After successful completion of the course, students are able to understand operator equations with non local operators analytically and numerically.

Especially students learn to cope with integral equations and the fractional Laplacian.

This lecture deals with non-local operators. Classical examples are integral operators with singular kernel (of convolution type) that arise from reformulations of PDEs as integral equations. We will at first study some analytic properties of these operators (e.g. mapping properties in Sobolev spaces) and afterwards discuss numerical methods (the boundary element method) for these equations as well as their advantages and disadvantages.

As a second model example of a non-local operator, we consider the fractional Laplacian, which is a non-integer power of the classical Laplacian. Such operators appear naturally in more refined models of anomalous diffusion in physics, biology or finance. Mathematically, there are various definition of the fractional Laplacian, which are not equivalent on bounded domains. In this lecture, we will consider two different definitions - the spectral and integral fractional Laplacian - and discuss numerical methods for these two definitions.

Lecture on the blackboard

Oral examination, make an appointment via e-mail

Lecture notes are created parallel to the course.