Two Kinds Of Regularization Methods And Error Estimation For The Cauchy Problem Of Helmholtz Equation

In this paper,the cauchy of Helmholtz equation is considered on the domain of 0<x<1,y∈R.the Cauchy data at x= 0 is given,and the solution of the Cauchy problem of Helmholtz equation in the interval of 0<x<1 is found by using the Fourier transform technique.the solution of the Cauchy problem of Helmholtz equation in the given Cauchy data is unstable.In this paper,a new filtering regularization method and a modified Tikhonov regularization method are used to construct the stable regularization approximate solution.Secondly,the posterior regularization parameters are selected according to the Morozov deviation principle respectively,and the convergence error estimates in the sense of L2 norm between the regularization approximate solution and the exact solution are obtained respectively.We use discrete fast Fourier transform(FFT)and discrete fast inverse Fourier transform(IFFT)to realize numerical simulation.Numerical examples show that these two kinds of regularization methods are effective and feasible.