**The optimal method of solving the partial difference equations in the computer algebra systems**

**Abstract:**

The analytical approximate methods of solving the initial and linear boundary-value problem and for linear
partial difference equations of order *k* with polynomial coefficients are described.
These methods are based on on Lanczos *t*-method and V. K. Dzyadyk a-method of solving the linear differential
equations of order *k* with polynomial coefficients.
We proved the equivalence of these methods to the projective method of solving the integral equation equivalent to the
original problem.
We constructed the algorithm of solving the linear boundary-value problem for linear partial difference equations of
order 2 with polynomial coefficients and boundary conditions on the edges of the rectangular region *D*.
The algorithm is written as a specification in terms of the algebraic programming system (APS) operators.
The APLAN-language procedure is equivalent to the algorithm.
It computes the algebraic polynomial of given order *n*.
This polynomial optimally approximates the original problem exact solution within the region *G*.