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

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 p...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-05, Vol.12 (10), p.1499
Main Authors: Koleva, Miglena N., Vulkov, Lubin G.
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary value problems
difference scheme
Dirichlet problem
disjoint domains
Finite difference method
Heat equation
Hilbert space
integral observation
inverse problem
Inverse problems
Mathematical analysis
Mathematical research
Numerical analysis
Partial differential equations
Saul’yev method
Thermodynamics
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12101499