We consider the Cauchy problem for the Helmholtz equation defined in a rectangular domain. The Cauchy data are prescribed on a part of the boundary and the aim is to find the solution in the entire domain. The problem occurs in applications related to acoustics and is illposed in the sense of Hadamard. In our work we consider regularizing the problem by introducing a bounded approximation of the second derivative by using Cubic smoothing splines. We derive a bound for the approximate derivative and show how to obtain stability estimates for the method. Numerical tests show that the method works well and can produce accurate results. We also demonstrate that the method can be extended to more complicated domains.

我们考虑在矩形域中定义的亥姆霍兹方程的柯西问题。柯西数据规定在边界的一部分上，目的是在整个域中找到解。该问题发生在与声学相关的应用中，并且在哈达玛的意义上是不合适的。在我们的工作中，我们考虑通过使用三次平滑样条引入二阶导数的有界近似来正则化问题。我们推导出近似导数的界限，并展示如何获得该方法的稳定性估计。数值试验表明，该方法效果良好，结果准确。我们还证明了该方法可以扩展到更复杂的领域。