# TMA4305 Partielle differensialligninger våren 2010

Kursbeskrivelse finnes i studiehåndboka.

A partial differential equation is an equation involving a function in several variables and its partial derivatives. Many natural laws come in the form of differential equations. Thus the partial differential equations are fundamental in science, in particular in physics, astronomy,and mathematical modelling. They also occur in pure mathematics. The course aims at giving the student a good understanding of the basic methods in this interesting field of mathematical analysis.

## Beskjeder

- The first lecture is on Thursday the 14th of January 2010. The exercises begin on Tuesday the 19th.

- Example §2.2: 3 seems to be wrong. The second derivatives do not exist on the line x = y and hence it cannot be a solution there. However, the given function is a
*weak*solution in the whole plane

- The calculation in § 3.3b (Domain of Dependence) is far too complicated. A succinct approach is in Evans's book §2.4.3b.

- The chapter about Green's functions in Strauss's book is easy to absorb.

- L^p -spaces and Sobolev Spaces. Somewhat too advanced notes.

- L^p -spaces and Sobolev Spaces. Simpler version, edited.

## Faginformasjon

### Lectures

- Thursday 13:15-15:00 in F6
- Friday 10:15-12:00 in F2

### Exercises

- Tuesday 16:15 -17:00 ??

### Professor

- Peter Lindqvist, lqvist [at] math [dot] ntnu [dot] no Room 1152, SB II.

### Text book

*Partial Differential Equations*, Robert McOwen (Prentice Hall) – 2nd edition

### Syllabus ("Pensum")

- All the EXERCISES
- Chapter 1 §1.1, §1.2 + notes on "Weak solutions to Laws of Conservation", Rankine-Hugoniot. Not §1.3.
- Chapter 2 (not §2.2c,d. Not §2.3b)
- Chapter 3
- Chapter 4
- Chapter 5 (not §5.3, not §5.4)
- Chapter 6. Can be replaced by notes on"L^p spaces and Sobolev Spaces" (simpler version). In particular, weak compactness, weak convergence, Rellich-Kondratschev.
- "Calulus of Variations, Euler-Lagrange Equation". -The direct method for the Dirichlet problem, Poisson's eqn; how to find the E-L Equation.
- Maximum Principles (§8.3a, §11.1a)

### Examination

Eksamensdato er 4. juni 2010, tillatte hjelpemidler: C. Tillatt hjelpemiddel er et A4-ark der man kan skrive hva man vil. Bare offisielle ark er godkjent. (Disse fås siden på instituttkontoret i 7. etasje, Sentralbygg II.)

EXAM. 4.VI.2010

A YELLOW Piece OF PAPER (size A4) STAMPED AT the INSTITUTE IS ALLOWED, you may in advance write whatever pleases you on this sheet.

## Øvinger

Tusday 16-17 in F4.

Øving | Oppgave | Dato |
---|---|---|

1 | §1.1: 4b,5b,6a,6b,9b | Tirsdag 19 januari |

2 | §1.2: 2,3,4,6. §2.1: 1,7. | Tirsdag i uke 4 |

3 | §2.2: 1a,2b,3,4. §2.3: 8. | Read §2.3: 3,15. |

4 | §2.3: 3,10a,b. §3.2: 1,2a | |

5 | §3.2: 3,5. §3.3: 1,2,4. | Initial data required in 3.3/2 |

6 | §3.4: 6. §4.1: 4,5,6. § 4.2: 9 | |

7 | §4.2: 10,11. §4.3: 1,2,3 | |

8 | §4.4: 2,4,5,7. §5.2: 2,11 | |

9 | §5.2: 4,5,7. §5.1: 6,7. | Read §5.3: 6 |

10 | §8.3: 2,7. §6.1: 2,3. | Take c= 0. |

11 | Øving 11 | 13 April |

12 | §6.2: 1,2,3,4b. Find the min.surf.eqn from the area integral. | 20 April |

13 | Øving13 | 27 April |

## Further reading

Følgende bøker kan dessuten anbefales:

L. C. Evans: "Partial Differential Equations".–One of the most used today. More advanced than McOwen.

J. Jost: "Partielle Differentialgleichungen".– A good one!

J. Jost: "Partial Differential Equations".

W. Strauss: "Partial Differential Equations: An Introduction". –Clear and easy to read. More elementary than McOwen. Helpful.

F. John: "Partial Differential Equations" – An advanced account of the classical theory. Not easy for modern students.

H. Holden & N. Risebro: "Front Tracking for Hyperbolic Conservation Laws".–Chapter 1 is an excellent elementary introduction.

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

J. Logan: "An Introduction to Nonlinear Partial Differential Equations".–Informative, interesting, and easy to read.