{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:58:02Z","timestamp":1762516682291,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we establish a new fractional integral identity linked to the 4-point Lobatto quadrature rule within the Riemann\u2013Liouville fractional calculus framework. Building on this identity, we derive several Lobatto-type inequalities under convexity assumptions, yielding error bounds that involve only first-order derivatives, thereby improving practical applicability. A numerical example with graphical illustration confirms the theoretical findings and demonstrates their accuracy. We also present applications to special means, highlighting the utility of the obtained inequalities. The integration of fractional analysis, quadrature theory, and numerical validation provides a robust methodology for refining and analyzing high-order integration rules.<\/jats:p>","DOI":"10.3390\/axioms14110823","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:41:47Z","timestamp":1762515707000},"page":"823","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fractional Error Bounds for Lobatto Quadrature: A Convexity Approach via Riemann\u2013Liouville Integrals"],"prefix":"10.3390","volume":"14","author":[{"given":"Li","family":"Liao","sequence":"first","affiliation":[{"name":"School of Mathematical and Computer Science, Yichun University, Yichun 336000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2943-2678","authenticated-orcid":false,"given":"Abdelghani","family":"Lakhdari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Umuttepe Campus, Kocaeli 41001, T\u00fcrkiye"},{"name":"National Higher School of Technology and Engineering, Laboratory of Industrial Systems Technologies, Annaba 23005, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1019-9485","authenticated-orcid":false,"given":"Muhammad Uzair","family":"Awan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3928-3664","authenticated-orcid":false,"given":"Hongyan","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Suqian University, Suqian 223800, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0156-7864","authenticated-orcid":false,"given":"Badreddine","family":"Meftah","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u201cACED\u201d, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/s00466-025-02619-z","article-title":"Cut spectral BFS plate element with Lobatto basis for wave propagation analysis","volume":"76","author":"Ambati","year":"2025","journal-title":"Comput. Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"14205","DOI":"10.1007\/s13369-024-08808-x","article-title":"A novel approach to linear and nonlinear time-history analysis of structures: Gauss\u2013Lobatto\u2013Hermite 4-point (GLH-4P) method","volume":"49","author":"Atasoy","year":"2024","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Shan, Y., and Liu, W. (2025). Space-time Legendre-Gauss-Lobatto collocation method for the two-dimensional Schr\u00f6dinger equation. Numer. 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