Inverse Neumann Boundary Flux Identification in Two-Dimensional Heat Transfer via an Explicit Integral Transform Approach

Inverse Neumann Boundary Flux Identification in Two-Dimensional Heat Transfer via an Explicit Integral Transform Approach | 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 Inverse Neumann Boundary Flux Identification in Two-Dimensional Heat Transfer via an Explicit Integral Transform Approach André J. P. Oliveira, Mousa J. Huntul, Luiz A. S. Abreu, Diego C. Knupp This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8330655/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 29 Apr, 2026 Read the published version in Boundary Value Problems → Version 1 posted 7 You are reading this latest preprint version Abstract This work addresses the inverse identification of a Neumann-type boundary heat flux, simultaneously dependent on the spatial coordinate and time, in a two-dimensional transient conduction problem in a rectangular domain. The direct problem is solved by means of the Classical Integral Transform Technique, which provides an explicit modal representation for the temperature field in terms of normalized eigenfunctions associated with the Neumann eigenvalue problem. This formulation allows , in the transformed space, the separation of the contributions of the initial condition and the boundary flux, leading to an analytical expression for the transformed temperatures as a function of the time-dependent modal coefficients of the unknown flux. In the inverse problem, the boundary flux is approximated by a truncated expansion in the eigenfunctions in the transverse direction, and the temporal coefficients of this expansion are explicitly recovered from transformed temperature data, through a centered finite-difference discretization of the transformed modal ordinary differential equations. A functional sensitivity analysis shows that, for each transverse mode, the choice of the first longitudinal mode maximizes the sensitivity of the transformed temperatures with respect to the flux coefficients, providing a simple and robust criterion for selecting the most informative mode. Additionally, the modal truncation level in the transverse direction is automatically determined from an adapted version of Morozov’s discrepancy principle, formulated in terms of the mean squared error between reconstructed temperatures and noisy data. The proposed methodology is numerically assessed with synthetic data, considering three flux profiles and different noise levels. The results show that the explicit formulation is able to reconstruct Neumann fluxes with good accuracy, while remaining simple to implement and computationally inexpensive. Neumann boundary flux Inverse problem Integral transforms Explicit scheme Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 29 Apr, 2026 Read the published version in Boundary Value Problems → Version 1 posted Editorial decision: Revision requested 07 Apr, 2026 Reviews received at journal 24 Mar, 2026 Reviewers agreed at journal 05 Mar, 2026 Reviewers invited by journal 29 Dec, 2025 Editor assigned by journal 17 Dec, 2025 Submission checks completed at journal 16 Dec, 2025 First submitted to journal 10 Dec, 2025 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-8330655","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":566747744,"identity":"5c588199-c775-44c6-88f2-2e23c183edf1","order_by":0,"name":"André J. P. Oliveira","email":"","orcid":"","institution":"Federal Institute of Northern Minas Gerais, IFNMG","correspondingAuthor":false,"prefix":"","firstName":"André","middleName":"J. 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The direct problem is solved by means of the Classical Integral Transform Technique, which provides an explicit modal representation for the temperature field in terms of normalized eigenfunctions associated with the Neumann eigenvalue problem. This formulation allows , in the transformed space, the separation of the contributions of the initial condition and the boundary flux, leading to an analytical expression for the transformed temperatures as a function of the time-dependent modal coefficients of the unknown flux. In the inverse problem, the boundary flux is approximated by a truncated expansion in the eigenfunctions in the transverse direction, and the temporal coefficients of this expansion are explicitly recovered from transformed temperature data, through a centered finite-difference discretization of the transformed modal ordinary differential equations. A functional sensitivity analysis shows that, for each transverse mode, the choice of the first longitudinal mode maximizes the sensitivity of the transformed temperatures with respect to the flux coefficients, providing a simple and robust criterion for selecting the most informative mode. Additionally, the modal truncation level in the transverse direction is automatically determined from an adapted version of Morozov’s discrepancy principle, formulated in terms of the mean squared error between reconstructed temperatures and noisy data. The proposed methodology is numerically assessed with synthetic data, considering three flux profiles and different noise levels. 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