Solving the Helmholtz Equation Together with the Cauchy Boundary Conditions by a Modified Quasi-Reversibility Regularization Method

Abstract

The Quasi-Reversibility Regularization Method (Q-RRM) provides stable approximate solution of the Cauchy problem of the Helmholtz equation in the Hilbert space by providing either additional information in the Laplace-type operator in the Helmholtz equation or the imposed Cauchy boundary conditions on the Helmholtz equation. To help bridge this gap in the literature, a Modified Quasi-reversibility Regularization Method (MQ-RRM) is introduced to provide additional information in both the Laplace-type operator occurring in the Helmholtz equation and the imposed Cauchy boundary conditions on the Helmholtz equation, resulting in a strong stable solution and faster convergence of the solution of the Helmholtz equation than the regularized solutions provided by Q-RRM and its variants methods.