Some Regularization Methods For The Cauchy Problem For The Helmholtz Equation In The Rectangular Region

In this paper the Cauchy problem for the Helmholtz equation in the rectangular region is considered. We want to get the value on the region 0 < x≤1 from the given Cauchy data at x = 0 .It is severely ill-posed problem, and the ill-posedness becomes sharper as the unknown solutions are closer to the boundary point. So some effective regularization methods to restore the stability of the solutions are not only extensive for practice applications but also very important for theoretical research. In the paper we consider the optimal error estimate for the Cauchy problem with only nonhomogeneous Dirichlet data on the boundary, and also use truncated regularization method, discrete regularization method to solve this problem. In addition, truncated regularization method is used to deal with the Cauchy problem with only nonhomogeneous Neumann data on the boundary. The convergence estimates are obtained for all these regularization methods. Especially,all the methods are convergent at the boundary point. Meanwhile, numerical example for truncated regularization method shows that the method works well.