Inverse time-dependent source problem for the heat equation with a nonlocal Wentzell-Neumann boundary condition

Fermín S. V. Bazán; Luciano Bedin; Mansur I. Ismailov; Leonardo S. Borges; Fermín S. V. Bazán; Luciano Bedin; Mansur I. Ismailov; Leonardo S. Borges

doi:10.3934/nhm.2023076

Networks and Heterogeneous Media

2023, Volume 18, Issue 4: 1747-1771. doi: 10.3934/nhm.2023076

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Research article

Inverse time-dependent source problem for the heat equation with a nonlocal Wentzell-Neumann boundary condition

1.
Department of Mathematics, Federal University of Santa Catarina, Florianópolis/SC 88040-900, Brazil
2.
Department of Mathematics, Gebze Technical University, Gebze-Kocaeli 41400, Turkey

Received: 29 June 2023 Revised: 17 August 2023 Accepted: 08 October 2023 Published: 18 October 2023

In this work, we consider the problem of recovering the heat source term for the heat equation with a nonlocal Wentzell-Neumann boundary condition subject to an integral overdetermination condition. Conditions for the existence and uniqueness of the classical solution of the inverse problem are revisited, and a numerical method for practical source reconstruction is introduced. Unlike all of the source reconstruction methods found in literature, the method introduced in this work computes regularized solutions from a triangular linear system arising from a semi-discretization in the space of the continuous model. Regularization is introduced by applying the generalized singular value decomposition of a proper matrix pair along with truncation. Numerical results illustrate the effectiveness of the method.
- inverse heat transfer,
- Wentzell boundary conditions,
- Morozov's discrepancy principle,
- generalized cross validation,
- minimum product rule
Citation: Fermín S. V. Bazán, Luciano Bedin, Mansur I. Ismailov, Leonardo S. Borges. Inverse time-dependent source problem for the heat equation with a nonlocal Wentzell-Neumann boundary condition[J]. Networks and Heterogeneous Media, 2023, 18(4): 1747-1771. doi: 10.3934/nhm.2023076

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Abstract

In this work, we consider the problem of recovering the heat source term for the heat equation with a nonlocal Wentzell-Neumann boundary condition subject to an integral overdetermination condition. Conditions for the existence and uniqueness of the classical solution of the inverse problem are revisited, and a numerical method for practical source reconstruction is introduced. Unlike all of the source reconstruction methods found in literature, the method introduced in this work computes regularized solutions from a triangular linear system arising from a semi-discretization in the space of the continuous model. Regularization is introduced by applying the generalized singular value decomposition of a proper matrix pair along with truncation. Numerical results illustrate the effectiveness of the method.

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