A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation

A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation | 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 robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation Muhammad Nadeem, Zitian Li, Devendra Kumar, Yahya Alsayaad This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3854609/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Feb, 2024 Read the published version in Scientific Reports → Version 1 posted 10 You are reading this latest preprint version Abstract In ocean engineering, the Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. The equation is used to describe how waves, such as sound waves or electromagnetic waves, propagate through the ocean. This paper presents the Elzaki transform residual power series method (ET-RPSM) for the analytical treatment of fractional-order Helmholtz equation. To develop this scheme, we combine the Elzaki transform (ET) with the residual power series method (RPSM). The fractional derivatives are described in the Caputo sense. The ET is capable of handling the fractional order and turning the problems into a recurrence form. We implement the RPSM in such a way that this recurrence relation generates the results in the form of an iterative series. Some numerical applications are considered to demonstrate the efficiency and authenticity of this scheme. The obtained series are determined very quickly and converge to the exact solution only after a few iterations. Graphical plots and absolute error are shown to observe the authenticity of this suggested approach. Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Software Elzaki transform Fractional derivative Helmholtz equation Residual power series method Analytical results Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Feb, 2024 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 05 Feb, 2024 Reviews received at journal 04 Feb, 2024 Reviews received at journal 03 Feb, 2024 Reviewers agreed at journal 25 Jan, 2024 Reviewers agreed at journal 25 Jan, 2024 Reviewers invited by journal 25 Jan, 2024 Editor assigned by journal 25 Jan, 2024 Editor invited by journal 18 Jan, 2024 Submission checks completed at journal 18 Jan, 2024 First submitted to journal 11 Jan, 2024 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-3854609","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":268354113,"identity":"8b246038-0701-429e-b2db-e5a2c53ec26d","order_by":0,"name":"Muhammad Nadeem","email":"","orcid":"","institution":"Qujing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"","lastName":"Nadeem","suffix":""},{"id":268354114,"identity":"1d4f3135-bfbd-4dc2-a2b0-5708b5bd326a","order_by":1,"name":"Zitian Li","email":"","orcid":"","institution":"Qujing Normal University","correspondingAuthor":false,"prefix":"","firstName":"Zitian","middleName":"","lastName":"Li","suffix":""},{"id":268354115,"identity":"01a20461-43d1-4d55-bae7-b9249326984c","order_by":2,"name":"Devendra Kumar","email":"","orcid":"","institution":"University of Rajasthan","correspondingAuthor":false,"prefix":"","firstName":"Devendra","middleName":"","lastName":"Kumar","suffix":""},{"id":268354116,"identity":"281bc1d8-7711-4fef-a46b-df1418b264d6","order_by":3,"name":"Yahya Alsayaad","email":"data:image/png;base64,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","orcid":"","institution":"Hodeidah University","correspondingAuthor":true,"prefix":"","firstName":"Yahya","middleName":"","lastName":"Alsayaad","suffix":""}],"badges":[],"createdAt":"2024-01-11 19:59:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3854609/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3854609/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-024-54870-8","type":"published","date":"2024-02-20T15:01:15+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":51648314,"identity":"277d5082-ee7d-41f4-b62a-707e5ed3fe78","added_by":"auto","created_at":"2024-02-26 15:12:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1981393,"visible":true,"origin":"","legend":"","description":"","filename":"LatexFile.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3854609/v1_covered_742ab1f4-8be8-4bd0-b165-6c56ac7e981e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation","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":"Elzaki transform, Fractional derivative, Helmholtz equation, Residual power series method, Analytical results","lastPublishedDoi":"10.21203/rs.3.rs-3854609/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3854609/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In ocean engineering, the Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. 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