Affiliation:
1. SMAD Team, Department of Mathematics Polydisciplinary Faculty of Larache Abdelmalek Essaadi University MOROCCO
Abstract
The Cauchy problem associated with the Helmholtz equation is an ill-posed inverse problem that is challenging to solve due to its instability and sensitivity to noise. In this paper, we propose a metaheuristic approach to solve this problem using Genetic Algorithms in conjunction with Tikhonov regularization. Our approach is able to produce stable, convergent, and accurate solutions for the Cauchy problem, even in the presence of noise. Numerical results on both regular and irregular domains show the effectiveness and accuracy of our approach.
Publisher
World Scientific and Engineering Academy and Society (WSEAS)
Reference30 articles.
1. Kirsch, A. (2011). An introduction to the mathematical theory of inverse problems (Vol. 120). New York: Springer.
2. Delillo, T., Isakov, V., Valdivia, N. & Wang, L. (2001). The detection of the source of acoustical noise in two dimensions. SIAM Journal On Applied Mathematics. 61, 2104-2121.
3. Colton, D., Kress, R. & Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory.
4. Hall, W. S., & Mao, X. Q. (1995). A boundary element investigation of irregular frequencies in electromagnetic scattering. Engineering Analysis with Boundary Elements, 16(3), 245-252.
5. Hadamard, J. (1923). Lectures on Cauchy’s problem in linear partial differential equations (Vol. 15). Yale university press.