{"paper":{"title":"Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Siwei Duo, Yanzhi Zhang","submitted_at":"2018-04-08T17:08:25Z","abstract_excerpt":"In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\\Delta)^{\\frac{\\alpha}{2}}$ ($0 < \\alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\\alpha \\to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\\Delta$. We prove that our methods are consistent if $u \\in C^{\\lfloor\\alpha\\rfloor, \\alpha-\\lfloor\\alpha\\rfloor+\\epsilon}({\\mathbb R}^d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02718","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1804.02718/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}