{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T18:42:03Z","timestamp":1778611323355,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T00:00:00Z","timestamp":1728000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council","award":["112-2221-E-239-022"],"award-info":[{"award-number":["112-2221-E-239-022"]}]},{"name":"National Science and Technology Council","award":["3322M224"],"award-info":[{"award-number":["3322M224"]}]},{"name":"National Science and Technology Council","award":["330"],"award-info":[{"award-number":["330"]}]},{"name":"National United University","award":["112-2221-E-239-022"],"award-info":[{"award-number":["112-2221-E-239-022"]}]},{"name":"National United University","award":["3322M224"],"award-info":[{"award-number":["3322M224"]}]},{"name":"National United University","award":["330"],"award-info":[{"award-number":["330"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Fractional calculus extends the conventional concepts of derivatives and integrals to non-integer orders, providing a robust mathematical framework for modeling complex systems characterized by memory and hereditary properties. This study enhances the convergence rate of the Caputo-based Newton\u2019s solver for solving one-dimensional nonlinear equations. By modifying the order to 1+\u03b7, we provide a thorough analysis of the convergence order and present numerical simulations that demonstrate the improved efficiency of the proposed modified fractional Newton\u2019s solver. The numerical simulations indicate significant advancements over traditional and existing fractional Newton-type approaches.<\/jats:p>","DOI":"10.3390\/axioms13100689","type":"journal-article","created":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T04:49:02Z","timestamp":1728017342000},"page":"689","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Modified Fractional Newton\u2019s Solver"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9846-0694","authenticated-orcid":false,"given":"Chih-Wen","family":"Chang","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, National United University, Miaoli 360302, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7225-2309","authenticated-orcid":false,"given":"Sania","family":"Qureshi","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon"},{"name":"Department of Mathematics, Near East University, 99138 Mersin, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematics Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-4893-2631","authenticated-orcid":false,"given":"Khair Muhammad","family":"Saraz","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering & Technology, Jamshoro 76062, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6175-1455","authenticated-orcid":false,"given":"Evren","family":"Hincal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Near East University, 99138 Mersin, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,4]]},"reference":[{"key":"ref_1","first-page":"400","article-title":"Comparative study of methods of various orders for finding simple roots of nonlinear equations","volume":"9","author":"Chun","year":"2019","journal-title":"J. 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