{"paper":{"title":"A finite difference method for a two-point boundary value problem with a Caputo fractional derivative","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Jos\\'e Luis Gracia, Martin Stynes","submitted_at":"2013-12-18T15:54:09Z","abstract_excerpt":"A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order $\\delta \\in (1,2)$ is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution $u$ of the boundary value problem are established, showing that $u''(x)$ may be unbounded at the interval endpoint $x=0$. These bounds and a discrete comparison principle are used to prove pointwise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5189","kind":"arxiv","version":1},"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"}