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Applicable Analysis
An International Journal
Volume 87, 2008 - Issue 3
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Original Articles

A procedure for determining a spacewise dependent heat source and the initial temperature

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Pages 265-276 | Received 12 Oct 2007, Accepted 12 Nov 2007, Published online: 19 Mar 2008
 

Abstract

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.

Acknowledgements

T. Johansson would like to acknowledge the grant and financial support from the Wenner–Gren foundations. Both authors would like to thank G. Baravdish (former Bastay) for his useful comments on this work.

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