An algorithm for finding the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point using computer algebra is described. In the one-dimensional case, solving the Cauchy problem is easy; however, already in the two-dimensional case the number of unknowns rapidly increases at each step. To automate the process of computing the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a given point, an algorithm in MATLAB is developed in which the input data are the matrix of coefficients obtained on the basis of the structure of the two-dimensional polynomial difference equation, coordinates of the points that specify the structure and the size of the matrix of initial data, and the matrix of the initial data. The algorithm produces the solution to the Cauchy problem for the given two-dimensional difference equation at the given point.

In this paper, we propose an algorithm for solving the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point, based on its coefficients and the initial data for the Cauchy problem, which are defined in a “strip.” For this purpose, computer algebra methods are employed. To automate the process of computing the solution, the algorithm is implemented in MATLAB, with the input data being the matrix of coefficients of the two-dimensional polynomial difference equation, the coordinates of the initial data matrix (which defines the structure of the difference equation), and the coordinates of the point that determines the dimension of the initial data matrix. The output of the algorithm is a solution of the Cauchy problem (with the initial data defined in a “strip”) for the two-dimensional difference equation, which is a function value at the desired point.

An algorithm for solving the Cauchy problem for a three-dimensional difference equation with constant coefficients by given coefficients of the difference equation and initial data in a parallelepiped. The algorithm is based on the Apanovich and Leirnartas theorems characterizing well-posedness of Cauchy problems and uses computer algebra methods to achieve computational efficiency.The algorithm is implemented in Matlab.

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SOLVING THE CAUCHY PROBLEM FOR A THREE-DIMENSIONAL DIFFERENCE EQUATION IN A PARALLELEPIPED РЕШЕНИЕ ЗАДАЧИ КОШИ ДЛЯ ТРЕХМЕРНОГО РАЗНОСТНОГО УРАВНЕНИЯ В ПАРАЛЛЕЛЕПИПЕДЕ

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.