Weakly-singular formulation of the Fractional Laplacian operator

TL;DR

This paper introduces a novel weakly-singular formulation of the fractional Laplacian operator, enabling efficient numerical solutions by reducing the problem to a weakly singular integral equation, demonstrated with high-accuracy numerical examples.

Contribution

It proposes a new formulation expressing the fractional Laplacian as a composition of classical Laplacian and a weakly singular integral, facilitating efficient numerical methods.

Findings

High accuracy in numerical examples

Efficient computational performance

Reduction to weakly singular integral equation

Abstract

This paper presents a new formulation of the fractional Laplacian operator $(-\Delta)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-\Delta)^s$ as a composition of the classical Laplace differential operator and a weakly singular integral operator -- which can be used to reduce e.g. the Dirichlet problem for the fractional Laplacian to a weakly singular integral equation involving both volumetric and boundary integral operators. This reformulation is well suited for efficient and accurate numerical implementation. Although a full description of the associated high-order algorithm is deferred to a subsequent contribution, several numerical examples are included in this paper to demonstrate the high accuracy and computational efficiency achieved by the proposed approach.