Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method |
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Authors: | Chein-Shan Liu Chung-Lun Kuo |
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Affiliation: | 1. Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, China;2. Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan |
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Abstract: | In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise. |
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