Abstract
In this paper, an effective numerical method for the Dirichlet problem connected with the Helmholtz equation is proposed. We choose a single-layer potential approach to obtain the boundary integral equation with the density function, and then we deal with the weakly singular kernel of the integral equation via singular value decomposition and the Nystrom method. The direct problem with noisy data is solved using the Tikhonov regularization method, which is used to filter out the errors in the boundary condition data, although the problems under investigation are well-posed. Finally, a few examples are provided to demonstrate the effectiveness of the proposed method, including piecewise boundary curves with corners.
Highlights
We considered a Dirichlet problem connected with the Helmholtz equation by a singerlayer potential approach, in which we were able to obtain the numerical approximation of the solution on the boundary as well as on the interior of the whole domain
In order to show the effectiveness of the presented method, we present a comparative study about the proposed method and the method of fundamental solution (MFS)
We use a singer-layer potential approach to deal with the Dirichlet problem connected with the Helmholtz equation
Summary
Chen et al [12] introduced the Ficheras concept [13,14] into the indirect boundary element method (IBEM), which can solve the ill-posed problems resulting from a degenerate scale in the case of the twodimensional Laplace equation with a Dirichlet boundary condition. Based on the Ficheras concept, Chen et al [15] studied the field of both interior and exterior problems They added a complete base with a constant and an extra constraint to conventional BEM. We considered a Dirichlet problem connected with the Helmholtz equation by a singerlayer potential approach, in which we were able to obtain the numerical approximation of the solution on the boundary as well as on the interior of the whole domain. Some examples are given to illustrate the effectiveness of the method
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