Some implications of a new definition of the exponential function on time scales
Abstract
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 Padé approximants, we obtain Padé-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2010
- DOI:
- 10.48550/arXiv.1008.4911
- arXiv:
- arXiv:1008.4911
- Bibcode:
- 2010arXiv1008.4911C
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Dynamical Systems;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Primary: 33B10;
- 26E70. Secondary: 34N05;
- 65L12
- E-Print:
- 12 pages. Presented at 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications