A Theoretical and Computational Study of the Fractional Brusselator System with Neural Network Validation

A Theoretical and Computational Study of the Fractional Brusselator System with Neural Network Validation | 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 Article A Theoretical and Computational Study of the Fractional Brusselator System with Neural Network Validation Sadique Ahmad Ahmad, Israr Ahmad Ahmad, Ala Saleh Alluhaidan Alluhaidan, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8533671/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted 14 You are reading this latest preprint version Abstract This work introduces the time-space fractional Brusselator (TSFB) model, a generalisation of the classical reaction-diusion system that describes autocatalytic chemical oscillations and pattern formation. The model consists of two fractional parameters, σ and η, that incorporate memory and non-local spatial eects, respectively. A complete qualitative analysis is presented, including existence, uniqueness, and UlamHyers (UH) stability. A nite dierence scheme is developed for numerical analysis. A neural network (NN) approach is employed to validate accuracy and reliability. Detailed simulations illustrate how the fractional parameters σ and η inuence the solutions behaviour in two and three dimensions. The principal novelty of this work lies in the combined application of fractional operators, a nite dierence method, and neural computing to the TSFB system, oering a unied framework for analysing complex nonlinear fractional dynamical systems. Physical sciences/Mathematics and computing Physical sciences/Physics Fractional Brusselator model Neural network validation Stability Finite difference method Pattern formation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 22 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 16 Feb, 2026 Reviewers agreed at journal 15 Feb, 2026 Reviewers agreed at journal 15 Feb, 2026 Reviews received at journal 14 Feb, 2026 Reviewers agreed at journal 13 Feb, 2026 Reviewers agreed at journal 02 Feb, 2026 Reviews received at journal 25 Jan, 2026 Reviewers agreed at journal 19 Jan, 2026 Reviewers agreed at journal 16 Jan, 2026 Reviewers invited by journal 14 Jan, 2026 Editor assigned by journal 14 Jan, 2026 Editor invited by journal 14 Jan, 2026 Submission checks completed at journal 12 Jan, 2026 First submitted to journal 12 Jan, 2026 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-8533671","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":575826515,"identity":"bb91fd29-4e85-4f12-8a69-d52a558be4a9","order_by":0,"name":"Sadique Ahmad Ahmad","email":"","orcid":"","institution":"EIAS: Data Science and Blockchain Laboratory, College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia","correspondingAuthor":false,"prefix":"","firstName":"Sadique","middleName":"Ahmad","lastName":"Ahmad","suffix":""},{"id":575826518,"identity":"fa3c649f-45d8-417f-89ea-2188b2ca1a86","order_by":1,"name":"Israr Ahmad Ahmad","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7ElEQVRIiWNgGAWjYBACNgh1AIiTDxgAScYGBgZmYrWkJRCnhQGhJQekgwgtfPxnH36uqLkTzd+e86HgB4ON7IYDvIcN8DpMIt1Y8syxZ7kzzrzdYNjDkGa84QBfcgJ+LWwMkg1sh3MbbuRuMGZgOJy44QCP8QG8WviPMf9s+Hc4d/6NnAdALf+J0MKQxibZ2HY4d8ONHAaglgNgLQQclsZm2dh3OHfjmWcGhj0GycYzD/Ml4/W+fP8x5psN3w7nzjue/MzgR4WdbN/x3sMS+LSg2GjAADKemYdYDUC1DyA0CVpGwSgYBaNgRAAADSROfqiwGdMAAAAASUVORK5CYII=","orcid":"","institution":"Department of Mathematics, Government Post Graduate Jahanzeb College, Swat 19130, Khyber Pakhtunkhwa, Pakistan","correspondingAuthor":true,"prefix":"","firstName":"Israr","middleName":"Ahmad","lastName":"Ahmad","suffix":""},{"id":575826519,"identity":"31877a42-e5b1-451a-9654-2d7c959d74e3","order_by":2,"name":"Ala Saleh Alluhaidan Alluhaidan","email":"","orcid":"","institution":"Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia","correspondingAuthor":false,"prefix":"","firstName":"Ala","middleName":"Saleh Alluhaidan","lastName":"Alluhaidan","suffix":""},{"id":575826522,"identity":"a758f2a3-53e5-4dbe-b744-6dcc50577a81","order_by":3,"name":"Mohammed A. ElAffendi ElAffendi","email":"","orcid":"","institution":"EIAS: Data Science and Blockchain Laboratory, College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia","correspondingAuthor":false,"prefix":"","firstName":"Mohammed","middleName":"A. ElAffendi","lastName":"ElAffendi","suffix":""},{"id":575826527,"identity":"0b8b9026-3703-44ce-b374-c9ad85744171","order_by":4,"name":"Manel Hleili Manel","email":"","orcid":"","institution":"University of Tabuk","correspondingAuthor":false,"prefix":"","firstName":"Manel","middleName":"Hleili","lastName":"Manel","suffix":""},{"id":575826532,"identity":"58844d21-c1c9-4c3d-a607-64514ef81d58","order_by":5,"name":"Maryam Salem Alatawi Alatawi","email":"","orcid":"","institution":"University of Tabuk","correspondingAuthor":false,"prefix":"","firstName":"Maryam","middleName":"Salem Alatawi","lastName":"Alatawi","suffix":""}],"badges":[],"createdAt":"2026-01-06 