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This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier–Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem connected with the Laplace equation in a smooth bounded domain. The harmonic polynomial method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the scattering problem of a polygonal-line arc. We solve this polygonal-line arc-scattering problem by a least-squares finite element method. In the method, Fourier–Bessel functions is used to capture the singularities around tips and corners. A combination of fundamental solutions is used to represent the scattered field towards infinity. We also analyse the convergence and give an error estimate of the method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The inverse problem considered in this article is to determine the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of some far field patterns. Based on the optimization method due to Kirsch and Kress for the inverse scattering problem, we propose a new scheme by reformulating the cost functional via a technique of piecewise integration with respect to incident directions. Convergence analysis of this method is given. Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions for the full-aperture problems. The method is extended to the limited-aperture case by weighting the total fields with special factors. Numerical examples for limited-aperture problems are also presented which show that our method produces satisfactory results efficiently in the illuminated regions. [ABSTRACT FROM AUTHOR]