An improved non‐local boundary value problem method for a cauchy problem of the Laplace equation

In this paper, we propose an improved non‐local boundary value problem method to solve a Cauchy problem for the Laplace equation. It is known that the Cauchy problem for the Laplace equation is severely ill‐posed, i.e., the solution does not depend continuously on the given Cauchy data. Convergence estimates for the regularized solutions are obtained under a‐priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.