The Landweber Iterative Regularization Method for Solving the Cauchy Problem of the Modified Helmholtz Equation

Abstract

In this manuscript, the Cauchy problem of the modified Helmholtz equation is researched. This inverse problem is a serious ill-posed problem. The classical Landweber iterative regularization method is designed to find the regularized solution of this inverse problem. The error estimations between the exact solution and the regularization solution are all obtained under the a priori and the a posteriori regularization parameter selection rule. The Landweber iterative regularization method can also be applied to solve the Cauchy problem of the modified Helmholtz equation on the spherically symmetric and cylindrically symmetric regions.

Keywords

• modified Helmholtz equation
• ill-posed problem
• error estimation
• Landweber iterative method