Nach positiver Absolvierung der Lehrveranstaltung sind Studierende in der Lage, fraktionale Differerntialgleichungen hinsichtlich ihrer analytischen Eigenschaften zu bewerten und geeignete numerische Verfahren zu entwickeln.
Fractional differential operators arise in many applications including the
description of anamalous diffusion, finance, material sciences, image processing.
In this lecture, we discuss the properties of fractional powers of elliptic operators
("fractional Laplacian") and their numerical approximation. Different ways to define
the fractional Laplacian include Fourier techniques, singular integrals, semigroup
techniques, reduction to elliptic problems by increasing the dimension. We will
discuss these techniques and show how they lead to different numerical methods. These numerical methods will be analyzed with respect to their convergence properties and implementation aspects.
For time-fractional operators (of Riemann-Liouville or Caputo type), various discretization
techniques have been proposed. We will discuss some of the them focussing in particular
on the technique of Convolution Quadrature, which is a general technique to discretize
convolution integrals.
While there are no textbooks for the topics, the current research is reflected in
the following references:
M. Kwasnicki. Ten equivalent definitions of the fractional Laplace operator. Fract. Calc.
Appl. Anal., 20(1):7–51, 2017.
https://arxiv.org/abs/1507.07356
A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otárola, and A.J. Salgado. Numerical
methods for fractional diffusion. Comput. Vis. Sci., 19(5-6):19–46, 2018.
https://arxiv.org/abs/1707.01566
Lischke et al.: What is the fractional Laplacian? J. Comput. Phys. 404 (2020), 109009, 62 pp.
https://arxiv.org/abs/1801.09767
