Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms

Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms | 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 Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms Weiguo Rui This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1155271/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach which combined with the separation method of semi-fixed variables and dynamical system method is introduced. As example, a time-fractional reaction-diffusion equation with higher-order terms is studied under the different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems which derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional models are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed and the graphs of them are illustrated. PACS: 02.30.Jr; 02.30.Oz; 02.70.-c; 02.70.Mv; 02.90.+p; 04.20.Jb; 05.10.-a Separation method of semi-fixed variables Dynamical system method Nonlinear time-fractional PDEs Time-fractional reaction-diffusion equations Existence and dynamical property of solution Full Text Supplementary Files Highlights.doc Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Major revisions 13 Jan, 2022 Reviews received at journal 19 Dec, 2021 Reviewers invited by journal 19 Dec, 2021 Editor assigned by journal 11 Dec, 2021 First submitted to journal 08 Dec, 2021 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. 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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-1155271","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":71209515,"identity":"9be01ceb-2c00-47e4-bc4f-e0675e2a8c76","order_by":0,"name":"Weiguo Rui","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYFACxgYQmWDAwHyA4YEBaVrYEkAk8QComMeAIYEYpQbHm5tf81TcyzOXyPn4IaHgcOJ2BuaHj27g03LmYJs1z5niYssZuZslEgwOJ+5sYDM2zsGjxexGYpsxb1tC4oYbuRvAWjYc4GGTxqvl/kOYlpzHP4jTcoOx+TFUCxtxttifSWxjnHMGqOXMMzOLBIN04w2HCfhFsv344w9vKoBajic/vvHhj7XshuPNDx/j0wIEbBJInGYGBmb8ykGA+QMSp46w+lEwCkbBKBhxAAAQuVe0ZiNxPgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-4472-0811","institution":"Chongqing Normal University","correspondingAuthor":true,"prefix":"","firstName":"Weiguo","middleName":"","lastName":"Rui","suffix":""}],"badges":[],"createdAt":"2021-12-09 09:23:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1155271/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1155271/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":16668211,"identity":"0a0962e1-5209-4e8e-a410-c8282f4f0a90","added_by":"auto","created_at":"2021-12-22 00:25:47","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1182633,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptRui.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1155271/v1_covered.pdf"},{"id":16668191,"identity":"405c1acb-5629-47a5-92bf-6a58d52fa184","added_by":"auto","created_at":"2021-12-22 00:25:35","extension":"doc","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":15360,"visible":true,"origin":"","legend":"","description":"","filename":"Highlights.doc","url":"https://assets-eu.researchsquare.com/files/rs-1155271/v1/a16131985f1966dd1ea96da0.doc"}],"financialInterests":"","formattedTitle":"\u003cp\u003eSeparation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms\u003c/p\u003e","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1155271/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"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":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Separation method of semi-fixed variables, Dynamical system method, Nonlinear time-fractional PDEs, Time-fractional reaction-diffusion equations, Existence and dynamical property of solution","lastPublishedDoi":"10.21203/rs.3.rs-1155271/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1155271/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIt is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. 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