In this paper we consider the the initial boundary value problem for the Helmholtz equation. The well-possedness of the Cauchy problem for the Helmholtz equation is investigated and the uniqueness of the solution of the original problem in the class of functions represented as Fourier series is proved. For a direct problem, a proof of the stability theorem of the generalized solution is presented, and an estimate of the stability of the generalized solution is obtained. To solve the problem, the original problem is reduced to the inverse problem and is written in operator form. We reduce this problem to the functional minimization problem, the Landweber iteration method is used in the minimization problem. A numerical study of the stability of the direct and initial problems is carried out. The results of the study are shown in tables and figures.
keywords: inverse problem, Cauchy problem for Helmholtz equation, well-posedness, stability, numerical solution.
