Applied Hypercomplex Analysis I

 Time and place of the lecture:
Wednesday 14.05 h - 15.35 h, Room P1.408
Start: October 14, 2009
Office hours:
     Friday 10:15-11:15 a.m.

Contents

1. Quaternions

2. Generalized concept of holomorphicity and basic tools of a function theory in H

3. Integral theorems and formulas

4. Quaternionic integral operators and function spaces 

5. Partial differential equations and boundary value problems from harmonic analysis

6. The Stokes- and Navier-Stokes system

7. Coupling of the Navier-Stokes systems with the heat equation

Description

Partial differential equations describe highly complex processes in physics and modern engineering. Precise statements about the existence and uniqueness of non-linear fluid dynamical processes which are described by the Navier-Stokes equations belong to the unsolved millenium problems.

In particular, one is very interested in studying combinations of the Navier-Stokes system with the heat equation (in the framework of  the micro-chip development) and with the Maxwell equations. The combination with the latter ones is of high current interest in the plasma research.

A better and deeper understanding of the mathematical theory behind these phenomena would provide a significant boost in the development and improvement of efficient and stable computation and simulation methods. 

The already existing methods often only provide useful results in some very special cases. For example in the simulation of real existing magnetohydrodynamic phenomena in the solar plasma the long distance scales represent a serious problem, since the currently existing methods use very small time steps.

In this lecture we give a consise introduction to a new approach to treat such complex systems of differential equations. We apply recently developed methods from hypercomplex function theory which represents an innovative and rapidly growing research domain on world wide level.

The use of hypercomplex differential and integral operators leads to new theoretical insight into the structure and about the regularity of the solutions to the systems we want to study. Furthermore, we obtain explicit representations for the solutions in terms of these operators. These in turn can be used to set up new computation methods and algorithms.

This lecture introduces the student into this field of topics. In particular we deal with the modeling of the physical processes. At the end of this lecture, the student is up to date with the current state of research in this field.

Requirements

1. Introductory lecture on functions of one complex variable and basic knowledge in real analysis of functions in several real variables

2. Linear algebra I

Teaching language

German. If the students prefer, this lecture can also be presented in English.

Bibliography

1. K. Gürlebeck, K. Habetha and W. Sprößig: Holomorphic functions in the plane and n-dimensional space (Translation of the German Version "Funktionentheorie in der Ebene und im Raum"), Birkhäuser, Basel, 2008.

2. K. Gürlebeck, W. Sprößig: Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser, Basel 1990

3. K. Gürlebeck, W. Sprößig: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, Chichester-New York, 1997.

4. R. S. Kraußhar: Generalized Automorphic Forms in Hypercomplex Spaces, Birkhäuser, Basel, 2004

Upfollowing course

Applied Hypercomplex Analysis II (Spring semester 2010)

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