Open Access

ARTICLE

Using a Lie-Group Adaptive Method for the Identification of a Nonhomogeneous Conductivity Function and Unknown Boundary Data

Chein-Shan Liu¹

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan E-mail: liucs@ntu.edu.tw

Computers, Materials & Continua 2011, 21(1), 17-40. https://doi.org/10.3970/cmc.2011.021.017

Abstract

Only the left-boundary data of temperature and heat flux are used to estimate an unknown parameter function α(x) in T_t(x,t) = ∂(α(x)T_x)/∂x + h(x,t), as well as to recover the right-boundary data. When α(x) is given the above problem is a well-known inverse heat conduction problem (IHCP). This paper solves a mixed-type inverse problem as a combination of the IHCP and the problem of parameter identification, without needing to assume a function form of α(x) a priori, and without measuring extra data as those used by other methods. We use the one-step Lie-Group Adaptive Method (LGAM) for the semi-discretizations of heat conduction equation, respectively, in time domain and spatial domain to derive algebraic equations, which are used to solve α(x) through a few iterations. To test the stability of the present LGAM we also add a random noise in the initial data. When α(x) is identified, a sideways approach is employed to recover the unknown boundary data. The convergence speed and accuracy are examined by numerical examples.

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Keywords

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