### Computer Assisted Proofs for Partial Differential Equations (Summer Semester 2012)

- Lecturer: Dr. Kaori Nagato-Plum
- Classes: Lecture (0156600)
- Weekly hours: 4

Examination on 27.09.2012 at 3A-03 (Allianz-Gebäude)

Time Matrikelnummer

13:00-13:30 1430406

13:30-14:00 1431181

14:00-14:30 1444219

14:30-15:00 1508261

15:00-15:30 1517182

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The lectures will be given in English.

The lecture on 4th May is shifted to

**8th May** (**17:30--19:00**, Room: **1C-02**).

The lecture on 7th May is as usual (14:00-15:30).

**Prüfung**

27.09.2012 13:00--

Please choose the following title on the registration:

*Computerunterstützte analytische Methoden für Rand- und Eigenwertprobleme*

Deadline of the registration: **25.07.2012**!

Schedule | |||
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Lecture: | Monday 14:00-15:30 | 1C-04 | Begin: 16.4.2012 |

Friday 9:45-11:15 | 1C-04 |

Lecturers | ||
---|---|---|

Lecturer | Dr. Kaori Nagato-Plum | |

Office hours: | ||

Room 2.029 Kollegiengebäude Mathematik (20.30) | ||

Email: kaori.nagatou@kit.edu |

Mathematical models in form of differential equations play an essential role in science and engineering, and investigating their solutions is of high importance in various respects, including the basic concepts of industrial technologies. Many analytical methods have been (and are being) developed over the centuries which give answers to questions concerning existence, multiplicity, and qualitative and quantitative properties of solutions to various classes of differential equation problems, such as variational methods, index and degree theory, monotonicity methods, fixed-point methods, semi-group methods, and more. Nevertheless, many important problems from theory and practice are still lacking satisfactory analytical results. On the other hand, numerical methods for computing approximate solutions to differential equation problems have enjoyed a huge development over the last decades, ranging from finite differences and projective spectral methods to finite elements and finite volumes, often in combination with multigrid schemes and specialized numerical linear algebra.

So many differential equation problems allow very "stable" numerical computations of approximate solutions, but are lacking results which are reliable in a strict mathematical sense. The field of "computer-assisted proofs" or "verified computations", which has attracted much attention in recent years, exploits the knowledge of "good" numerical approximate solutions to produce, by additional analytical arguments, strict mathematical results e.g. in form of existence statements together with rigorous error bounds. This field is by nature located near the borderline between analysis and numerical mathematics, which makes it very interesting and, since strengths of both sides are combined, also powerful.

The lecture course aims at the introduction of basic principles of computer-assisted proofs for differential equations based on a functional analytical setting, and also at rigorous computational techniques by use of interval arithmetic.

The lecture course addresses students in their sixth semester or higher, with basic knowledge in functional analysis and partial differential equations. It is suitable for students of mathematics, and for students of other subjects who have strong mathematical interests.

Material on 16.04.2012 Material on 23&27.04.2012 Material on 27.04.2012

Material on 20.07.2012

# Examination

A written or oral examination (depending on the number of attendees) will be held in the end of the semester.

# References

(More papers will be introduced during the lecture course.)

1. Robert A. Adams, John J. F. Fournier, SOBOLEV SPACES, second edition, Academic Press, 2003.

2. Götz Alefeld and Jürgen Herzberger, INTRODUCTION TO INTERVAL COMPUTATIONS, Academic Press, 1983.

3. Michael Plum, Computer-assisted Proofs for Semilinear Elliptic Boundary Value Problems. Japan Journal for Industrial and Applied Mathematics, Vol. 26, No. 2-3 (2009).