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Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

Author

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  • Miglena N. Koleva

    (Department of Mathematics, Faculty of Natural Sciences and Education, University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria)

  • Lubin G. Vulkov

    (Department of Applied Mathematics and Statistics, Faculty of Natural Sciences and Education, University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria)

Abstract

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

Suggested Citation

  • Miglena N. Koleva & Lubin G. Vulkov, 2024. "Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations," Mathematics, MDPI, vol. 12(10), pages 1-18, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1499-:d:1392535
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    References listed on IDEAS

    as
    1. Somayeh Pourghanbar & Jalil Manafian & Mojtaba Ranjbar & Aynura Aliyeva & Yusif S. Gasimov, 2020. "An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, November.
    2. Miglena N. Koleva & Lubin G. Vulkov, 2023. "Weak and Classical Solutions to Multispecies Advection–Dispersion Equations in Multilayer Porous Media," Mathematics, MDPI, vol. 11(14), pages 1-18, July.
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