{"paper":{"title":"Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Jiwei Zhang, Qian Zhang, Shidong Jiang, Zhimin Zhang","submitted_at":"2015-11-11T11:08:15Z","abstract_excerpt":"We present an efficient algorithm for the evaluation of the Caputo fractional derivative $_0^C\\!D_t^\\alpha f(t)$ of order $\\alpha\\in (0,1)$, which can be expressed as a convolution of $f'(t)$ with the kernel $t^{-\\alpha}$. The algorithm is based on an efficient sum-of-exponentials approximation for the kernel $t^{-1-\\alpha}$ on the interval $[\\Delta t, T]$ with a uniform absolute error $\\varepsilon$, where the number of exponentials $N_{\\text{exp}}$ needed is of the order\n  $O\\left(\\log\\frac{1}{\\varepsilon}\\left(\n  \\log\\log\\frac{1}{\\varepsilon}+\\log\\frac{T}{\\Delta t}\\right)\n  +\\log\\frac{1}{\\De"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03453","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"}