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# Asymptotic Analysis

Lecture (2 SWS) with Tutorial (1 SWS) in summer term 2014 at TU Berlin

Dates
Time
Room
Lecturer
Lectures
Wed, 12.15 pm - 1.45 pm
MA 542
Dr. Kersten Schmidt
Exercises
Wed, 2.15 pm - 3.45 pm
MA 542
Dr. Anastasia Thöns-Zueva

## Content

In mathematical models in natural sciences as well as in technological devices very different time or length scales are often present. The presence of the different scales can then be represented by a small parameter (denoted by ε for example). The parameter has often a singular character and cannot simply be set to zero. The solution of the model with ε = 0 differs then much from the solution with small, but non-vanishing parameter and the application of standard methods leads often to utterly wrong results.

To study and solve such so called singularly perturbed problems the asymptotic analysis and asymptotic expansions can be helpful. The original problem is replaced by a series of problems, which are much easier to treat, and whose solutions give (in sum) an approximation to the original problem. There exist special analytical methods like the method of matched asymptotics or the multiscale method and specially adapted numerical methods.

The lecture consists of two parts. In the first part in winter term 2013/14 we studied the asymptotic analysis and asymptotic expansions for integrals and ordinary differential equations (ODEs) including homogenisation. In this second part we will study the asymptotic methods for singularly perturbed partial differential equations (PDEs) including homogeniyation as well as numerical methods inspired by results from asymptotic analysis.

## References

• H.J.J. Roessel and J.C. Bowman, Asymptotic Methods, lecture notes, University of Alberta, Edmonton, Canada, 2012.
http://www.math.ualberta.ca/~bowman/m538/m538.pdf
• C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999
• J. A. Murdock, Perturbations: Theory and Methods, SIAM, 1987.
• W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, 1979.
• J. Kevorkian und J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, Applied Mathematical Sciences 114, 1996.

## Tutorials

• Serie 1,  be prepared to present your results on May 7th, 2014, 12.15am in MA 542.

## Audience

Students at bachelor, master level or diploma students in mathematics (incl. Techno-, Wirtschaftsmathematik and Scientific Computing), as well as doctoral students (incl. BMS). Students in physics and engineering disciplines with interest in theory are welcome.

## Requirements

Analysis I-II, Linear Algebra.

Helpful: Asymptotic Analysis I and basic knowledge of differential equations.

## Auxiliary Functions

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TU Berlin
Institute of Mathematics
sec. MA 6-4
Strasse des 17. Juni 136
D-10623 Berlin

## How to find us

Maths Building (MA)
6th floor
rooms 363, 365 and 379