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Mathematical Problems in Engineering
Volume 2012, Article ID 808913, 28 pages
Research Article

Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem

Clear/Coppe/Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, RJ, Brazil

Received 27 July 2011; Revised 30 November 2011; Accepted 8 December 2011

Academic Editor: Cristian Toma

Copyright © 2012 Marcelo L. Rainha and Nilson C. Roberty. 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.


The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.