Applied Mathematics Seminar

Speaker: Hans-Werner van Wyk

Title: A novel finite difference method for the fractional Laplacian

Abstract: In this talk we discuss a novel finite difference method to discretize the fractional Laplacian operator over a bounded domain. Our approach is based on splitting the hypersingular kernel function in its integral representation into two parts, allowing some of the singularity to be absorbed by the operator’s input function \(u\). We then approximate the resulting weakly singular integral by a weighted trapezoidal rule. Our method is simple to implement and has a higher accuracy than the existing finite difference approximations. For smooth enough \(u\) and a judicious choice of the splitting parameter, we obtain second order accuracy, uniformly for any diffusion order.