16:53:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8533671/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8533671/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-026-47451-4","type":"published","date":"2026-04-22T15:59:46+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":100538686,"identity":"a33c39b4-a772-4d37-802f-13887410d1ad","added_by":"auto","created_at":"2026-01-19 04:45:59","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6086555,"visible":true,"origin":"","legend":"","description":"","filename":"Paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8533671/v1/e905d5812aaf6776ff75d8a6.pdf"},{"id":100538670,"identity":"8f871ecd-2857-489b-a57a-f8104e967c53","added_by":"auto","created_at":"2026-01-19 04:45:40","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6686,"visible":true,"origin":"","legend":"","description":"","filename":"d2d3cbfdda944ab0975e83e9473bd048.json","url":"https://assets-eu.researchsquare.com/files/rs-8533671/v1/56a2d04d3c40279adb1bf51e.json"},{"id":107929440,"identity":"3ca36745-0655-44f5-a3b2-4eb7e4de9088","added_by":"auto","created_at":"2026-04-27 16:15:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5456623,"visible":true,"origin":"","legend":"","description":"","filename":"Paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8533671/v1_covered_b44b6aff-29c6-40e2-9bd1-6c17e5afec13.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Theoretical and Computational Study of the Fractional Brusselator System with Neural Network Validation","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Fractional Brusselator model, Neural network validation, Stability, Finite difference method, Pattern formation","lastPublishedDoi":"10.21203/rs.3.rs-8533671/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8533671/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This work introduces the time-space fractional Brusselator (TSFB) model, a generalisation of the classical reaction-diusion system that describes autocatalytic chemical oscillations and pattern formation. The model consists of two fractional parameters, σ and η, that incorporate memory and non-local spatial eects, respectively. A complete qualitative analysis is presented, including existence, uniqueness, and UlamHyers (UH) stability. A nite dierence scheme is developed for numerical analysis. A neural network (NN) approach is employed to validate accuracy and reliability. Detailed simulations illustrate how the fractional parameters σ and η inuence the solutions behaviour in two and three dimensions. The principal novelty of this work lies in the combined application of fractional operators, a nite dierence method, and neural computing to the TSFB system, oering a unied framework for analysing complex nonlinear fractional dynamical systems.","manuscriptTitle":"A Theoretical and Computational Study of the Fractional Brusselator System with Neural Network Validation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-19 04:45:34","doi":"10.21203/rs.3.rs-8533671/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-16T08:12:10+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"312027830601195947875166394147136610920","date":"2026-02-16T04:17:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"212038647717792500684584677406891300034","date":"2026-02-16T04:06:44+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-14T14:44:24+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"69546380587364787406642507608790738987","date":"2026-02-14T03:53:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"40143647847523406932802661507614111856","date":"2026-02-03T03:17:41+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-01-25T12:29:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"298922837177443222401484374005260147490","date":"2026-01-19T13:34:04+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"92981635941910500138191622517251924324","date":"2026-01-16T21:04:02+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-14T13:12:48+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-14T12:51:48+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-01-14T12:41:33+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-12T11:32:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-01-12T11:26:26+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"8732f30f-4435-4287-86bd-dd61a145e14e","owner":[],"postedDate":"January 19th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":61272923,"name":"Physical sciences/Mathematics and computing"},{"id":61272924,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-04-27T16:15:32+00:00","versionOfRecord":{"articleIdentity":"rs-8533671","link":"https://doi.org/10.1038/s41598-026-47451-4","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-04-22 15:59:46","publishedOnDateReadable":"April 22nd, 2026"},"versionCreatedAt":"2026-01-19 04:45:34","video":"","vorDoi":"10.1038/s41598-026-47451-4","vorDoiUrl":"https://doi.org/10.1038/s41598-026-47451-4","workflowStages":[]},"version":"v1","identity":"rs-8533671","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8533671","identity":"rs-8533671","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}