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Scattering Problems (Winter Semester 2007/08)

The course is targeted at students of mathematics or physics after the Vordiplom. Basic knowledge of functional analysis is helpfull but not necessary to follow the course. There will be weekly problem sheets that can be solved in teams during the problem classes.

If required, the course will be held in English.

Lecture: Tuesday 8:00-9:30 Seminarraum 31
Thursday 8:00-9:30 Seminarraum 31
Problem class: Tuesday 9:45-11:15 Seminarraum 009 (20.30)
Lecturer PD Dr. Tilo Arens
Office hours: Wednesday 11:00-12:00 Uhr
Room 1.047 Kollegiengebäude Mathematik (20.30)
Email: tilo.arens@kit.edu
Problem classes Dr. Kai Sandfort
Office hours:
Room Allianz-Gebäude (05.20)

In a scattering problem, in incident (acoustic) wave hits an obstacle. Through interaction of obstacle and incident wave, a scattered field is generated. This is to be determined.

Mathematically, scattering problems can be formulated as exterior boundary value problems for the Helmholtz equation. On the boundary of the obstace, a boundary condition is prescribed which corresponds to the physical properties of the scatterer. In addition, at infinity, a suitable condition has to be formulated which guarantees uniqueness of solution. This condition is called a radiation condition.

In the first part of the lecture, we will consider solutions of the Helmholtz equation that can be obtained through separation of variables. In this way, we can formulate series expansions for general solutions. In addition, we obtain some useful identities which we will use later on. With Green's indentities and Green's representation formula we provide important tools for working with the Helmholtz equation.

In the second part of the course we investigate weak formulation of boundary value problems for the Helmholtz equation. We will use Sobolev spaces and various theorem from functional analysis. We formulate conditions that guarantee existence and uniqueness of solutions.

Finally, in the third part of the course, we consider the numerical solution of scattering problems through boundary element methods. The plan is to consider integral equations of both the first and the second kind and to analyse various approaches for their solution.