Correctness of a Two-dimensional Cauchy Problem for a Polynomial Difference Operator with Constant Coefficients

Abstract

The correctness of Cauchy problem for a polynomial difference operator is studied. An easily verifiable sufficient condition for correctness of a two-dimensional Cauchy problem for an operator with constant coefficients is proved.

Solving the Cauchy problem for a three-dimensional difference equation in a parallelepiped

“…In the theory of difference schemes, such problems are multilayer implicit difference scheme. In [21] the well-posedness of problem (3) -(4) for n = 2 is investigated and an easily verifiable sufficient condition for correctness is proved. In [22] for n = 3, an easily verified sufficient condition for the solvability of the Cauchy problem (3) -( 4) is proved.…”

The Cauchy Problem for Multidimensional Difference Equations and the Preservation of the Hierarchy of Generating Functions of its Solutions

“…Conditions on the cone and the point m that ensure solvability, that is, existence and uniqueness, have been studied in papers [2,5,8,13].…”

“…This problem has a unique solution (see [2], Theorem 1) for any fixedφ(x) andg(x). We consider the solutions u(x) of the Cauchy problem (12)- (13) for arguments x lying in the original cone K.…”

An Approach to Multidimensional Discrete Generating Series

“…arises in a wide class of combinatorial analysis problems [3], for instance, in lattice path problems [4], the theory of digital recursive filters [14], and the wavelet theory [15]. The question about correctness and well-posedness of (2) was considered in [16][17][18]. We equip (1) with initial data on a set named X m , which is used often enough.…”