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Title Some implications of a new definition of the exponential function on time scales

Name Jan L Cie\'{s}li\'{n}ski
Country Poland
Email janek@alpha.uwb.edu.pl
Submit Time 2010-02-28 16:23:45
Special Session 48: Differential, Difference, and Dynamic Equations
We present new defintions of exponential, hyperbolic and trigonometric functions on time scales, motivated by the Cayley transformation. These functions approximate more accurately the corresponding continuous functions and preserve most of their qualitative properties. In particular, Pythagorean trigonometric identities hold exactly on any time scale. \par Dynamic equations satisfied by the Cayley-motivated functions have a natural similarity to the corresponding differential equations but, in contrast to the standard approach, they are implicit. For instance, the Cayley-exponential function satisfies the equation $$ x^\Delta (t) = \alpha (t) \frac{ x (t) + x (\sigma (t)) }{2}$$ where $\alpha = \alpha (t)$ is a given function on a time scale. It suggests a new natural correspondence between differential equations and dynamic systems on time scales. \par Our approach is strongly motivated by numerical methods. The delta calculus corresponds to the forward (explicit) Euler scheme, the nable calculus corresponds to the implicit Euler scheme, and the presented Cayley approach is related to the trapezoidal rule (and to the discrete gradient methods). An important conclusion is that there are no unique `natural' time scales analogues of differential equations and it is worthwhile to consider and apply different numerical schemes in the context of the time scales approach.