On the approximate solution of the Cauchy problem for the Helmholtz equation on the plane

In this paper, approximate solutions of the Cauchy problem for the Helmholtz equation on a two-dimensional bounded region are found. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. When solving applied problems, it is necessary to find not only an approximate solution but also a derivative of the approximate solution. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the initial Cauchy data, their continuous approximations with a given error in the uniform metric are given. Stability estimates for the solution of the Cauchy problem in the classical sense are obtained.