Partial difference equations and their application in systems theory

Abstract

This paper is a survey of basic results of the theory of partial difference equations (PDE's) and its application in multi dimensional systems theory. Existence and uniqueness theorems of solutions of initial value problems, some boundary value problems, fundamental solutions for linear PDE's are presented. Most results are extended to systems of linear PDE's. Recursive solutions play an important role not only in these theorems but can also be used to fnd grow the estimates and formulate further qualitative properties of PDE's. Application of these results to input output relations of linear multidimensional systems (called also nD-systems) enables to introduce concepts analogous to time invariane, ausality, weight functions, impulse response and similar ones, well known from (one dimensional) systems theory. In this paper fundamental results concerning some PDE are described. Some generalizations of earlier published results are introduced.

Keywords: Difference equations, partial difference equations, n-dimen- sional sequences, initial value problems, boundary value problems, recursive solution, system theory, discrete system, shift invariant system, translation invariant system, BlBO stability, initial state stable system, inputs table system