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Mathematical Problems in Engineering
Volume 2017, Article ID 2472060, 16 pages
https://doi.org/10.1155/2017/2472060
Research Article

On Inverse Problems for Characteristic Sources in Helmholtz Equations

1CEMAT-IST, Instituto Superior Técnico, Lisboa, Portugal
2Universidade Federal do Rio de Janeiro, Campus Macaé, Macaé, RJ, Brazil
3Programa de Engenharia Nuclear, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil
4CEMAT-IST, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal

Correspondence should be addressed to Roberto Mamud; rb.jrfu.eacam@dumamr

Received 8 December 2016; Accepted 24 January 2017; Published 27 February 2017

Academic Editor: Guido Ala

Copyright © 2017 Carlos J. S. Alves 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.

Abstract

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.