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Smooth and discrete systems – algebraic, analytic, and geometrical representations. (English) Zbl 1077.34008

Summary: 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 on old and recent results and mainly new approaches to problems concerning smooth and discrete representations based on analytic, algebraic, and geometrical tools is presented.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
39A12 Discrete version of topics in analysis
35A25 Other special methods applied to PDEs
34A45 Theoretical approximation of solutions to ordinary differential equations
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