# Lecture plan/topics

This plan will be updated throughout the semester. Chapters refer to the book by Borthwick.

If you spot any mistakes in some of the notes below, please let us know, so that we can correct them.

Day | Topic | Chapter | Repetition | Assumed knowledge from other courses |
---|---|---|---|---|

22.8.23 | Introduction (selv study) | 1 | Repetition | |

ODE theory and needed notation | 2.5-2.6 | |||

Method of characteristics | 3.2 | |||

25.8.23 | Method of characteristics | 3.2-3.4, Notes H,Notes K | Repetition | |

29.8.23 | Method of characteristics | 3.2-3.4, Notes H, Notes K | Repetition | |

Wave equation | 4.1-4.4 | Even and odd extensions of functions | ||

1.9.23 | Wave equation | 4.5-4.6 | Repetition | Surface integrals (over spheres). Remark: 4.5 not treated in class. |

5.9.23 | Wave equation | 4.6-? | Repetition | double and triple integrals, surface integrals |

Vector calculus | 2.6 | |||

8.9.23 | Wave equation | 4.6-4.7 | Repetition | |

12.9.23 | Domain of Dependence | (6.1). 6.2-6.3 | Repetition | 6.1 self study. |

Wave Equation. Heat Equation. | ||||

15.9.23 | Heat equation. Section 9.5 (or notes).Parabolic. Max.Pr.. One-dim. | 6.4, Notes H | Repetition | |

19.9.23 | Heat equation | Notes H 6.4 | Repetition | |

22.9.23 | Burgers's Eqn. with viscosity (short remark). The Laplace equation | 9.2, Notes H | Repetition | |

26.9.23 | Laplace equation | 9.2, Notes H | Repetition | |

\(L^p\) spaces | 7.1-7.4, Notes K | 7.1: inner product space, normed vector spaces | ||

29.9.23 | \(L^p\) spaces | 7.1-7.4, Notes K | Repetition | |

3.10.23 | \(L^p\) spaces | 7.1-7.4, Notes K | Repetition | |

The proof of Theorem 12.9 was sketched, page 253. | ||||

6.10.23 | Conservation laws. Weak derivatives (Sobolev deriv.) | 10.1, 10.2Weak Solutions and Shocks | Repetition | Green's theorem |

10.10.23 | Conservation laws | Repetition | ||

Sobolev spaces | ||||

13.10.23 | Weak solutions | Chapter 10 | Repetition | Chapter 10. (Section 10.4 without proofs). |

17.10.23 | Weak solutions | Repetition | ||

Dirichlet's Problem | 11.1 -11.3. | dElementary notes | 11.4 without proof. | |

20.10.23 | Poisson equation. Eigenvalues. Variational Principles. | 11.3, 11.5. Not regularity issues. | Repetition | |

Harald Hanche-Olsen lectured the following two weeks (four lectures) | ||||

24.10.23 | Distributions | 12.1-3, Notes K | Repetition | There are several common notations for distributions: \(u(\phi)\), \(\langle u,\phi\rangle\), and \((u,\phi)\). |

27.10.23 | Distributions | 12.1-3, Notes K | Repetition | |

31.10.23 | Fundamental solutions | 12.4, Notes H | ||

14.11.23 | Classification of 2nd order equarions | 12.5, Notes H | F. John. Partial Differential Equations, Ch. 2.1. Springer | |

17.11.23 | Elliptic 2nd order Equations: Maximum Principle. | Section 9.4. See also Weak Maximum Principle | ||

21.11.23 and 24.11.23 | Mostly repetiotion and examples. | Repetitio mater studiorum est | ||

Repetition | Eigenvalues and eigenvectors | |||

10.11.23 and 7.11.23 | Calculus of Variations. Dirichlet Eigenvalues. | 11.5 and 11.8 11.7 | Orthonormal basis, Properties of Hilbert spaces | |

Classification of linear PDEs | Chapter 4.4 | You must be inside the NTNU network, to have access. | ||

12.12.23 | Exam. 2023. | Focus: The problems you are most intersted in. |