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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 Helmholtz equation in a smooth-bounded domain. The Fourier–Bessel 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 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 Cauchy problem connected with the Laplace equation. We investigate the denseness of harmonic polynomials and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with a suitable choice of the regularization method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the Cauchy problem of elliptic operators. Through the use of a single-layer potential function, we devise a numerical method for approximating the solution of the Cauchy problem of elliptic operators, which are well known to be highly ill-posed in nature. The method is based on the denseness of single-layer potential functions. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. It has been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we establish new density results for the equilibrium equations. Based on the denseness result of the elastic potential functions, the Cauchy problem for the equilibrium equations is investigated. For this ill-posed problem, we construct a regularizing solution using the single-layer potential function. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. The method combines minimum norm solution with Morozov discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the discretisation about the integral equations on pseudo-boundary and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples. [Copyright &y& Elsevier]