Weakly-singular formulation of the Fractional Laplacian operator
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914339664953344 |
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| author | Bruno, Oscar P. Sachan, Sabhrant |
| author_facet | Bruno, Oscar P. Sachan, Sabhrant |
| contents | This paper presents a new formulation of the fractional Laplacian operator $(-Δ)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-Δ)^s$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weakly-singular formulation of the Fractional Laplacian operator Bruno, Oscar P. Sachan, Sabhrant Analysis of PDEs Numerical Analysis This paper presents a new formulation of the fractional Laplacian operator $(-Δ)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-Δ)^s$ 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. |
| title | Weakly-singular formulation of the Fractional Laplacian operator |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2602.17970 |