A Hybrid Adomian–Runge–Kutta Method for Solving Nonlinear Reaction–Diffusion Equations

A Hybrid Adomian–Runge–Kutta Method for Solving Nonlinear Reaction–Diffusion 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 A Hybrid Adomian–Runge–Kutta Method for Solving Nonlinear Reaction–Diffusion Equations Sathish Marakonda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7080046/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 paper presents a novel hybrid numerical framework that combines the Adomian Decomposition Method (ADM) with the classical fourth-order Runge–Kutta (RK4) scheme for solving nonlinear reaction–diffusion equations. The hybrid approach analytically handles nonlinear terms via Adomian polynomials and employs RK4 for stable and accurate time integration. This strategy addresses the limitations of both ADM and RK4 when used independently. The proposed method is applied to a range of benchmark problems, including the exponential-type, Ginzburg–Landau, Gray–Scott, and Zeldovich reaction–diffusion equations. In each case, the hybrid ADM–RK4 method demonstrates enhanced accuracy and stability compared to pure RK4. Numerical experiments confirm low absolute errors (as small as 10−4 to 10−3) and strong agreement between hybrid and standard RK4 solutions. The results are validated through detailed tabulated data and surface plots, showcasing the hybrid method’s effectiveness in capturing the dynamics of nonlinear reaction–diffusion systems over short time intervals 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-7080046","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":482719311,"identity":"99cb5832-7946-413f-a145-ff20b0339afa","order_by":0,"name":"Sathish Marakonda","email":"data:image/png;base64,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","orcid":"","institution":"Anurag University","correspondingAuthor":true,"prefix":"","firstName":"Sathish","middleName":"","lastName":"Marakonda","suffix":""}],"badges":[],"createdAt":"2025-07-09 05:38:08","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7080046/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7080046/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86376498,"identity":"c8453e2e-d315-475f-b787-93ae034ca877","added_by":"auto","created_at":"2025-07-10 03:07:22","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":494828,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7080046/v1_covered_c94e6e2d-b375-4f80-9586-6f0e32a97a46.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Hybrid Adomian–Runge–Kutta Method for Solving Nonlinear Reaction–Diffusion Equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"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":"","lastPublishedDoi":"10.21203/rs.3.rs-7080046/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7080046/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper presents a novel hybrid numerical framework that combines the Adomian Decomposition Method (ADM) with the classical fourth-order Runge–Kutta (RK4) scheme for solving nonlinear reaction–diffusion equations. 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