{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T01:41:46Z","timestamp":1775180506162,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T00:00:00Z","timestamp":1565049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-095896-B-C22"],"award-info":[{"award-number":["PGC2018-095896-B-C22"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003359","name":"Generalitat Valenciana","doi-asserted-by":"publisher","award":["PROMETEO\/2016\/089"],"award-info":[{"award-number":["PROMETEO\/2016\/089"]}],"id":[{"id":"10.13039\/501100003359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev\u2019s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann\u2013Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of     3 \u03b1    -th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of    \u03b1    close to one and almost any initial estimation.<\/jats:p>","DOI":"10.3390\/sym11081017","type":"journal-article","created":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T03:09:08Z","timestamp":1565147348000},"page":"1017","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["A Variant of Chebyshev\u2019s Method with 3\u03b1th-Order of Convergence by Using Fractional Derivatives"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"first","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ivan","family":"Girona","sequence":"additional","affiliation":[{"name":"Facultat de Matem\u00e0tiques, Universitat de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9893-0761","authenticated-orcid":false,"given":"Juan R.","family":"Torregrosa","sequence":"additional","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2017). Fractional and Multivariable Calculus. Model Building and Optimization Problems, Springer. Springer Optimization and Its Applications 122.","DOI":"10.1007\/978-3-319-59993-9"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BFb0067096","article-title":"A brief history and exposition of the fundamental theory of fractional calculus","volume":"Volume 457","author":"Ross","year":"1975","journal-title":"Fractional Calculus and Its Applications"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Atanackovic, T.M., Pilipovic, S., Stankovic, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles, Wiley.","DOI":"10.1002\/9781118909065"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"134","DOI":"10.3934\/Math.2019.1.134","article-title":"The dynamics of Zika virus with Caputo fractional derivative","volume":"4","author":"Khan","year":"2019","journal-title":"AIMS Math."},{"key":"ref_5","unstructured":"Brambila, F., and Torres, A. (2017). Fractional Newton-Raphson Method. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.aml.2019.06.028","article-title":"A fractional Newton method with 2th-order of convergence and its stability","volume":"98","author":"Cordero","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_7","unstructured":"Traub, J.F. (1964). 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Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1017\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:09:00Z","timestamp":1760188140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1017"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,6]]},"references-count":12,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["sym11081017"],"URL":"https:\/\/doi.org\/10.3390\/sym11081017","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,6]]}}}