Advances in Difference Equations
Volume 2004 (2004), Issue 2, Pages 111-120
doi:10.1155/S1687183904401034
Smooth and discrete systems—algebraic, analytic, and geometrical representations
Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno 616 62, Czech Republic
Received 12 January 2004
Copyright © 2004 František Neuman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
What is a differential equation? Certain objects may have
different, sometimes equivalent representations. By using
algebraic and geometrical methods as well as discrete relations,
different representations of objects mainly given as analytic
relations, differential equations can be considered. Some
representations may be suitable when given data are not
sufficiently smooth, or their derivatives are difficult to obtain
in a sufficient accuracy; other ones might be better for
expressing conditions on qualitative behaviour of their solution
spaces. Here, an overview of old and recent results and mainly
new approaches to problems concerning smooth and discrete
representations based on analytic, algebraic, and geometrical
tools is presented.