{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T10:32:35Z","timestamp":1778409155204,"version":"3.51.4"},"reference-count":49,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2501"],"award-info":[{"award-number":["IMSIU-DDRSP2501"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We propose a hybrid Caputo\u2013Lagrange Discretization Method (CLDM) for the fractional-order modeling of glucose\u2013insulin dynamics. The model incorporates key physiological mechanisms such as glucose suppression, insulin activation, and delayed feedback with memory effects captured through Caputo derivatives. Analytical results establish positivity, boundedness, existence, uniqueness, and Hyers\u2013Ulam stability. Numerical simulations confirm that the proposed method improves accuracy and efficiency compared with the Residual Power Series Method and the fractional Runge\u2013Kutta method. Sensitivity analysis highlights fractional order \u03b8 as a biomarker for metabolic memory. The findings demonstrate that CLDM offers a robust and computationally efficient framework for biomedical modeling with potential applications in diabetes research and related physiological systems.<\/jats:p>","DOI":"10.3390\/axioms14110800","type":"journal-article","created":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T05:28:43Z","timestamp":1761888523000},"page":"800","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Hybrid Euler\u2013Lagrange Approach for Fractional-Order Modeling of Glucose\u2013Insulin Dynamics"],"prefix":"10.3390","volume":"14","author":[{"given":"Muflih","family":"Alhazmi","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Northern Border University, Arar 73213, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-1385-3208","authenticated-orcid":false,"given":"Safa M.","family":"Mirgani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5790-3222","authenticated-orcid":false,"given":"Sayed","family":"Saber","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Baha University, Alaqiq 65779, Saudi Arabia"},{"name":"Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 2722165, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"American Diabetes Association (2018). 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