### Page Content

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 Pert*urbations, 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.