New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials

New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials | 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 New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials Jumah Aswad Zarnan, Wafaa Mustafa Hameed This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3865732/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 Linear and nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. They are inspired by problems in diverse fields such as economics, biology, fluid dynamics, physics, differential geometry, engineering, control theory, materials science, and quantum mechanics. This special issue aims to highlight recent developments in methods and applications of linear and nonlinear differential equations. In addition, there are papers analyzing equations that arise in engineering, classical and fluid mechanics, and finance. In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Volterra integral equation. The obtained solutions are novel, and previous literature lacks such derivations. For this aim, we derive a simple and efficient matrix formulation using Bernstein polynomials as trial functions. Numerical examples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the existing methods available in the literature. The Volterra integral equation can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field. It is shown that the numerical results are excellent. The efficiency of the proposed numerical technique is exhibited through graphical illustrations and results drafted in tabular form for specific values of the parameters to validate the numerical investigation. Volterra integral equation Bernstein polynomials method approximate solution 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-3865732","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":267738463,"identity":"d5f8d86d-4b0d-4dfc-92fd-899697a03b02","order_by":0,"name":"Jumah Aswad Zarnan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVRIiWNgGAWjYBACxgYGBmYGBhs5efYGEIN4LWnGhj0HiNQCAkCVhxMbbiQQqYV5Rnbi5wIGZsbGmW8MPxdU2DDwt3cn4HfYjNzN0jMY2JjZpXOMpWecSWOQOHN2AyEtG6R5GHjYGGfnGEjzth1mMJDIJahl828eBgkehptnjH8Tq2Ub0BYDCYYbPGZE2tLzdps1D0OCgWFPWpk1z5k0HoJ+MWzP3Xybh+F//Xz2w0BGhY0cf3svAS0NIKv+gZgcBiCSB69yEJBHMNkfEFQ9CkbBKBgFIxMAAKouQQ9obPFqAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Jumah","middleName":"Aswad","lastName":"Zarnan","suffix":""},{"id":267738464,"identity":"f06b786d-3c63-45d4-97a4-f1d04b8f24bf","order_by":1,"name":"Wafaa Mustafa Hameed","email":"","orcid":"","institution":"Cihan University Sulaimaniya","correspondingAuthor":false,"prefix":"","firstName":"Wafaa","middleName":"Mustafa","lastName":"Hameed","suffix":""}],"badges":[],"createdAt":"2024-01-15 07:44:43","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3865732/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3865732/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":86379147,"identity":"cf986e9d-bad8-4acd-abd6-3c80a3303b75","added_by":"auto","created_at":"2025-07-10 03:46:43","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":403027,"visible":true,"origin":"","legend":"","description":"","filename":"ResearchPaper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3865732/v1_covered_5c004363-7331-465c-b608-f84053b4be20.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"New Solution Method for Solving Volterra Integral Equation Using Bernstein Polynomials","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":"Volterra integral equation, Bernstein polynomials method, approximate solution","lastPublishedDoi":"10.21203/rs.3.rs-3865732/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3865732/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eLinear and nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. 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