## TMA4305 Partielle Differensialligninger 2014

## TMA4305 Partial Differential Equations Fall 2014

### Messages

### Office hours before the exam:

Wednesday the 26th from 14 to 16 in room 1152, SB II

Thursday the 27th from 12 to 14 in room 1152, SB II

#### See below about the Exam.

* The lectures are in K26. (Kjemi 4, first floor)

About Ch. 2: Random walk almost omitted for Heat Eqn.

Week | Section | Comments | |
---|---|---|---|

34 | 4.5.1, 4.5.2, 3.1, 3.2, ? | The beginning of Ch.4 later. Different kinds of equations and problems. Laplace's Eq | |

35 | Ch 3. 109–118, 124–133 | Laplace's Eq | |

36 | Ch.3. 132–133, 139–155 Chapter 2.1,2.2 | Thm 3.11 with proof. Lemma 3.2 by an example.Easy proof of Parabolic Maximum Principle Simplest case | |

37 | From Ch. 2 | The Heat Equation, Maximum principles, Heat kernel via Fourier Transf (Ex. 2.12, p.99). Initial values attained (rigorous proof). Duhamels Principle with verification. Einstein 1905 | Fouriertransf. |

38 | Ch. 2. Ch. 5 (?) | Heat Eqn. Tychonov's Thm. Backwards uniqueness. Porous Medium Eqn. | |

39 | Ch. 5 | Wave Eqn., Energy, D'Alembert, Duhamel's Principle,Weak Solutions | |

40 | Ch. 5 | Kirchhoffs formula. Huygen's Principle | |

41 | Ch. 5 | Duhamel (again!). Method of Stationary Phase. Equations of 2nd order | Classification, Characterisitics |

42 | Ch. 5 | Characteristic coordinates. Eqns of 2nd order. Symmetric matrices | |

43 | ch. 5 | Elliptic Equations again A Proof | Weak Solutions and Shocks |

44 | Ch. 4 | ||

45 | Ch. 4 | Breaking time with smooth data. Why the Atom Does not Collapse. Comments on Existence | Perron's Method |

46 | Repetiton with examples | Elliptic eqs. | |

47 | Repetition with Examples | Planned 2006/3,4. 2007/1,3,4. 2010/2. 2011/1. 2009/2,4. |

### Information about the course

#### Textbook

Sandro Salsa: "Partial Differential Equations in Action", Springer.

##### Misprints in the book

errata_salsa.pdf —Abundant dulcibus vitiis !

Habent sua fata libelli pro captu lectoris.

### Lectures

Tuesday 12–14 in K26

Thursday 12–14 in K26

### Teacher

Peter Lindqvist (SB II room 1152)

### Exercises

*Monday 13 –14 S 265

Week | Problems | Comments |
---|---|---|

35 | Exercise 1 | 3/2-1/2(xx+yy) misprinted |

36 | 3.1, 3.2. 3.3, 3.5 | Example 3.5 Example 3.3 |

37 | 3.8, 3.11, 2.2 and these | ?Is 2.2 misprinted? |

38 | 2.1, 2.8, 2.12, 2.14, 2.15 | 2.12 was lectured, done.Soln 2.14 |

39 | 2.13, 2.16, 2.19 | Ex. 2.19 comment |

40 | 3.21, 5.5a, 5.4, Svag lo"sn | Ex. 5.4 |

41 | 5.3, 5.16, 5.17 | Ex.: verify that the solution in Kirchhoff's formula takes the initial data |

42 | Exercises | 5.17 focus14.pdf. Ex.3: bx-2ay,x. Ex.2: invariant |

43 | 5.8, 5.9, 5.10, 5.12, Example | confusing: non-characteristic initial values means that these be given on a characteristic curve !? In 5.9 the other coordinate can be \eta = x |

44 | 4.1, 4.2, Examples | |

45 | 4.3, 4.6, 4.9, 4.13 | Ex. 4.6Answer |

46 | Exercises | Solution with vanishing viscosity. The question in 2.16 confusing: u = y/2. (The obtained formula cannot be ext.) |

47 | Last exercises |

# Syllabus (Pensum)

##### ALL THE EXERCISES (Øvinger)

##### Chapter 2

2.1, 2.2, 2.3, not 2.4, not 2.5.1, 2.5.2, not 2.5.3, not 2.6, 2.8, not 2.9, 2.10.1, not 2.10.2.

##### Chapter 3

3.1, 3.2, not 3.3.1, 3.3.2 - 3.3.5. Not 3.36, not 3.3.7. 3.4, 3.5.1, not 3.5.2, 3.5.3, not 3.5.4, 3.6, 3.7.1 (Only Gauss's Lemma), not 3.7.2.

##### Chapter 4

4.1 - 4.5 (not 4.4.1, not 4.5.4), not 4.6.

##### Chapter 5

Not 5.6, not 5.8.2, not 5.9.1, not 5.10, but 5.10.6 (Stationary Phase) in the simplest case included.

Sections 5.1 and 5.4.5 serve as orientation.

How to find the Euler-Lagrange Equation of a Variational Integral. Dirichlet's Principle.

# The End

### Extra literature

P. Olver: "Introduction to Partial Differential Equations. Springer. –Available on Springer link. Interesting.
L. C. Evans:"Partial Differential Equations". –One of the most used standard texts today. Advanced.

W. Strauss: "Partial Differential Equations: an Introduction". –Clear and easy to read.

A. Tveito & R. Winther:" Introduction to Partial Differential Equations (A Computational Approach)". –An excellent, but very elementary account.

### Exam 1.XII.2014

The exam (written) is in English only. The students may answer in Norwegian or English. To the exam you are allowed to bring one A4-sized sheet of yellow paper on which you may write whatever you want in advance. It must be pre-stamped at the Department of Mathematics (SB II, 7th floor), where empty sheets can be acquired. No other aids are permitted in the exam.

Exam 1.XII.2014.Misprint 3b. Read u(x,y,z,0) = f(x,y,z). Solutions 1.XII.2014

### NOTES, LINKS

- Dirichlet's Principle Notes
- The temperature of the earth earth.pdf
- Perron's Method perron.pdf
- Poisson's formula from Cauchy'scauchy.pdf

#### Referansegruppe

Håkon Bølviken | haakosb [at] stud [dot] ntnu [dot] no |

Emil Hansen | emilhan [at] stud [dot] ntnu [dot] no |

Stine Vennemo | vennemo [at] stud [dot] ntnu [dot] no |