Analytical Approaches to Nonlinear Systems: The Itoh-Kupershmidt Equation via HPM and VIM

Analytical Approaches to Nonlinear Systems: The Itoh-Kupershmidt Equation via HPM and VIM | 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 Analytical Approaches to Nonlinear Systems: The Itoh-Kupershmidt Equation via HPM and VIM Muhammad Rashid, Muhammad Rashid Iqbal, Muqaddas Naeem, Dr Muhammad Zafar, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6702800/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 Modern approaches like the HPM and the VIM are investigated by researchers because non-linear ODEs and PDEs frequently elude simple analytical solutions. We investigate a well-known non-linear PDE, the Itoh-Kupershmidt equation, which has numerous applications in mathematical physics. Although HPM is not necessarily compact, it is well known for its ability to provide approximations of solutions by series expansion. It is still a helpful tool, though, particularly for non-linear ODE treatments and semi-analytical PDE. The Itoh-Kupershmidt equation has both exact and approximate solutions, which may or may not be in non-compact forms. Our objective is to solve the equation using HPM. Simultaneously, we examine the VIM, a widely used method for reliably and accurately solving a wide range of nonlinear PDEs. Applying VIM to the Itoh-Kupershmidt problem is expected to yield more convenient solutions than HPM. There will be a comparison study to assess how well the HPM and VIM techniques solve the Itoh-Kupershmidt equation. Predictably, both methods should yield promising results demonstrating their effectiveness on a wide range of non-linear PDEs. Additionally, numerical simulations will be performed to show the benefits and drawbacks of each method. Lastly, an accuracy assessment will be performed by contrasting the approaches of the two methods to exact solutions, with particular focus on how well the solutions fulfill the specified boundary criteria. We'll look closely at the underlying causes of any variation or errors. In summary, this work aims to show how effective VIM and HPM are at solving the Itoh-Kupershmidt equation, stressing their applicability to a larger class of non-linear PDEs and providing a critical evaluation of their performance relative to exact solutions and numerical benchmarks. Partial differential equation Tricomi equation Nonlinear partial differential equation BKDVE Homotopy perturbation method Homotopy Analysis method Variational iteration method Full Text Additional Declarations The authors declare potential competing interests as follows: Declaration of Interests I have no competing financial or non-financial interests to declare. My purpose is to provide information and assist with inquiries to the best of my abilities, without any personal or organizational biases. 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. 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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-6702800","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":459025068,"identity":"e1c28311-c989-4074-955a-156dc317ed6a","order_by":0,"name":"Muhammad Rashid","email":"","orcid":"","institution":"Department of Mathematics University of Layyah ,Layyah ,Punjab ,Pakistan.","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"","lastName":"Rashid","suffix":""},{"id":459025069,"identity":"a9354624-3ed8-4b26-ac9f-9c6a99279275","order_by":1,"name":"Muhammad Rashid Iqbal","email":"data:image/png;base64,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","orcid":"","institution":"Institute of southern Punjab Multan","correspondingAuthor":true,"prefix":"","firstName":"Muhammad","middleName":"Rashid","lastName":"Iqbal","suffix":""},{"id":459025070,"identity":"08804f32-8e25-4e7e-b627-adeb05ca0cf0","order_by":2,"name":"Muqaddas Naeem","email":"","orcid":"","institution":"University of agriculture Faisalabad","correspondingAuthor":false,"prefix":"","firstName":"Muqaddas","middleName":"","lastName":"Naeem","suffix":""},{"id":459025071,"identity":"22450cff-1a27-45bf-9ed6-2535fda10df9","order_by":3,"name":"Dr Muhammad Zafar","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"Dr","firstName":"Muhammad","middleName":"","lastName":"Zafar","suffix":""},{"id":459025072,"identity":"8bfc6eea-cb1a-41a1-a3f3-cbf41c969012","order_by":4,"name":"Sania sarfaraz","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Sania","middleName":"","lastName":"sarfaraz","suffix":""}],"badges":[],"createdAt":"2025-05-20 02:17:45","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":true,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-6702800/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6702800/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[],"financialInterests":"The authors declare potential competing interests as follows: Declaration of Interests\nI have no competing financial or non-financial interests to declare. 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