{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T20:48:15Z","timestamp":1772830095537,"version":"3.50.1"},"reference-count":67,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T00:00:00Z","timestamp":1727395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Free University of Bozen-Bolzano","award":["IN200Z"],"award-info":[{"award-number":["IN200Z"]}]},{"name":"Free University of Bozen-Bolzano","award":["I53C22002100003"],"award-info":[{"award-number":["I53C22002100003"]}]},{"name":"Provincia Autonoma di Bolzano\/Alto Adige\u2014Ripartizione Innovazione, Ricerca, Universit\u00e0 e Musei","award":["IN200Z"],"award-info":[{"award-number":["IN200Z"]}]},{"name":"Provincia Autonoma di Bolzano\/Alto Adige\u2014Ripartizione Innovazione, Ricerca, Universit\u00e0 e Musei","award":["I53C22002100003"],"award-info":[{"award-number":["I53C22002100003"]}]},{"name":"INdAM-GNCS","award":["IN200Z"],"award-info":[{"award-number":["IN200Z"]}]},{"name":"INdAM-GNCS","award":["I53C22002100003"],"award-info":[{"award-number":["I53C22002100003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of 2\u03c8+2. The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability.<\/jats:p>","DOI":"10.3390\/axioms13100671","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T07:49:24Z","timestamp":1727423364000},"page":"671","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["An Efficient and Stable Caputo-Type Inverse Fractional Parallel Scheme for Solving Nonlinear Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2980-5801","authenticated-orcid":false,"given":"Mudassir","family":"Shams","sequence":"first","affiliation":[{"name":"Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100 Bozen-Bolzano, Italy"},{"name":"Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0516-0033","authenticated-orcid":false,"given":"Bruno","family":"Carpentieri","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100 Bozen-Bolzano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2563","DOI":"10.1016\/j.nonrwa.2009.09.001","article-title":"Dynamics of a new Lorenz-like chaotic system","volume":"11","author":"Liu","year":"2010","journal-title":"Nonlinear Anal. 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