A Simple Regularization Method For The Cauchy Problem Of Helmholtz-type Equations

In this paper,we consider two typical ill-posed problems:the Cauchy problem for Helmholtz equation and the modified Helmholtz equation,which are both defined in an infinite strip plane 0＜x≤1,y∈R.When Cauchy data are given for x=0,the solution for 0＜x≤<1 is sought.We deal with the Cauchy problem for Helmholtz equation and the modified Helmholtz equation by using a boundary modification regularization method.Then, under the priori bound and the appropriate parameter,we get some convergence estimates.The paper is organized as follows:In the chapter 2 of this thesis,we consider the Cauchy problem for the modified Helmholtz equation.At first,we give a statement for the ill-posedness of the problem by using the Fourier transform.Then we use a boundary modification regularization method to get the regularization solution for the problem.The convergence estimates under the suitable choice of the regularization parameters for the cases of 0＜x＜1 and x=1 are proved. Finally,numerical results show that our proposed method is effective and accurate. In the chapter 3,we consider the Cauchy problem for Helmholtz equation by the same way.