Weakly-singular formulation of the Fractional Laplacian operator

This paper presents a new formulation of the fractional Laplacian operator $(-\mathrm{\Delta }{)}^s$ in $n$-dimensional space ($n\ge 1$). The proposed formulation expresses $(-\mathrm{\Delta }{)}^s$ 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.