In this paper, Spectral Galerkin Method is applied for Cauchy problem of Helmholtz and Laplace equations in the regular domains. It is well known that these problems have severely ill-posed solutions. Accordingly, regularization methods are required to overcome the ill-posedness issue. In this paper, we utilize the regularization method based upon mapped methods. These methods include Tikhonov and truncated singular value decomposition methods and additionally several new filters of regularization which are introduced. Finally, some test examples are given to demonstrate the capability and efficiency of the proposed method.

中文翻译：

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亥姆霍兹方程和拉普拉斯方程的柯西问题的映射正则化方法

本文将谱Galerkin方法应用于正则域中Helmholtz和Laplace方程的柯西问题。众所周知，这些问题有严重的不适解决方案。因此，需要正规化方法来克服不适性问题。在本文中，我们利用基于映射方法的正则化方法。这些方法包括Tikhonov和截断的奇异值分解方法，另外还引入了几种新的正则化滤波器。最后，给出了一些测试示例，以证明该方法的能力和效率。