Table of Contents
ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 150979, 11 pages
http://dx.doi.org/10.5402/2011/150979
Research Article

The Cauchy Problem for the Laplace Equation and Application to Image Inpainting

1University of Dammam, College of Science, P.O. Box 838, Dammam 31113, Saudi Arabia
2Tunis El Manar University, ENIT-LAMSIN, BP 37, 1002 Le Belvédère, Tunis, Tunisia

Received 4 August 2011; Accepted 11 September 2011

Academic Editors: G. Garcea and C. Zhu

Copyright © 2011 Lamia Jaafar Belaid 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

The moment approach to solve the Cauchy problems is investigated. First, we consider the Cauchy problem for the Laplace equation, and we present a moment method for solving it in the case of a flat boundary. Second, we consider the reciprocity gap concept used to solve the problem of crack detection, as a stopping criterion and we study the case of noise data. Finally, we propose an application to the Cauchy problem for the Laplace equation, for the inpainting problem. Some numerical results showing the efficiency of the method proposed are also given.