A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation

Abstract

This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice strategies of the regularization parameter are proposed, and error estimations of the corresponding regularization solutions are also presented. Finally, two numerical examples are introduced to show the effectiveness of the general mollification regularization method.

Keywords

• modified Helmholtz equation
• Cauchy problem
• mollification method
• regularization parameter
• regularization solution
• error estimation