Numerical solution of forward and inverse problems of heat conduction in multi-layered media

Yu Xu; Youjun Deng; Dong Wei; Yu Xu; Youjun Deng; Dong Wei

doi:10.3934/math.2025280

AIMS Mathematics

2025, Volume 10, Issue 3: 6144-6167. doi: 10.3934/math.2025280

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Numerical solution of forward and inverse problems of heat conduction in multi-layered media

Yu Xu ¹,
Youjun Deng ^{1
,
,},
Dong Wei ²

1.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
2.
China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China

Received: 20 January 2025 Revised: 02 March 2025 Accepted: 05 March 2025 Published: 19 March 2025
MSC : 65M08, 65M32

The primary objective of this paper was to delve into the exploration of numerical methods for solving forward and inverse problems related to heat conduction in one-dimensional multi-layered media. To address the non-differentiability at multilayer medium interfaces that prevents direct discretization, this paper employed the finite volume method to construct finite difference schemes. Compared with traditional difference methods, the proposed method improved accuracy by considering coefficient variations near interfaces. For the ill-posed initial value problem in inverse heat conduction of multilayer media, we transformed the inverse problem into an operator equation using the finite volume method for forward problems. The Landweber iterative regularization method combined with the Morozov discrepancy principle was then applied to obtain iterative sequences. Numerical simulations demonstrate the algorithm's superior accuracy and noise resistance compared with conventional methods through comparative studies and sensitivity analyses.
- heat conduction,
- multi-layered media,
- forward and inverse problems,
- finite difference method,
- iterative regularization
Citation: Yu Xu, Youjun Deng, Dong Wei. Numerical solution of forward and inverse problems of heat conduction in multi-layered media[J]. AIMS Mathematics, 2025, 10(3): 6144-6167. doi: 10.3934/math.2025280

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Abstract

The primary objective of this paper was to delve into the exploration of numerical methods for solving forward and inverse problems related to heat conduction in one-dimensional multi-layered media. To address the non-differentiability at multilayer medium interfaces that prevents direct discretization, this paper employed the finite volume method to construct finite difference schemes. Compared with traditional difference methods, the proposed method improved accuracy by considering coefficient variations near interfaces. For the ill-posed initial value problem in inverse heat conduction of multilayer media, we transformed the inverse problem into an operator equation using the finite volume method for forward problems. The Landweber iterative regularization method combined with the Morozov discrepancy principle was then applied to obtain iterative sequences. Numerical simulations demonstrate the algorithm's superior accuracy and noise resistance compared with conventional methods through comparative studies and sensitivity analyses.

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