In this paper, two Cauchy problems of Helmholtz equation in a three-dimensional case are considered. To address these problems, a mollification method with bivariate Dirichlet kernel is proposed. Stable errors estimates are obtained based on appropriate a priori choices of mollification parameters. Convergence estimates show that the regularization solution depends continuously on the data and wavenumber. Numerical examples of our interest show that Dirichlet kernel is more effective than the Gaussian kernel under the same parameter selection rule, and our procedure is stable with respect to perturbations noise in the data.

中文翻译：

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二维亥姆霍兹方程两个柯西问题的一种基于狄利克雷核的磨粒正则化方法

本文考虑了三维情况下亥姆霍兹方程的两个柯西问题。为了解决这些问题，提出了一种具有双变量 Dirichlet 核的软化方法。基于缓和参数的适当先验选择获得稳定误差估计。收敛估计表明正则化解决方案持续依赖于数据和波数。我们感兴趣的数值例子表明，在相同的参数选择规则下，狄利克雷核比高斯核更有效，并且我们的程序对于数据中的扰动噪声是稳定的。