# TMA4305 Partielle Differensialligninger 2016

# TMA4305 Partial Differential Equations Fall 2016

"Everything should be made as simple as possible - but no simpler [than that]" A. EINSTEIN

## Messages

This is NOT an internet course. Necessary details and calculations, missing from the text in the book, are lectured. The exercises are essential. If you cannot participate, you had better arrange so that somebody present provides you with notes!

Solutions to the exam. are given below.

## Office hours before the exam.

***Friday 9.XII 15-16, Thursday 15.XII 14-16, Friday 16.XII 15-16 (SB II room 1152)***

The first lecture will take place on Wednesday the 24th of August 08:15 - 10:00 in KJL3) The first exercises will take place on Monday, the 29th of August (14:15 - 15:00 in KJL3)

### See below about the Exam.

* The lectures are in K26. (Kjemi 4, first floor) and KJL3

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

34 | Ch.1, Ch.2, Ch. 3.1 | 2016. Different kinds of equations and problems. Characteristics. First order eqs. Classification of 2nd order eqs | Characteristics 2nd order eqs | ||

35 | Ch.3, Ch.4 | 2016. Breaking time. One-dimensional Wave eqn, d"Alembert's soln. Duhamel's Principle | Breaking timeEx. 3.15 | ||

36 | Ch 4. Ch. 6.3 | The Wave Equation. Energy considerations. Kirchhoff's formula. | Omitted calculations in the treatment of Kirchhoff's formula in the textbook are provided | ||

37 | Ch. 4. Ch. 2.3 | The wave eqn. The two-dimensinal case. Duhamels formula (retarded potential). Cauchy-Kowalewskaja (2.3) | Retarded potential not in the book. Domain of Dependence/Influence | ||

38 | Ch. 5 | The Heat Equation. | Basic facts of the Fourier Transform used. | ||

39 | Ch. 5 | Heat Equation. Parabolic Maximum Principle. Weierstrass's Approximation Thm proved with the Heat Eqn. | One dimension. Parabolic Boundary. Simple proof. Schrodinger's Eqn. | ||

40 | Ch. 8 | Backwards Heat Eqn.—Laplace's Eqn. Newtons potential. Green's formulas | Potential of a ball | ||

41 | Ch. 8, Sec. 9.3 | Laplace's Eqn, Poisson's formula, Liouville's Thm. Strong Max. Principle | 2016 | ||

42 | Ch. 9, Ch. 10 | Distributions, Dirac's delta, Sobolev's space | 2016 | ||

43 | Ch. 10, Ch. 11 | Poincare Ineq., Maximum Principle for 2nd order elliptic equations. | See notes below about max.princ. | ||

44 | Dirichlet's Principle. Calculus of Variations | Dirichlet's Principle | Optional: See the direct method | ||

45 | Ch. 13, 14 | Conservation Laws, Shocks | 2016 | ||

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

47 | Repetition. Exercises | ? |

## Information about the course

### Textbook

M. Shearer & R. Levy: "Partial Differential Equations: an Introduction to Theory and Applications", Princeton University Press 2015.

The lectures usually contain more details and examples than the book. The syllabus is according to the lectures. This is not an internet course! (Habent sua fata libelli pro captu lectoris.)

### Lectures

Wednesday 08–10 in KJL3

Thursday 14–16 in K26

### Teacher

Peter Lindqvist (SB II room 1152)

### Exercises

*Monday 14 -15 aud. KJL3.

"That never any knowledge was delivered in the same order it was invented, not in the mathematic,.." Sir Francis Bacon (1561-1626)

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

35 | 1.4, 1.5, 1.6, 1.8, 2.1, 3.1, 3.2, 3.8 | 2016 | |

36 | 3.12, 4.2, 4.8, 4.10 | 2016 | |

37 | Exercises | Ex. 4 is essential | |

38 | Exercises | Pay attention to Leibnitz' rule in ex. 4 | 2016 |

39 | 5.5, 5.6, 5.7, 5.8, 5.9 | Solutions | 2016 |

40 | Exercises | Parab. comparison in unbounded domain | |

41 | 8.1, 8.3, 8.4, 8.5, 8.6 | 2016 | |

42 | 8.7, 9.1, and exercises | 2016 | |

43 | 9.8, 9.10, 9.11, 9.12 pde2016uke43.pdf | 2016 | |

44 | Exercises | 2016 | |

45 | Exercises | Misprint in the denominator of Ex.2. | |

46 | Exercises | 16x=(3t+1)(t+3) | |

47 | Last Exercises | 21.XI.2016. Soln of ex.3 |

—————————————————————————————————–

## Preliminary Syllabus (Pensum) 2016

#### ALL THE EXERCISES (Øvinger)

#### Chapter 1

#### Chapter 2

#### Chapter 3

#### Chapter 4

#### Chapter 5

Chapter 6 & 7. Only separation of variables

#### Chapter 8

#### Chapter 9

#### Chapter 10

#### Chapter 11

#### Ch. 13.1, 13.2, 13.5 (Cole-Hopf)

#### Ch. 14.2.7 (Scalar case)

#### 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.

P. Olver: " Introduction to Partial Differential Equations". Springer 2014. –Helpful for those who cannot attend to the lectures.

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 19.XII.2016

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. 19. XII.2016. Solutions. Misprint in 6.

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

- Weak solutions and shocks weaksolutions14.pdf
- Elliptic Equations Maximum Principle Again
- Dirichlet's Principle Notes
- Vibrating membranes (Not the Bessel functions)
- Isospectral Domains Optional picture.
- The temperature of the earth earth.pdf
- Perron's Method perron.pdf. Optional
- Poisson's formula from Cauchy'scauchy.pdf Optional
- Black-Scholes's Formula Optional (r and sigma constants).

### EXTRA NOTES

- Advanced Notes on Sobolev Spaces Sobolev

### NOTE OF INTEREST?

- Isospectral Domains isospectral.pdf*
- HalloweenDead class