Boundary value problems for partial difference equations
The paper deals with linear partial difference equations of the form \[ \sum_{p\in B}b(p,l)y(l-p)= x(l),\quad l\in A\subseteq\mathbb{Z}^d, \] where \(\mathbb{Z}^d\) is the set of \(d\)-dimensional vectors with integer coordinates, \(A\) is an arbitrary non-empty subset of \(\mathbb{Z}^d\) and \(B\) is a finite subset of \(\mathbb{Z}^d\) containing the zero vector as well as at least one non-zero vector. The functions \(b(p,l)\) and \(x(l)\) are complex-valued. For certain boundary value problems conditions are formulated ensuring the uniqueness of solutions. Using the concept of fundamental solutions also results on the existence and construction of solutions are proved. Various examples illustrate the results of this paper.
scientific article; zbMATH DE number 890723