Numerical Method for a Cauchy Problem for Multi-Dimensional Laplace Equation with Bilateral Exponential Kernel

Abstract

This study examined a Cauchy problem for a multi-dimensional Laplace equation with mixed boundary. This problem is severely ill-posed in the sense of Hadamard. To solve this problem, a mollification approach is suggested based on a bilateral exponential kernel and this is a new approach. The stable error estimates are obtained under the priori and posteriori rule, in which the numerical findings are much influenced by the unknown a priori information. An error estimate between the exact and regular solution is given. A numerical experiment of interest reveals that our procedure is efficient and stable for perturbation noise in the data.

Keywords

• multi-Laplace equation
• ill-posed
• bilateral exponential kernel
• Cauchy problem
• error estimate
• mollification method