三维Laplace方程柯西问题具有Dirichlet核的软化正则解

A Mollification Method with Dirichlet Kernel to Solve the Cauchy Problem for the Three-dimensional Laplace Equation

研究一类三维Laplace方程Cauchy问题,该问题是严重不适定的.为了获得其稳定的数值解,利用二维Dirichlet核构造软化算子,得到正则逼近解的显式形式,在先验参数的选取规则之下,给出正则近似解与精确解之间的误差估计,并通过数值实验检验方法的有效性和稳定性.

The Cauchy problem for the Laplace equation is a severely ill-posed problem.In this paper,to obtain the stable numerical solution for this problem,a mollification method with the two-dimensional Dirichlet kernel is proposed to construct regularization approximation solution,and the error estimate between the regularization approximation solution and the exact solution is given.Finally,a numerical example is presented to show the effectiveness of the proposed method.