Natural Transform Method with Modified ADM for Nonlinear Time Fractional PDEs with Proportional Delay

Natural Transform Method with Modified ADM for Nonlinear Time Fractional PDEs with Proportional Delay | 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 Natural Transform Method with Modified ADM for Nonlinear Time Fractional PDEs with Proportional Delay Alemu Senbeta Bekela, Mesfin Mekuria Woldaregay This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7741477/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 practical approach for solving nonlinear partial differential equations with both time fractional and proportional delay. These equations appear in many real-world situations, such as viscoelasticity, earthquake, population dynamics, volcanic eruption, and control theory. These problems are challenging, and the fractional operator and the nature of the delay add another layer of difficulty. Because of this, there is a need for efficient numerical methods. The study uses the natural transform along with a Modified Adomian Decomposition Method. The Caputo fractional derivative helps manage the memory effects present in fractional systems. We effectively handle the nonlinear parts using modified Adomian polynomials, and examine our method’s convergence and stability in the Banach sense. To show that our method works well, we test it on carefully chosen benchmark problems involving nonlinear fractional dynamics with proportional delay. These examples demonstrate our method’s capability to manage the challenges of nonlinearity, fractional order, and delay terms. The analysis of absolute, relative errors and and statistical performance measure confirms the accuracy and reliability of the technique, even with few iterations. We also discuss the method’s convergence behavior and how it compares to other numerical methods in terms of efficiency. The results demonstrated that the suggested approach provides accurate results with a limited number of terms and performs better than the other numerical techniques in the literature. The novelty of this work lies in integrating the natural transform with a modified decomposition method designed for fractional-delay systems. We also discuss the limitations and possible extensions of the method, offering insights for future research directions. Time fractional partial differential equation Caputo fractional derivative natural transform proportional delay modified Adomian decomposition method 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-7741477","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":528954247,"identity":"4b67f7c0-96c8-4347-af15-5c524c61de6c","order_by":0,"name":"Alemu Senbeta Bekela","email":"","orcid":"","institution":"Adama Science and Technology University","correspondingAuthor":false,"prefix":"","firstName":"Alemu","middleName":"Senbeta","lastName":"Bekela","suffix":""},{"id":528954248,"identity":"965273ab-458e-4ed1-a153-f862cf368aa8","order_by":1,"name":"Mesfin Mekuria Woldaregay","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYFACxgcMDAcYePjZG4AcAwtitDAbgLTISfYcAGmRIF6LscGNBBCPCC26/YfZHt04cy+x4ebzqxt+FEgw8Ld3J+DVYnYjmd0450ZxYuPsnLKbPUCHSZw5u4GAFv5j0jkfEhKbpXPSbvAAtRhI5BLQcv4wG1hLm+SZtJt/iNJyIBmo5UaCMY8E+7HbxNlyA6TlTIKcBE8O220ZAwkewn4BO+xYAo/98ePPbr75YyPH396LXwsS4DEAk8QqBwH2B6SoHgWjYBSMghEEAOEOShtQYNY4AAAAAElFTkSuQmCC","orcid":"","institution":"Adama Science and Technology University","correspondingAuthor":true,"prefix":"","firstName":"Mesfin","middleName":"Mekuria","lastName":"Woldaregay","suffix":""}],"badges":[],"createdAt":"2025-09-29 11:23:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7741477/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7741477/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93645491,"identity":"48135247-9617-4ba5-9fda-4d623c42d2e2","added_by":"auto","created_at":"2025-10-16 04:09:43","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":562356,"visible":true,"origin":"","legend":"","description":"","filename":"NaturalModified.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7741477/v1/dc9a6a6da11f235e5970ed0e.pdf"},{"id":93645490,"identity":"367bd3b6-7075-48cb-bf5a-e3fb9bcee9ea","added_by":"auto","created_at":"2025-10-16 04:09:42","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4387,"visible":true,"origin":"","legend":"","description":"","filename":"e0c7f10e555142f1bf8c3f65ff9c521b.json","url":"https://assets-eu.researchsquare.com/files/rs-7741477/v1/7091bde26165de04ab1606f2.json"},{"id":95361612,"identity":"69db6811-abc3-4159-92d3-b83fb3046f76","added_by":"auto","created_at":"2025-11-07 07:39:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":425691,"visible":true,"origin":"","legend":"","description":"","filename":"NaturalModified.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7741477/v1_covered_1a0410f3-15d2-4bc2-abb0-3d2e3aad4e29.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Natural Transform Method with Modified ADM for Nonlinear Time Fractional PDEs with Proportional Delay","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":"Time fractional partial differential equation, Caputo fractional derivative, natural transform, proportional delay, modified Adomian decomposition method","lastPublishedDoi":"10.21203/rs.3.rs-7741477/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7741477/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper presents a practical approach for solving nonlinear partial differential equations with both time fractional and proportional delay. 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