Asymptotic Analysis

Lecture (2 SWS) with Tutorial (1 SWS) in summer term 2016 at TU Berlin

Link to university calendar [1]

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.

Topics:

• Asymptotic sequences und series'
• Asymptotics of integrals (Laplace integrals, Watson lemma, method of startionary phases, steepest descent methods)
• Asymptotics of (ordinary) differential equations (regular asymptotic expansions, singularly perturbed differential equations, method of matched asymptotic expansions, multiscale method, WKB method)

• 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 [5]
• 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.

The solutions can be handed in in paper form or as PDF document attached to an e-mail to Anastasia Thöns-Zueva.

• Series 1 [6], to be handed in by May 12th, 2016, 2.15 a.m.
• Series 2 [7], to be handed in by May 26th, 2016, 2.15 a.m.
• Series 3 [8], to be handed in by June 9th, 2016, 2.15 a.m.
• Series 4 [9], to be handed in by June 23d, 2016, 2.15 a.m.
• Series 5 [10], to be handed in by July 14th, 2016, 2.15 a.m.

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.

Analysis I-II, Linear Algebra.

Helpful: Basic knowledge of differential equations, Mathematical Modelling