Adaptive Spectral Homotopy Analysis with Convergence Control for High-Precision Solution of Bratu-Type Equations

Adaptive Spectral Homotopy Analysis with Convergence Control for High-Precision Solution of Bratu-Type Equations | 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 Adaptive Spectral Homotopy Analysis with Convergence Control for High-Precision Solution of Bratu-Type Equations Ujwal Warbhe This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7345294/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted 7 You are reading this latest preprint version Abstract Purpose : This study addresses the computational challenges in solving Bratutype equations, particularly near critical parameters where traditional methods fail.We develop a robust numerical framework to achieve machine-precision accuracy across the entire solution spectrum. Methods : An adaptive spectral homotopy analysis method (ASHAM) is proposed, integrating three innovations: (1) Chebyshev-Gauss-Lobatto spectral discretization for exponential convergence, (2) Homotopy deformation with Bell polynomial expansion of nonlinear terms, and (3) Adaptive ℏ-optimization via residual minimization using Brent’s method. Results : For Bratu’s boundary value problem, ASHAM achieves 10 −12 maximum absolute error with λ = 2 in 0.15 seconds, outperforming state-of-the-art methods by 3 orders of magnitude. Near criticality (λ c = 3.513830719), it maintains 5.7×10 −10 accuracy where existing techniques diverge. The method also demonstrates 98% computational speedup versus wavelet approaches. Conclusion : ASHAM provides unprecedented accuracy and efficiency for Bratutype equations through its convergence-optimized spectral framework. The methodology establishes a new paradigm for singular nonlinear boundary value problems with broad applications in combustion theory and nanotechnology. Bratu’s problem Spectral homotopy analysis Convergence control Nonlinear differential equations Chebyshev spectral methods Adaptive optimization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 22 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 27 Sep, 2025 Reviews received at journal 26 Sep, 2025 Reviewers agreed at journal 13 Aug, 2025 Reviewers invited by journal 13 Aug, 2025 Editor assigned by journal 13 Aug, 2025 Submission checks completed at journal 13 Aug, 2025 First submitted to journal 11 Aug, 2025 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. 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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-7345294","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":500241364,"identity":"e9c7253c-bd1f-4534-803d-ae939b9f9f22","order_by":0,"name":"Ujwal Warbhe","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYJACZsYGBjk2EAnhEqnFGKilsYEkLYlA5TBrCACDG8nPPhfusEnvY2duf1zAYCfPwM57gICWNOPZM8+k5bYBHdY8gyHZsIGZLwG/ltsJxsy8bYchWngYmBMYmHkMCGhJ/wzU8j+dDaKlnhgtOSBbDiRAtRwmrEXy/pti5pltyYYgh83mMTgOZBDQwnfm+GbmwjY7efn+4w8+81RUy/Pzn8GvBd2dDAxspKgfBaNgFIyCUYAdAADsFDjP1qXmIwAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Information Technology, Nagpur","correspondingAuthor":true,"prefix":"","firstName":"Ujwal","middleName":"","lastName":"Warbhe","suffix":""}],"badges":[],"createdAt":"2025-08-11 10:23:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7345294/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7345294/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02250-2","type":"published","date":"2025-10-22T16:16:40+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":94490063,"identity":"0f853ea1-ec59-4eaa-9763-145147b8b1ac","added_by":"auto","created_at":"2025-10-27 17:07:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":748830,"visible":true,"origin":"","legend":"","description":"","filename":"AdaptiveSpectralHomotopyAnalysiswithConvergenceControlforHighPrecisionSolutionofBratuTypeEquations3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7345294/v1_covered_9cf2ddd5-5162-4972-9be4-e895a7232dbb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Adaptive Spectral Homotopy Analysis with Convergence Control for High-Precision Solution of Bratu-Type Equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Bratu’s problem, Spectral homotopy analysis, Convergence control, Nonlinear differential equations, Chebyshev spectral methods, Adaptive optimization","lastPublishedDoi":"10.21203/rs.3.rs-7345294/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7345294/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003ePurpose\u003c/strong\u003e: This study addresses the computational challenges in solving Bratutype equations, particularly near critical parameters where traditional methods fail.We develop a robust numerical framework to achieve machine-precision accuracy across the entire solution spectrum.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods\u003c/strong\u003e: An adaptive spectral homotopy analysis method (ASHAM) is proposed, integrating three innovations: (1) Chebyshev-Gauss-Lobatto spectral discretization for exponential convergence, (2) Homotopy deformation with Bell polynomial expansion of nonlinear terms, and (3) Adaptive ℏ-optimization via residual minimization using Brent’s method.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults\u003c/strong\u003e: For Bratu’s boundary value problem, ASHAM achieves 10\u003csup\u003e−12\u003c/sup\u003e maximum absolute error with λ = 2 in 0.15 seconds, outperforming state-of-the-art methods by 3 orders of magnitude. Near criticality (λ\u003csub\u003ec\u003c/sub\u003e = 3.513830719), it maintains 5.7×10\u003csup\u003e−10\u003c/sup\u003e accuracy where existing techniques diverge. The method also demonstrates 98% computational speedup versus wavelet approaches.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusion\u003c/strong\u003e: ASHAM provides unprecedented accuracy and efficiency for Bratutype equations through its convergence-optimized spectral framework. 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