多连通域上泊松方程柯西问题的一种新型无网格方法
Cauchy Problem of the Poisson Equation in a Multi-Connected Domain by a New Meshless Method
DOI: 10.12677/PM.2021.115094, PDF,    国家自然科学基金支持
作者: 王珊珊, 温 瑾：西北师范大学数学与统计学院，甘肃 兰州；王士娟：兰州交通大学交通运输学院，甘肃 兰州；兰州财经大学信息工程学院，甘肃 兰州
关键词: 泊松方程；柯西问题；基本解方法；径向基函数；多连通域；Poisson Equation； Cauchy Problem； Method of Fundamental Solutions； Radial Basis Function； Multi-Connected Domain
摘要: 本文提出了一种求解多连通域中泊松方程柯西问题的无网格数值方法。结合拉普拉斯方程的基本解和径向基函数得到了数值解。由于系数矩阵是不适定的，因此采用正则化方法来求解所得到的线性方程组。通过对正则化参数的适当选取和对柯西数据的先验假设，得到了上述问题的正则化解，并且利用数值例子验证了该方法的有效性和准确性。
Abstract: This paper presents a meshless numerical scheme to solve the Cauchy problem of the Poisson equation in a multi-connected domain. Fundamental solutions of Laplace’s equations and radial basis functions (RBFs) are used to obtain a numerical solution. Because the coefficient matrix is illposed, the Tikhonov regularization method is applied to solve the resulting system of linear equations. By the suitable choices of a regularization parameter and a priori assumption to the Cauchy data, the regularized solution to above problem is obtained. Several numerical examples are given to verify the efficiency and accuracy of the proposed method.
文章引用：王珊珊, 温瑾, 王士娟. 多连通域上泊松方程柯西问题的一种新型无网格方法[J]. 理论数学, 2021, 11(5): 802-813. https://doi.org/10.12677/PM.2021.115094

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