Advances in Difference Equations
Volume 2004 (2004), Issue 2, Pages 111-120

Smooth and discrete systems—algebraic, analytic, and geometrical representations

František Neuman

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.


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.