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Keywords = backward heat conduction problem (BHCP)

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17 pages, 3762 KiB  
Article
A Time–Space Numerical Procedure for Solving the Sideways Heat Conduction Problem
by Ching-Chuan Tan, Chao-Feng Shih, Jian-Hung Shen and Yung-Wei Chen
Mathematics 2025, 13(5), 751; https://doi.org/10.3390/math13050751 - 25 Feb 2025
Viewed by 510
Abstract
This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated with measurement noise. These perturbations may lead [...] Read more.
This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated with measurement noise. These perturbations may lead to significant oscillations in the solution. The uniqueness of the solution in this SHCP requires revaluation when boundary conditions (BCs) or initial conditions (ICs) are missing. First, the spatial gradient between two points resolves the missing BCs in the computational domain by a one-step Lie group scheme. Further, the SHCP can be transformed into a backward-in-time heat conduction problem (BHCP). The second-order backward explicit integration can be applied to determine the ICs using the two-point solution at each time step. The performance of the suggested strategy is demonstrated with three numerical examples. The exact solution and the numerical results correspond well, despite the absence of some boundary and initial conditions. The only method of preventing numerical instability in this study is to alter the direction of numerical integration instead of relying on regularization techniques. Therefore, a numerical formula with two integration directions proves to be more accurate and stable compared to existing methods for the SHCP. Full article
(This article belongs to the Special Issue Research on Applied Partial Differential Equations)
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17 pages, 14521 KiB  
Article
A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations
by Yung-Wei Chen, Jian-Hung Shen, Yen-Shen Chang and Chun-Ming Chang
Mathematics 2023, 11(19), 4052; https://doi.org/10.3390/math11194052 - 24 Sep 2023
Viewed by 1280
Abstract
In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome [...] Read more.
In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique. Full article
(This article belongs to the Topic Numerical Methods for Partial Differential Equations)
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