{"paper":{"title":"Discretized fractional substantial calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Minghua Chen, Weihua Deng","submitted_at":"2013-10-11T11:31:20Z","abstract_excerpt":"This paper discusses the properties and the numerical discretizations of the fractional substantial integral $$I_s^\\nu f(x)=\\frac{1}{\\Gamma(\\nu)} \\int_{a}^x{\\left(x-\\tau\\right)^{\\nu-1}}e^{-\\sigma(x-\\tau)}{f(\\tau)}d\\tau,\\nu>0, $$ and the fractional substantial derivative $$D_s^\\mu f(x)=D_s^m[I_s^\\nu f(x)], \\nu=m-\\mu,$$ where $D_s=\\frac{\\partial}{\\partial x}+\\sigma=D+\\sigma$, $\\sigma$ can be a constant or a function without related to $x$, say $\\sigma(y)$; and $m$ is the smallest integer that exceeds $\\mu$. The Fourier transform method and fractional linear multistep method are used to analyze t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.3086","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"}