{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:47:05Z","timestamp":1760143625239,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T00:00:00Z","timestamp":1708646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence rate. Furthermore, the study explores applications of the methods in both real-life and academic contexts. In particular, we examine the convergence of the methods by applying them to the problems, namely Van der Waals equation of state, Planck\u2019s law of radiation, the Manning equation for isentropic supersonic flow and some academic problems. Numerical results reveal that the proposed derivative-free methods are more efficient and consistent than existing methods.<\/jats:p>","DOI":"10.3390\/axioms13030143","type":"journal-article","created":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T10:47:30Z","timestamp":1708685250000},"page":"143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Optimal Fourth-Order Methods for Multiple Zeros: Design, Convergence Analysis and Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8471-5139","authenticated-orcid":false,"given":"Sunil","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 140413, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4627-2795","authenticated-orcid":false,"given":"Janak Raj","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering Technology, Longowal 148106, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8524-743X","authenticated-orcid":false,"given":"Lorentz","family":"J\u00e4ntschi","sequence":"additional","affiliation":[{"name":"Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania"},{"name":"Institute of Doctoral Studies, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,23]]},"reference":[{"key":"ref_1","unstructured":"Bradie, B. 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