August  2012, 6(3): 423-446. doi: 10.3934/ipi.2012.6.423

Probing for inclusions in heat conductive bodies

1. 

Aix-Marseille University, France

2. 

University of Tsukuba, Japan

3. 

Aix-Marseille Université, France

4. 

University of Helsinki, Finland, Finland

Received  September 2010 Revised  May 2012 Published  September 2012

This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1--3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data.
Citation: Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen, Janne Tamminen. Probing for inclusions in heat conductive bodies. Inverse Problems & Imaging, 2012, 6 (3) : 423-446. doi: 10.3934/ipi.2012.6.423
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show all references

References:
[1]

Ann. of Math, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar

[2]

Inverse Problems, 14 (1998), 1429-1453.  Google Scholar

[3]

Inverse Problems, 14 (1998), 853-867. doi: 10.1088/0266-5611/14/4/006.  Google Scholar

[4]

Heat Mass Transfer, 41 (2004), 95-103. Google Scholar

[5]

Comm. in Pure and Appl. Math., 60 (2007), 1415-1442. doi: 10.1002/cpa.20194.  Google Scholar

[6]

Appl. Anal., 86 (2007), 963-1005. doi: 10.1080/00036810701460834.  Google Scholar

[7]

M. Ikehata and M. Kawashita, An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method,, \arXiv{1002.4457}., ().   Google Scholar

[8]

Inverse Problems, 26 (2010), 15pp. doi: 10.1088/0266-5611/26/9/095004.  Google Scholar

[9]

SIAM J. Math. Anal., 28 (1997), 1389-1405. doi: 10.1137/S0036141096299375.  Google Scholar

[10]

Inverse Probl. Sci. Eng., 13 (2005), 65-78. doi: 10.1080/10682760410001710141.  Google Scholar

[11]

Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[12]

Topical Review, Inverse Problems, 24 (2008), 81pp.  Google Scholar

[13]

Engineering Analysis with Boundary Elements, 33 (2009), 225-232. doi: 10.1016/j.enganabound.2008.04.006.  Google Scholar

[14]

Inverse Problems in Science and Engineering, 17 (2009), 551-567. doi: 10.1080/17415970802231610.  Google Scholar

[15]

Inverse Problems and Imaging, 4 (2010), 547-569.  Google Scholar

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