Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain

Abstract

<p>In this paper, a regularized solution to the Cauchy problem for matrix factorization of the Helmholtz equation in a three-dimensional unbounded domain is constructed explicitly based on the Carleman matrix. When solving applied problems, in addition to an approximate solution, the derivative of the approximate solution is found. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with precisely specified Cauchy data. An explicit formula for continuing the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the original Cauchy data, their continuous approximations with a specified error in the uniform metric are given. As a result, the stability of the solution to the Cauchy problem in the classical sense is estimated.</p>