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 计算数学  2018, Vol. 40 Issue (3): 254-270    DOI:
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Helmholtz方程Cauchy问题的间接积分方程方法

INDIRECT BOUNDARY INTEGRAL EQUATION METHOD FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION
Sun Yao, Chen Bo
College of science, Civil Aviation University of China, Tianjin 300300, China
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Abstract： In this paper, we examine the data completion problem of the Helmholtz equation in two-and three-dimensionals. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of Cauchy problem. The proposed method is non-iterative and intrinsically handle the case of noisy and incompatible data. In order to cope with this ill-posed problem, our formulation is based on Tikhonov regularization in conjunction with the Morozov discrepancy principle associated with linear ill-posed inverse problems and leads to convergent scheme. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. Two-and three-dimensional examples are given for checking the effectiveness of the proposed method. The numerical convergence, accuracy, and stability with respect to the number of source points, the distance between the pseudo and real boundary, and decreasing the amount of noise added into the input data, respectively, are also analyzed.

 引用本文: . Helmholtz方程Cauchy问题的间接积分方程方法[J]. 计算数学, 2018, 40(3): 254-270. . INDIRECT BOUNDARY INTEGRAL EQUATION METHOD FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION[J]. Mathematica Numerica Sinica, 2018, 40(3): 254-270.

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