Smooth and discrete systems - Algebraic, analytic, and geometrical representations

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 obtainin 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.