Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 130834, 19 pages
Research Article

Moving Heat Source Reconstruction from Cauchy Boundary Data: The Cartesian Coordinates Case

1Programa de Engenharia Nuclear, Coppe, Universidade Federal do Rio de Janeiro, 21941-914 Rio de Janeiro, RJ, Brazil
2Department of Mathematics, ICEx-PUVR, Fluminense Federal University, Volta Redonda, RJ, Brazil

Received 21 December 2010; Accepted 17 August 2011

Academic Editor: Francesco Pellicano

Copyright © 2011 Nilson C. Roberty et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the problem of reconstruction of an unknown characteristic interval and block transient thermal source inside a domain. By exploring the definition of an Extended Dirichlet to Neumann map in the time space cylinder that has been introduced in Roberty and Rainha (2010a), we can treat the problem with methods similar to that used in the analysis of the stationary source reconstruction problem. Further, the finite difference θ-scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic interval and parallelepiped source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support. Numerical experiment for capture of an interval and an rectangular parallelepiped characteristic source inside a cubic box domain from boundary data are presented in threedimensional and one-dimensional implementations. The problem of centroid determination is addressed and questions are discussed from an computational points of view.