## TMA4305 Partielle Differensialligninger 2015

## TMA4305 Partial Differential Equations Fall 2015

### Messages

### Office hours before the Exam

Tuesday 14–16

Wednesday 15–16

Thursday 14–16.

The first lecture will take place on Tuesday the 18th of August (14:15 - 16:00 in K26) The first exercises will take place on Monday, the 24th of August (17:15 - 18 in F3)

#### See below about the Exam.

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

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

34 | Ch.1, Ch.2 | Different kinds of equations and problems. Characteristics. First order eqs. | ||

35 | Ch.2, Ch. 10.4 | Weak Solutions. Rankine-Hugoniot Shock Condition | Notice Lemma 10.12 | |

36 | Ch.2, Ch 4.1, 4.2 | One dimensional Wave and Heat Equations. Duhamel"s Principle. Energy considerations ex.4.2.35 &36 | Duhamel's Principle not in the textbook | |

37 | Ch. 4 | Equations of 2nd order in two variables. Characteristic curves. Poisson's formula | Characteristics | |

38 | Ch. 6 | Distributions. Fundamental solutions | ? | |

39 | Ch. 7, Ch. 8 | Fourier transforms, the Heat Equation | Only selected parts | |

40 | Ch. 6, Ch. 8, Sec. 12.3 | Green's function. Duhamel's Principle. Newtons potential. Green's formulas | Section 12.3 is helpful (in space). Potential of a ball | |

41 | Sec. 8.3, Sec. 12.6 | Parabolic Maximum Principle, Kirchhoff's formula, Huygens's Principle | Simple proof, One dimension, parabolic boundary | |

42 | Ch. 12 | Duhamel's principle and the retarded potential for wave eqn., Nonlinear Diffusion, Dispersion | An example with Cole-Hopf cole.pdf. Local energy consideration. Uniqueness. Light cone | |

43 | See notes below. Ch. 9.3 | Dirichlet's Principle. Variational Integrals. Euler-Lagrange Equations | See the "direct method" | |

44 | Sobolev Spaces. Existence Proofs | Weak compactness, weak convergence | ||

45 | Ch. 9.4 | Eigenvalues, Rayleigh quotient | Merely for Helmholtz"s equation | Vibrating Membranes |

46 | Some minor additions. Repetition | ? | ||

47 | Repetition. Exercises | ? |

### Information about the course

#### Textbook

Peter Olver: "Introduction to Partial Differential Equations", Springer 2014. Also available on Springer link.

The lectures are usually more succinct than the book. (Habent sua fata libelli pro captu lectoris.)

### Lectures

Tuesday 14–16 in K26

Thursday 14–16 in K26

### Teacher

Peter Lindqvist (SB II room 1152)

### Exercises

*Monday 17 -18 aud. F3.

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

35 | 1.12, 1.13, 1.14, 2.1.4, 2.1.6, 2.2.4, 2.2.11a | read ex. 1.16 for your information.Some answers |

36 | 2.3.2, 2.3.3, 2.3.5, 2.3.15, 2.3.17, 2.3.22 | ? |

37 | 4.1.5, 4.1.16a,b, 4.1.17, 4.1.18, 4.2.9a,b,c, 4.2.35, 4.2.36 | Notice that 4.2.35&36 were done in class. Slns |

38 | Problems 4.3: 1a, 25a, 26, 32, 46;. 4.4.5 | Comments |

39 | 6.1: 1f, 2b, 5b, 7, 10 Examples | NO CLASS 21.IX. Solutions |

40 | 6.3: 9, 17, 29. 8.1: 17, 20. | Solution |

41 | 6.3.16, 8.1.14, 8.1.19 (misprint?), Exercises | Replace i by -i in 8.1.19. Scrodinger |

42 | 8.3: 4,5,6. 12.6.1 a,d, 12.6.2, 12.6.3 | Solutions |

43 | Exercises | Misprint in 1, WAVE eqn meant. Use rather cos(az)v(x,y,t) in 3. Solutions |

44 | 8.4.1, 8.4.11, 8.5.3a,b. Exercises | ? |

45 | 9.3: 18, 19, 20, 25. | Some Solutions |

46 | 9.4.5c, 9.4.6, 9.5.18 Exercises | Torsional Creep. More solutions |

47 | Exercises | Last ex. too long; do it for f = 1. Misprint: - sign in the root |

—————————————————————————————————–

# Preliminary Syllabus (Pensum)

##### ALL THE EXERCISES (Øvinger)

##### Chapter 1

##### Chapter 2

##### Chapter 4

##### Chapter 6

##### Chapter 8

##### Chapter 9.3, 9.4

##### Chapter 10.4

##### Chapter 12

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

Weak solutions (with test functions under the integral sign). Shocks.

# The End

### Extra literature

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.

S. Salsa: "Partial Differential Equations in Action". –Contains relevant information but is sometimes confusing.

### Exam 30.XI.2015

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. (A simple calculator is permitted.) No other aids are permitted in the exam.

Exam. 30.Xi.2015 Solutions 30.XI.2015

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

### NOTES, LINKS

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

### EXTRA NOTES

- Advanced Notes on Sobolev Spaces Sobolev

### NOTE OF INTEREST?

- Isospectral Domains isospectral.pdf*
- HalloweenDead class

#### Referansegruppe

Lene Hagen | lenehag [at] stud [dot] ntnu [dot] no |

Andre' Flakke | flakke [at] stud [dot] ntnu [dot] no |

August Sonne | Apsonne [at] stud [dot] ntnu [dot] no |