Numerical study of arbitrary order Foam-Drainage equation with cubic B-spline hybrid method

Numerical study of arbitrary order Foam-Drainage equation with cubic B-spline hybrid 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 study of arbitrary order Foam-Drainage equation with cubic B-spline hybrid method Alina Yousafzai, Sirajul Haq, Abdul Ghafoor, Kamal Shah, Thabet Abdeljawad This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3867027/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 work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2 , and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulationsthat the proposed method gives quite accurate solutions. Non-linear Models CBS Finite difference approximation Stability analysis Full Text Additional Declarations No competing interests reported. Supplementary Files AE32epsconvertedto.pdf AE32.eps Bibtex.bib E1.eps E2.eps E3.eps E1epsconvertedto.pdf E2epsconvertedto.pdf E3epsconvertedto.pdf EVA2epsconvertedto.pdf EVA3epsconvertedto.pdf EVA11epsconvertedto.pdf EVA12epsconvertedto.pdf EVA3.eps EVA11.eps EVA12.eps SA1epsconvertedto.pdf SA2epsconvertedto.pdf SA1.eps SA2.eps SA31.eps SA32.eps SA33.eps SA331.eps SA332.eps SA34.eps SA31epsconvertedto.pdf SA32epsconvertedto.pdf SA33epsconvertedto.pdf SA331epsconvertedto.pdf SA332epsconvertedto.pdf SA34epsconvertedto.pdf SE1.eps SE2.eps SE331.eps SE1epsconvertedto.pdf SE2epsconvertedto.pdf SE331epsconvertedto.pdf SE332epsconvertedto.pdf SE332.eps SubBVPs.aux SubBVPs.bbl SubBVPs.log SubBVPs.synctex SubBVPs.tex SubBVPsd1.pdf 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-3867027","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":267329710,"identity":"09295366-446f-40fb-a688-d537ee832c89","order_by":0,"name":"Alina Yousafzai","email":"","orcid":"","institution":"GIK Institute","correspondingAuthor":false,"prefix":"","firstName":"Alina","middleName":"","lastName":"Yousafzai","suffix":""},{"id":267329711,"identity":"8a5f994b-b029-4774-aecd-fc1de21e6fcd","order_by":1,"name":"Sirajul Haq","email":"","orcid":"","institution":"GIK 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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":"Non-linear Models, CBS, Finite difference approximation, Stability analysis","lastPublishedDoi":"10.21203/rs.3.rs-3867027/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3867027/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e∞\u003c/sub\u003e norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulationsthat the proposed method gives quite accurate solutions.\u003c/p\u003e","manuscriptTitle":"Numerical study of arbitrary order Foam-Drainage equation with cubic B-spline hybrid method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-17 06:52:33","doi":"10.21203/rs.3.rs-3867027/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"e3a6ec1d-c4db-4fb6-8934-72f53d25e93f","owner":[],"postedDate":"January 17th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-01-21T02:44:12+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-17 06:52:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3867027","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3867027","identity":"rs-3867027","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}