{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:39:10Z","timestamp":1760143150535,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,19]],"date-time":"2024-01-19T00:00:00Z","timestamp":1705622400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f\u2032(r), f\u2033(r), and f\u2034(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22\/3=1.587 with three function evaluations and over the bound 21\/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dz\u02c7unic\u00b4 method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.<\/jats:p>","DOI":"10.3390\/sym16010120","type":"journal-article","created":{"date-parts":[[2024,1,19]],"date-time":"2024-01-19T03:33:41Z","timestamp":1705635221000},"page":"120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6366-3539","authenticated-orcid":false,"given":"Chein-Shan","family":"Liu","sequence":"first","affiliation":[{"name":"Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9846-0694","authenticated-orcid":false,"given":"Chih-Wen","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National United University, Miaoli 360302, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chung-Lun","family":"Kuo","sequence":"additional","affiliation":[{"name":"Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,19]]},"reference":[{"key":"ref_1","first-page":"654","article-title":"Some fourth-order modifications of Newton\u2019s method","volume":"197","author":"Chun","year":"2008","journal-title":"Appl. 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