Abstract: A Cauchy problem for the Helmholtz equation with Neumann boundary conditions in a rectangle domain is discussed in this paper. This problem is a serious ill-posed inverse problem of partial differential equations, that is, the solution does not depend continuously on the input data. Based on the classical Tikhonov regularization method using the idea of modifying the kernel function by self-designed filters subsets, a new regularization method is presented for approximating the solution to this problem based on themethod of separating variables. After performing error analysis on the exact and approximate solutions under both a priori and a posteriori selection rules of the regularization parameters, the Hlder-type error estimates satisfying convergence and stability are obtained.

Key words: Cauchy problem for Helmholtz equation, Neumann boundary condition, ill-posed problem, regularization, a posteriori parameter choice rule