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In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in Rm,(m=2k,k≥2). The corresponding theorems on the stability of the solution of problems are proved.

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.

In this paper we consider an inverse problem for determiningthe source function in fractional equation with Riemann-Liouville derivative.Using the classical Fourier method, we prove the uniqueness and the existencetheorem for this inverse problem.

In this paper, using the construction of the Carleman matrix, we explicitly find a regularized solution of the Cauchy problem for matrix factorizations of the Helmholtz equation in a three-dimensional unbounded domain.

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space Rm. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions.

There are many applications of mathematical physics in several fields of basic science and engineering [...]

In the paper it is considered the regularization of the Cauchy problem for systems of elliptic type equations of the first order with constant coefficients factorisable Helmholtz operator in three-dimensional bounded domain.Using the results of [1-6; 19,20,21,22], we construct in explicit form Carlemanmatrix and, based on the regularized solution of the Cauchy problem.

Two-variable interpolation by polynomials is investigated for thegiven f : R2 ! R. The new idea is to compute for the points on the twosides of a rectangle. In this paper, we present a generalization of the Newtondivided interpolation polynomials in two dimension. The only bet is that in(2n + 1) distinct points of function have similar quantities.

In this paper, we consider the problem of recovering solutionsfor matrix factorizations of the Helmholtz equation in a three-dimensionalbounded domain from their values on a part of the boundary of this domain, i.e., the Cauchy problem. An approximate solution to this problem isconstructed based on the Carleman matrix method.