{"paper":{"title":"A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Changpin Li, Hengfei Ding","submitted_at":"2016-05-07T12:05:51Z","abstract_excerpt":"Compared to the the classical first-order Gr\\\"unwald-Letnikov formula at time $t_{k+1} (\\textmd{or}\\, t_{k})$, we firstly propose a second-order numerical approximate scheme for discretizing the Riemann-Liouvile derivative at time $t_{k+\\frac{1}{2}}$, which is very suitable for constructing the Crank-Niclson technique applied to the time-fractional differential equations. The established formula has the following form $$ \\begin{array}{lll} \\displaystyle \\,_{\\mathrm{RL}}{{{\\mathrm{D}}}}_{0,t}^{\\alpha}u\\left(t\\right)\\left|\\right._{t=t_{k+\\frac{1}{2}}}= \\tau^{-\\alpha}\\sum\\limits_{\\ell=0}^{k} \\var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02177","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":""},"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"}