A fundamental sequences method for an inverse boundary value problem for the heat equation in double-connected domains

Ihor Borachok; Roman Chapko

doi:10.3934/ipi.2024043

Article Contents

2025, Volume 19, Issue 3: 523-538. Doi: 10.3934/ipi.2024043

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A fundamental sequences method for an inverse boundary value problem for the heat equation in double-connected domains

Ihor Borachok^1, , and
Roman Chapko^1,

1.
Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Ukraine

^*Corresponding author: Ihor Borachok
^*Corresponding author: Ihor Borachok

Received: July 2024

Revised: November 2024

Early access: November 13, 2024

Published online: November 13, 2024

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Abstract

The application of the fundamental sequences method for reconstructing the inner part of the boundary of a double-connected domain from the overdetermined Cauchy data of the solution of the heat conduction equation on the outer part of the boundary is considered. The nonlinear ill-posed problem is numerically solved by the regularized Newton's method, at each step of which direct problems for the heat equation are solved. Using Rothe's method, each direct problem is reduced to a sequence of elliptic Dirichlet problems for the inhomogeneous modified Helmholtz equation. Which, in turn, is fully discretized by the fundamental sequences method. The results of numerical examples in both two- and three-dimensional domains confirm the accuracy of the proposed method with negligible computational effort.

Keywords:

Mathematics Subject Classification: Primary: 35K05, 65N80, 49M15, 47J06; Secondary: 80M20, 65M30.

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Figure 1. Reconstructed (solid line) and exact (dashed line) boundary curves $ \Gamma_1 $ for exact and $ 5\% $ noisy data for the example 1

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Figure 2. Reconstructed (solid line) and exact (dashed line) boundary curves $ \Gamma_1 $ for exact and $ 5\% $ noisy data for the example 2

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Figure 3. Domains used in examples 3 and 4

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Figure 4. Exact (a) and reconstructed boundary surfaces $ \Gamma_1 $ for exact (b) and $ 5\% $ noisy (c) data for the example 3

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Figure 5. Reconstructed (solid blue line) and exact (dashed blue line) sections of boundary surfaces $ \Gamma_1 $ at $ x_1 = 0 $, for exact and $ 5\% $ noisy data for the example 3

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Figure 6. Exact (a) and reconstructed boundary surfaces $ \Gamma_1 $ for exact (b) and $ 5\% $ noisy (c) data for the example 4

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Figure 7. Reconstructed (solid blue line) and exact (dashed blue line) sections of boundary surfaces $ \Gamma_1 $ at $ x_1 = 0 $, for exact and $ 5\% $ noisy data for the example 4

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