution. Traffic flow. Flood waves. Shallow water. | (PDF) |

13 | Shallow water and higher order terms. Traveling waves, shocks, and the effects of dispersion. Solitons. Small dispersion limit. | (PDF) |

14 | PDE and propagation of information. Equations that allow weak singularities. Examples. | (PDF) |

15 | Hyperbolicity and weak singularities. Examples: Hamilton-Jacobi equation and characteristic form. Eikonal equation. Multiple values. | (PDF) |

16 | Continue with Hamilton-Jacobi equation. Characteristics, strips, and Monge cones. Eikonal as characteristic equation for wave equation in 2-D and 3-D. | (PDF) |

17 | Eikonal. Focusing and caustics. Description of the caustic. Breakdown of approximation. Derivation of amplitude equation. | (PDF) |

18 | Eikonal. Amplitude and curvature along rays. Behavior near caustic. Caustic expansion. WKBJ review. Turning points. Conneccion formulas and Airy functions. Matching. | (PDF) |

19 | First order 1-D systems of equations. Classification. Hyperbolic systems and characteristics. Domains of dependence and influence. Examples. | (PDF) |

20 | Examples of first order 1-D hypebolic systems. Linear acoustics. Wave equation. D'Alembert solution. Simple waves. Wave breaking. Shocks and shock conditions. Examples | (PDF) |

21 | Gas dynamics in 1-D. Characteristics, simple waves, Riemann Invariants, rarefaction waves, shocks and shock conditions. Riemann problem. Generalizations to N by N systems. | (PDF) |

22 | Continue with Lecture 21. | |

23 | Linear equations. Superposition. Normal modes and impulse problems (Green's functions). Heat equation in 1-D examples: various initial and boundary value problems. Method of images. | (PDF) |

24 | Green's functions for signaling and source terms. Heat equation examples. Generalized functions. | (PDF) |

25 | Generalized functions. Green's functions for heat equation in multi-D. | (PDF) |

26 | Green's function. Poisson equation. Stokes equation. Example: stokes drag on a sphere. | (PDF) |