48 research outputs found
Some implications of a new definition of the exponential function on time scales
We present a new approach to exponential functions on time scales and to
timescale analogues of ordinary differential equations. We describe in detail
the Cayley-exponential function and associated trigonometric and hyperbolic
functions. We show that the Cayley-exponential is related to implicit midpoint
and trapezoidal rules, similarly as delta and nabla exponential functions are
related to Euler numerical schemes. Extending these results on any Pad\'e
approximants, we obtain Pad\'e-exponential functions. Moreover, the exact
exponential function on time scales is defined. Finally, we present
applications of the Cayley-exponential function in the q-calculus and suggest a
general approach to dynamic systems on Lie groups.Comment: 12 pages. Presented at 8th AIMS International Conference on Dynamical
Systems, Differential Equations and Applications; Dresden, 25-28.05.201