Cauchy problems of Laplaceʹs equation by the methods of fundamental solutions and particular solutions

The Cauchy problems of Laplaceʹs 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.