{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:17:58Z","timestamp":1775081878630,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,23]],"date-time":"2015-06-23T00:00:00Z","timestamp":1435017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.<\/jats:p>","DOI":"10.3390\/e17064439","type":"journal-article","created":{"date-parts":[[2015,6,23]],"date-time":"2015-06-23T10:19:06Z","timestamp":1435054746000},"page":"4439-4453","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":249,"title":["Analysis of the Keller\u2013Segel Model with a Fractional Derivative without Singular Kernel"],"prefix":"10.3390","volume":"17","author":[{"given":"Abdon","family":"Atangana","sequence":"first","affiliation":[{"name":"Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences,  University of the Free State, 9300 Bloemfontein, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Badr","family":"Alkahtani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Colleges of Sciences, King Saud University, P.O. 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