Optimizing Nonlinear Problem Solving: Novel Third to Eighth-Order Iterative Schemes via Homotopy Perturbation Method

Optimizing Nonlinear Problem Solving: Novel Third to Eighth-Order Iterative Schemes via Homotopy Perturbation Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Optimizing Nonlinear Problem Solving: Novel Third to Eighth-Order Iterative Schemes via Homotopy Perturbation Method Muhammad Raza, Mashood Ul haq, Najma Abdul Rehman This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7439516/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper presents a comparison of the newly proposed third, sixth, seventh, and eighth-order iterative schemes (NM3, NM6, NM7, NM8) with existing methods for solving nonlinear problems. The methods are evaluated against several established schemes, including those by Amat et al. (AM), Kim (KI), Maheshwari (MA), and others. Real life problems are solved using these iterative schemes. The analysis includes a detailed numerical comparison, convergence behavior, and the effectiveness of these methods. Results indicate that proposed methods demonstrate superior performance in terms of convergence speed and accuracy. The newly proposed methods also generate larger basins of attraction, indicating better stability and reliability in solving nonlinear equations. Math classification 2020: 65H20, 90C39 Applied Mathematics Computational Mathematics Nonlinear equations homotopy perturbation method order of convergence dynamical behavior basin of attraction stability Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7439516","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":504500958,"identity":"f3ab03a4-e109-412a-852b-1bde39d40560","order_by":0,"name":"Muhammad Raza","email":"","orcid":"https://orcid.org/0000-0002-1026-0856","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"","lastName":"Raza","suffix":""},{"id":504500959,"identity":"ec532aec-ff69-4e0a-b937-472f370c9a39","order_by":1,"name":"Mashood Ul haq","email":"","orcid":"","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":false,"prefix":"","firstName":"Mashood","middleName":"Ul","lastName":"haq","suffix":""},{"id":504500960,"identity":"26ee0910-6eb1-4c23-8cc8-3e8cb534e94c","order_by":2,"name":"Najma Abdul Rehman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYHCCBDDJLwGmJGSI1yI5g4GxAaiFh3i7DG6AtTAQ1mLOfuDphh8MdvnGt5uPP7pRY8HDwH746AZ8Wix7EtJu9jAkW267cyyxOecY0GE8aWk38LrnQELaDR4GZgOzGzmGzTlsQC0SPGb4tZx/kHbzD0O9gfEMkJZ/xGi5kZB2m4fhsIGBBFBLbhtRWh6k3ZZhOG4gcSMtcXZunwQPG0G/nM9Ju/mGodqAf0bygc853+rk+NkPH8OrBRgRCQyM/5D4bPiVgwD7AcJqRsEoGAWjYGQDAJrER7z3AihlAAAAAElFTkSuQmCC","orcid":"","institution":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus","correspondingAuthor":true,"prefix":"","firstName":"Najma","middleName":"Abdul","lastName":"Rehman","suffix":""}],"badges":[],"createdAt":"2025-08-23 07:46:06","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7439516/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7439516/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89870395,"identity":"f0b0d02d-ba9a-4018-abf4-6797a111b930","added_by":"auto","created_at":"2025-08-26 02:20:11","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1601102,"visible":true,"origin":"","legend":"","description":"","filename":"paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7439516/v1_covered_5aa498b2-9f71-48d5-aa1b-add14600633a.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eOptimizing Nonlinear Problem Solving: Novel Third to Eighth-Order Iterative Schemes via Homotopy Perturbation Method\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Nonlinear equations, homotopy perturbation method, order of convergence, dynamical behavior, basin of attraction, stability","lastPublishedDoi":"10.21203/rs.3.rs-7439516/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7439516/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents a comparison of the newly proposed third, sixth, seventh, and eighth-order iterative schemes (NM3, NM6, NM7, NM8) with existing methods for solving nonlinear problems. 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