Development of radial basis functionmethods for solving the viscous Burgersequation with applications to diffusivecoagulation-fragmentation equations

Development of radial basis functionmethods for solving the viscous Burgersequation with applications to diffusivecoagulation-fragmentation 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 Development of radial basis functionmethods for solving the viscous Burgersequation with applications to diffusivecoagulation-fragmentation equations Farideh Ghoreishi, Changiz Goli Keshavarzi, Rezwan Ghaffari This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8442296/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract This paper develops a novel numerical scheme for solving the two-dimensional generalized Burgers equation, with direct applications to diffusive coagulation-fragmentation models. The method combines a second-order Crank-Nicolson/leapfrog time-stepping scheme with a shifted surface spline radial basis function collocation method (RBFCM) for spatial discretization. This approach delivers exceptional computational efficiency while maintaining conditional stability—a property rigorously established through a novel matrix-based analysis that employs Lagrange polynomial approximations of the radial basis functions. We derive a priori error estimates demonstrating $\mathcal{O}(\tau h^m)$ convergence, revealing that uniform node distributions significantly enhance accuracy compared to scattered configurations. Comprehensive numerical experiments, including challenging coagulation-fragmentation test cases, validate the theoretical framework and demonstrate the method's robustness, accuracy, and practical effectiveness across diverse problem regimes. 2000 Mathematics Subject Classification: 65M12; 65D12; 35L65; 82C22; 65N15. Radial basis functions Collocation method Diffusive Coagulationfragmentation equation. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 08 Feb, 2026 Editor assigned by journal 31 Jan, 2026 Submission checks completed at journal 30 Jan, 2026 First submitted to journal 24 Dec, 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. 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-8442296","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":589107748,"identity":"80ada9f5-a863-4a17-907c-45e5aa75a159","order_by":0,"name":"Farideh Ghoreishi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVRIie3NMQrCMBSA4VcCmaKuBcFeIZJRb+KSEKiLiOLiVApCXXSvk4foBVoe1KUHcHDzAoqLg4Opg4ND2tEh/5DhvXw8AJfrH/Prh4OkBPE7zNsRGmoA2ZrUf+lMfIm1YLIpb7dFNO9S9hDLJ0Jvm3u4tpDhpdSHlOOK0k6mU4ngVxKKykbSmSCM5yoJ9hkyQ+AMUMRN5MUjlVB2/ZCgiQS+IcBJTYiuCW8i3A+1t+NoCBWChVM2rFRsv5JqhOcrUseEXPtsPBoMToh365X8Z8AAPBswV+xrl8vlcpneJbxJsPgNpwMAAAAASUVORK5CYII=","orcid":"","institution":"K.N.Toosi University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Farideh","middleName":"","lastName":"Ghoreishi","suffix":""},{"id":589107749,"identity":"b89b6c8d-c338-4e84-a7c4-271852d9cdf8","order_by":1,"name":"Changiz Goli Keshavarzi","email":"","orcid":"","institution":"K.N.Toosi University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Changiz","middleName":"Goli","lastName":"Keshavarzi","suffix":""},{"id":589107750,"identity":"6807377e-6046-4d0b-946e-008a829fef9e","order_by":2,"name":"Rezwan Ghaffari","email":"","orcid":"","institution":"K.N.Toosi University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Rezwan","middleName":"","lastName":"Ghaffari","suffix":""}],"badges":[],"createdAt":"2025-12-24 11:38:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8442296/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8442296/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102746529,"identity":"22b458b2-82c4-49eb-9164-e1ef20d27612","added_by":"auto","created_at":"2026-02-16 08:58:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3907966,"visible":true,"origin":"","legend":"","description":"","filename":"Burger289404.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8442296/v1_covered_38f86e0e-6375-4079-b707-b92e91d84138.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Development of radial basis functionmethods for solving the viscous Burgersequation with applications to diffusivecoagulation-fragmentation equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"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":"Radial basis functions, Collocation method, Diffusive Coagulationfragmentation equation.","lastPublishedDoi":"10.21203/rs.3.rs-8442296/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8442296/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper develops a novel numerical scheme for solving the two-dimensional generalized Burgers equation, with direct applications to diffusive coagulation-fragmentation models. The method combines a second-order Crank-Nicolson/leapfrog time-stepping scheme with a shifted surface spline radial basis function collocation method (RBFCM) for spatial discretization. This approach delivers exceptional computational efficiency while maintaining conditional stability—a property rigorously established through a novel matrix-based analysis that employs Lagrange polynomial approximations of the radial basis functions.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eWe derive a priori error estimates demonstrating $\\mathcal{O}(\\tau h^m)$ convergence, revealing that uniform node distributions significantly enhance accuracy compared to scattered configurations. 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