{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T21:07:18Z","timestamp":1765487238350,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:00:00Z","timestamp":1760400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, we present a novel one-parameter family with fourth-order convergence to solve nonlinear systems, along with its convergence analysis. Several numerical experiments including a Bratu problem, a mixed Hammerstein integral equation, and nonlinear optimization problems (namely, the Broyden banded function and the Broyden tridiagonal function) as well as applications of differential equations, are analyzed using the proposed schemes to demonstrate their effectiveness. The results indicate that these methods produce more accurate approximations and exhibit greater efficiency compared to existing approaches.<\/jats:p>","DOI":"10.3390\/computation13100241","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T14:34:15Z","timestamp":1760452455000},"page":"241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Fourth-Order Parametric Iterative Approach for Solving Systems of Nonlinear Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1128-3910","authenticated-orcid":false,"given":"Sonia","family":"Bhalla","sequence":"first","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Gharuan, Mohali 140413, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9145-3715","authenticated-orcid":false,"given":"Gurjeet","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Gharuan, Mohali 140413, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2791-6230","authenticated-orcid":false,"given":"Higinio","family":"Ramos","sequence":"additional","affiliation":[{"name":"Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1505-8945","authenticated-orcid":false,"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics, Saveetha Institute of Medical and Technical Sciences Engineering College, Chennai 602105, Tamil Nadu, India"}]},{"given":"Hashim","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,14]]},"reference":[{"unstructured":"Regmi, S. (2020). Optimized Iterative Methods with Applications in Diverse Disciplines, Nova Science Publishers.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1016\/j.nonrwa.2009.08.003","article-title":"On locating all roots of systems of nonlinear equations inside a bounded domain using global optimization methods","volume":"11","author":"Tsoulos","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press.","key":"ref_3"},{"unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall.","key":"ref_4"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarratt\u2019s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algor."},{"key":"ref_6","first-page":"4548","article-title":"Efficient high-order methods based on the golden ratio for nonlinear systems","volume":"217","author":"Cordero","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"165452","DOI":"10.1155\/2012\/165452","article-title":"On a novel fourth-order algorithm for solving systems of nonlinear equations","volume":"2012","author":"Babajee","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_8","first-page":"133","article-title":"On a modification of the Newton method","volume":"19","author":"Samanskii","year":"1967","journal-title":"Ukr. Math. J."},{"key":"ref_9","first-page":"199","article-title":"Variants of Newton\u2019s method for functions of several variables","volume":"183","author":"Cordero","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_10","first-page":"771","article-title":"Third-order methods from quadrature formulae for solving systems of nonlinear equations","volume":"149","author":"Frontini","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1016\/j.cam.2011.08.008","article-title":"On the computational efficiency index and some iterative methods for solving systems of nonlinear equations","volume":"236","author":"Grau","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.cam.2003.12.041","article-title":"A modified Newton method with cubic convergence: The multivariable case","volume":"169","author":"Homeier","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"ref_13","first-page":"686","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"1678","article-title":"Super cubic iterative methods to solve systems of nonlinear equations","volume":"188","author":"Darvishi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_15","first-page":"630","article-title":"A third-order Newton-type method to solve systems of nonlinear equations","volume":"187","author":"Darvishi","year":"2007","journal-title":"Appl. Math. Comput."},{"unstructured":"Potra, F.A., and Ptak, V. (1984). Nondiscrete Induction and Iterative Processes, Pitman Publishing.","key":"ref_16"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"895","DOI":"10.3390\/a8040895","article-title":"On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations","volume":"8","author":"Babajee","year":"2015","journal-title":"Algorithms"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.cam.2009.04.015","article-title":"Iterative methods of order four and five for systems of nonlinear equations","volume":"231","author":"Cordero","year":"2009","journal-title":"J. Comput. Appl. Math."},{"doi-asserted-by":"crossref","unstructured":"Cordero, A., Leonardo-Sep\u00falveda, M.A., Torregrosa, J.R., and Vassileva, M.P. (2023). Enhancing the convergence order from p to p + 3 in iterative methods for solving nonlinear systems of equations without the use of Jacobian matrices. Mathematics, 11.","key":"ref_19","DOI":"10.20944\/preprints202309.0673.v1"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1007\/s11075-023-01631-9","article-title":"A highly efficient class of optimal fourth-order methods for solving nonlinear systems","volume":"95","author":"Cordero","year":"2024","journal-title":"Numer. Algor."},{"key":"ref_21","first-page":"129231","article-title":"Maximally efficient damped composed Newton-type methods to solve nonlinear systems of equations","volume":"492","author":"Cordero","year":"2025","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s10092-013-0097-1","article-title":"Efficient Jarratt-like methods for solving systems of nonlinear equations","volume":"51","author":"Sharma","year":"2014","journal-title":"Calcolo"},{"unstructured":"Wolfram, S. (2003). The Mathematica Book, Wolfram Media.","key":"ref_23"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"259371","DOI":"10.1155\/2013\/259371","article-title":"An efficient higher-order quasilinearization method for solving nonlinear BVPs","volume":"2013","author":"Alaidarous","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_25","first-page":"295","article-title":"Some problems in the theory of quasi-linear equations","volume":"2","author":"Gelfand","year":"1963","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1515\/IJNSNS.2004.5.1.5","article-title":"Thermo-electro-hydrodynamic model for the electrospinning process","volume":"5","author":"Wan","year":"2004","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1006\/jdeq.2001.4151","article-title":"The Liouville-Bratu-Gelfand problem for radial operators","volume":"184","author":"Jacobsen","year":"2002","journal-title":"J. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1868","DOI":"10.1016\/j.cpc.2010.08.004","article-title":"Non-polynomial spline method for solving Bratu\u2019s problem","volume":"181","author":"Jalilian","year":"2010","journal-title":"Comput. Phys. Commun."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"100095","DOI":"10.1016\/j.rinam.2020.100095","article-title":"Benchmarking results for the Newton\u2013Anderson method","volume":"8","author":"Pollock","year":"2020","journal-title":"Results Appl. Math."},{"unstructured":"Kamenetskii, F., and Al\u2019bertovich, D. (1969). Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press.","key":"ref_30"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"7987","DOI":"10.1002\/mma.9455","article-title":"Using decomposition of the nonlinear operator for solving non-differentiable problems","volume":"48","author":"Villalba","year":"2022","journal-title":"Math. Method Appl. Sci."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/10\/241\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T04:50:45Z","timestamp":1760590245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/10\/241"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,14]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["computation13100241"],"URL":"https:\/\/doi.org\/10.3390\/computation13100241","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,10,14]]}}}