Numerical Solution of Coupled Emden-FowlerEquations Using Haar Wavelet Collocation Method

Numerical Solution of Coupled Emden-FowlerEquations Using Haar Wavelet Collocation 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 Numerical Solution of Coupled Emden-FowlerEquations Using Haar Wavelet Collocation Method Arushi ., Pranay Goswami, Saad Althobaiti This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8936364/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 numerical scheme for solving coupled systems of Emden-Fowler equations based on the Haar wavelet collocation method (HWCM). This method converts the system of singular differential equations into a set of algebraic equations by approximating the higher-order derivatives using wavelet techniques. Newton's method is used to solve the resulting nonlinear system and obtain the Haar coefficients. Numerical experiments on several benchmark problems demonstrate the precision and effectiveness of the proposed approach. The results show excellent agreement with exact solutions and compare favorably with existing methods, including artificial neural network (ANN) techniques. The method is extended to coupled systems for the first time, achieving errors of order \(10^{-3}\) to \(10^{-6}\) with a theoretical convergence rate of \(O(2^{-3L/2})\). The method is implemented in Mathematica, and comprehensive error tables and graphical comparisons are provided. HWCM is shown to be a robust, deterministic alternative to data-driven methods for solving coupled singular differential systems in astrophysics and related fields. MSC: Primary 34A34, 34K37, 34K28, 65L60, 65T60. Coupled system of Emden-Fowler equations Numerical method Haar wavelet collocation method Deep neural network techniques. Full Text Additional Declarations No competing interests reported. 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-8936364","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":601775493,"identity":"5b78fc2f-5286-4676-8c25-1f0c81dc0b5a","order_by":0,"name":"Arushi .","email":"","orcid":"","institution":"Sri Venkateswara College","correspondingAuthor":false,"prefix":"","firstName":"Arushi","middleName":"","lastName":".","suffix":""},{"id":601775494,"identity":"7ce8d98f-963d-448d-8727-d91c1d419e70","order_by":1,"name":"Pranay Goswami","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVRIiWNgGAWjYPACGyBOADEkiNaSRrqWwzAtRABz6cPPHvxsOy9vcDyB8cMPBos8glos+9LMDXvbbhtuOPOAWbKHQaKYoBaDMwxmErxttxk33EhgkAb6JbGBsBb2b5J/287ZA7Uw/yZSC4+ZNG/bgUSgFjbibLHs4SmTljmXnDzzzMM2yx4DIrSY87Bvk3xTZmfbdzz58I0fFXVEOAxEMLKByQYolxgtDH+IUDkKRsEoGAUjFwAAwlE6ZDPgqP4AAAAASUVORK5CYII=","orcid":"","institution":"Ambedkar University Delhi","correspondingAuthor":true,"prefix":"","firstName":"Pranay","middleName":"","lastName":"Goswami","suffix":""},{"id":601775495,"identity":"f48d6c68-c59f-4c0c-b1af-0fbc7d708b42","order_by":2,"name":"Saad Althobaiti","email":"","orcid":"","institution":"Taif University","correspondingAuthor":false,"prefix":"","firstName":"Saad","middleName":"","lastName":"Althobaiti","suffix":""}],"badges":[],"createdAt":"2026-02-22 01:53:43","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8936364/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8936364/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108687212,"identity":"2315a967-d101-42d2-953a-c1c0bb181ed9","added_by":"auto","created_at":"2026-05-07 10:12:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1951913,"visible":true,"origin":"","legend":"","description":"","filename":"NumericalSolutionofCoupledEmdenFowlerEquationsUsingHaarWaveletCollocationMethod1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8936364/v1_covered_65dfd2ee-5ba6-472e-885a-3d16ec82dad3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Numerical Solution of Coupled Emden-FowlerEquations Using Haar Wavelet Collocation Method","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"Coupled system of Emden-Fowler equations, Numerical method, Haar wavelet collocation method, Deep neural network techniques.","lastPublishedDoi":"10.21203/rs.3.rs-8936364/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8936364/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents a numerical scheme for solving coupled systems of Emden-Fowler equations based on the Haar wavelet collocation method (HWCM). This method converts the system of singular differential equations into a set of algebraic equations by approximating the higher-order derivatives using wavelet techniques. Newton's method is used to solve the resulting nonlinear system and obtain the Haar coefficients. Numerical experiments on several benchmark problems demonstrate the precision and effectiveness of the proposed approach. The results show excellent agreement with exact solutions and compare favorably with existing methods, including artificial neural network (ANN) techniques. The method is extended to coupled systems for the first time, achieving errors of order \\(10^{-3}\\) to \\(10^{-6}\\) with a theoretical convergence rate of \\(O(2^{-3L/2})\\). The method is implemented in Mathematica, and comprehensive error tables and graphical comparisons are provided. 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