{"paper":{"title":"Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Adam Oberman, Yanghong Huang","submitted_at":"2013-11-29T20:56:42Z","abstract_excerpt":"The fractional Laplacian $(-\\Delta)^{\\alpha/2}$ is a non-local operator which depends on the parameter $\\alpha$ and recovers the usual Laplacian as $\\alpha \\to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.7691","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}