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

TL;DR

This paper develops a new algorithm for solving the Cauchy problem for three-dimensional linear difference equations with constant coefficients in a parallelepiped, advancing the theoretical and computational methods in this area.

Contribution

It introduces a novel algorithm for solving 3D difference equations, extending previous theoretical results and utilizing computer algebra for complex calculations.

Findings

New algorithm for 3D difference equations

Extension of solvability and well-posedness theorems

Application of computer algebra methods

Abstract

The aim of this article is further development of the theory of linear difference equations with constant coefficients. We present a new algorithm for calculating the solution to the Cauchy problem for a three-dimensional difference equation with constant coefficients in a parallelepiped at the point using the coefficients of the difference equation and Cauchy data. The implemented algorithm is the next significant achievement in a series of articles justifying the Apanovich and Leinartas' theorems about the solvability and well-posedness of the Cauchy problem. We also use methods of computer algebra since the three-dimensional case usually demands extended calculations.