Some Recent Advances in Partial Difference Equations

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In this review paper, the aim is to present a functional-analytic method developed relevantly recently by the authors for the study of partial difference equations in the spaces l1 and l2. The method is demonstrated using two illustrative examples. At the same time an effort is made in order top resent several other methods used by other researchers, which make use of functional analysis and operator theory.

Mathematics Subject Classification. 39-02, 39A05, 39All, 47N99

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.

In this study we investigate the connection of difference equations and numerical schemes through the study of a simple partial differential equation (pde). After an introduction to different numerical schemes, we use some well known finite differences schemes to discretize the simple linear pde ut+2ux = 0 with initial condition (x,0) = x. The different discretization schemes lead to different, consistent to the original pde, numerical schemes which constitute corresponding partial difference equations. The solution of the above mentioned pde is attained numerically as well as by analytic solution of the corresponding difference equations. The results show that the solution is always attained by using the analytic solution of difference equations where as, limitations should be taken into consideration when we try to achieve the solution numerically. These results indicate that the analytic solution of difference equations, resulting from application of numerical schemes, could be of extreme importan ce for the estimation of the solution of a pde.

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