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Cauchy problems of Laplace's equation by the methods of fundamental solutions and particular solutions

机译：基本解法和特殊解法的拉普拉斯方程的柯西问题

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摘要

The Cauchy problems of Laplace's equation are ill-posed with severe instability. In this paper, numerical solutions are solicited by the method of fundamental solutions (MFS) and the method of particular solutions (MPS). We focus on the analysis of the MFS, and derive the bounds of errors and condition numbers. The analysis for the MPS can also be obtained similarly. Numerical experiments and comparisons are reported for the Cauchy and Dirichlet problems by the MPS and the MFS. The Cauchy noise data and the regularization are also adopted in numerical experiments. Both the MFS and the MPS are effective to Cauchy problems. The MPS is superior in accuracy and stability; but the MFS owns simplicity of algorithms, and earns flexibility for a wide range of applications, such as Cauchy problems. These conclusions also coincide with [37]. The basic analysis of error and stability is explored in this paper, and applied to the Cauchy data. There are many reports on numerical Cauchy problems, see the survey paper in [12]; most of them are of computational aspects. The strict analysis of this paper may, to a certain degree, fill up the existing gap between theory and computation of Cauchy problems by the MFS and the MPS. Moreover, comprehensive analysis and compatible computation are two major characteristics of this paper, which may enhance the study of numerical Cauchy problems forward to a higher and advanced level.
机译：拉普拉斯方程的柯西问题由于严重的不稳定性而不适当地成立。本文通过基本解法（MFS）和特殊解法（MPS）寻求数值解。我们专注于MFS的分析，并得出错误和条件数的界限。 MPS的分析也可以类似地获得。 MPS和MFS报道了柯西和狄利克雷问题的数值实验和比较。数值实验还采用了柯西噪声数据和正则化方法。 MFS和MPS都可以有效解决柯西问题。 MPS的准确性和稳定性都很高；但是MFS拥有简单的算法，并为诸如柯西问题之类的广泛应用赢得了灵活性。这些结论也与[37]一致。本文探讨了误差和稳定性的基本分析，并将其应用于柯西数据。关于柯西数值问题的报道很多，参见[12]中的调查论文。他们大多数是在计算方面。对本文的严格分析可以在一定程度上弥补MFS和MPS对柯西问题的理论和计算之间的空白。而且，综合分析和兼容计算是本文的两个主要特征，可以将数值柯西问题的研究推向更高和更高的水平。

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