On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

Abstract

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.

Keywords

• integral formula
• regularization of the Cauchy problem
• approximate solution
• Carleman matrix
• family of vector functions
• Bessel and Hankel functions