Numerical Approximation of the logarithmic Laplacian via sinc-basis

Abstract

In recent works, the authors of this chapter have shown with co-authors how a basis consisting of dilated and shifted $\text{sinc}$-functions can be used to solve fractional partial differential equations. As a model problem, the fractional Dirichlet problem with homogeneous exterior value conditions was solved. In this work, we briefly recap the algorithms developed there and that -- from a computational point of view -- they can be used to solve nonlocal equations given through different operators as well. As an example, we numerically solve the Dirichlet problem for the logarithmic Laplacian $\log (-\Delta )$ which has the Fourier symbol $\log ({\left|\omega \right|}^2)$ and compute its Eigenvalues on disks with different radii in ${\mathbb{R}}^2$.

14 pages, 5 figures