Neural Network Method and Finite Difference Scheme for solving a Fractional Partial Differential Equation Arising in Chemistry

Neural Network Method and Finite Difference Scheme for solving a Fractional Partial Differential Equation Arising in Chemistry | 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 Neural Network Method and Finite Difference Scheme for solving a Fractional Partial Differential Equation Arising in Chemistry Vijay Kumar Patel This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9136268/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 In this paper, we have solved a fractional order partial differential equation arising from electromagnetic waves in dielectric media by using an artificial neural network (ANN) method and finite difference method (FDM). The proposed ANN method is based on single layer and the FDM is based on the Hermite formula. The Caputo’s fractional derivatives in time are approximated by trial function in ANN and discretized by a FDM of order O(k (3−α) ) & O(k (3−β) ), 1 < β < α < 2. The stability and the convergence analysis of the proposed methods are given by the standard von Neumann stability analysis under mild conditions for FDM. Also for fractional order partial differential equation, error analysis is investigated for ANN and accuracy of order O k (3−α) + k (3−β) + h 4 is investigated. Stability and convergence of FDM are also discussed. Finally, several numerical experiments with different fractional-order derivatives are provided and compared with the exact solutions to illustrate the accuracy and efficiency of the scheme. A comparative numerical study is also done to demonstrate the efficiency of the both method. Finite difference scheme fractional wave model Caputo fractional derivative Hermite formula stability and convergence analysis 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-9136268","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":627875636,"identity":"f1f159fa-2852-4afc-8a6d-c1c1ae4fe0d1","order_by":0,"name":"Vijay Kumar Patel","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFUlEQVRIie3RsUrEMBjA8a8G2iXaNYdDXiHF5Rz0XiUlECfhxoJwehTSJQ9wgw/hI/TocEseIMMN16WTg3JLQQdTvQOVtrNg/lNIvx8JDYDP9xdDwI+r5Y4z8uPTOMEQ5MyRL1OOkWOOqG78GxmIbaDeY9jSWbRU2W4+XdDVbb1/gWsK0WmvZBWIcwxNovFa2e5izN5ckBJE8oDOeB+Z5MAdqQJN0gMhEhxBHBBmA0S8OTLTtFbzjtCVRG0J94MkRiC7U1JNAgWff8zK0J1SjZLLR9YIbdKcODJ5Mk04NWyTqAESxkbY52x7VRTV+rV9X8S0kMhm2R2NY9NL3IPwE8zK37tuOOyfd0Vl0I4+nc/n8/37PgCOCFR2uWOxkAAAAABJRU5ErkJggg==","orcid":"","institution":"Indian Institute of Information Technology, Surat","correspondingAuthor":true,"prefix":"","firstName":"Vijay","middleName":"Kumar","lastName":"Patel","suffix":""}],"badges":[],"createdAt":"2026-03-16 10:08:49","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9136268/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9136268/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107870601,"identity":"f8891740-5b30-43c0-9275-e2a7e205fa8f","added_by":"auto","created_at":"2026-04-27 07:40:05","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1522373,"visible":true,"origin":"","legend":"","description":"","filename":"FDMforEMW.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9136268/v1_covered_e71e9dea-c646-4660-9191-a665f56163b2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Neural Network Method and Finite Difference Scheme for solving a Fractional Partial Differential Equation Arising in Chemistry","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":"Finite difference scheme, fractional wave model, Caputo fractional derivative, Hermite formula, stability and convergence analysis","lastPublishedDoi":"10.21203/rs.3.rs-9136268/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9136268/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we have solved a fractional order partial differential equation arising from electromagnetic waves in dielectric media by using an artificial neural network (ANN) method and finite difference method (FDM). The proposed ANN method is based on single layer and the FDM is based on the Hermite formula. The Caputo’s fractional derivatives in time are approximated by trial function in ANN and discretized by a FDM of order O(k \u003csup\u003e(3−α)\u003c/sup\u003e) \u0026amp; O(k \u003csup\u003e(3−β)\u003c/sup\u003e), 1 \u0026lt; β \u0026lt; α \u0026lt; 2. The stability and the convergence analysis of the proposed methods are given by the standard von Neumann stability analysis under mild conditions for FDM. Also for fractional order partial differential equation, error analysis is investigated for ANN and accuracy of order O k \u003csup\u003e(3−α)\u003c/sup\u003e + k \u003csup\u003e(3−β)\u003c/sup\u003e + h\u003csup\u003e 4\u003c/sup\u003e is investigated. Stability and convergence of FDM are also discussed. Finally, several numerical experiments with different fractional-order derivatives are provided and compared with the exact solutions to illustrate the accuracy and efficiency of the scheme. 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