Robin–Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain

• Published in: Journal of inverse and ill-posed problems Vol. 31; no. 5; pp. 653 - 667
• Main Authors: Achieng, Pauline, Berntsson, Fredrik, Kozlov, Vladimir
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.10.2023; Walter de Gruyter GmbH
• Subjects: 35J05; 35R30; Cauchy problem; Cauchy problems; Convergence; Dirichlet problem; Finite difference method; Helmholtz equation; Helmholtz equations; Inclusions; inverse problem ill-posed problem; Iterative methods; Operators; Helmholtz equation; inverse problem ill-posed problem; Cauchy problem
• ISSN: 0928-0219; 1569-3945; 1569-3945
• DOI: 10.1515/jiip-2020-0133

We consider the Cauchy problem for the Helmholtz equation with a domain in , with cylindrical outlets to infinity with bounded inclusions in Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz’ya [ ] proposed an alternating iterative method for solving Cauchy problems associated with elliptic, self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) [ ] for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters and , the Robin–Dirichlet alternating iterative procedure is convergent.