# American Institute of Mathematical Sciences

2011, 2011(Special): 302-311. doi: 10.3934/proc.2011.2011.302

## Some implications of a new approach to exponential functions on time scales

 1 Uniwersytet w Białymstoku, Wydział Fizyki, ul. Lipowa 41, 15-424 Białystok, Poland

Received  August 2010 Revised  March 2011 Published  October 2011

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary di erential 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é approximants, we obtain Pade-exponential functions. Moreover, the exact exponential function on time scales is de fined. Finally, we present applications of the Cayley-exponential function in the $q$-calculus and suggest a general approach to dynamic systems on Lie groups.
Citation: Jan L. Cieśliński. Some implications of a new approach to exponential functions on time scales. Conference Publications, 2011, 2011 (Special) : 302-311. doi: 10.3934/proc.2011.2011.302
 [1] Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231 [2] Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072 [3] Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008 [4] Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671 [5] Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 [6] Damiano Foschi. Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals. Communications on Pure & Applied Analysis, 2005, 4 (3) : 569-588. doi: 10.3934/cpaa.2005.4.569 [7] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [8] Peter E. Kloeden, Björn Schmalfuss. Lyapunov functions and attractors under variable time-step discretization. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 163-172. doi: 10.3934/dcds.1996.2.163 [9] Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407 [10] Yuk L. Yung, Cameron Taketa, Ross Cheung, Run-Lie Shia. Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 229-248. doi: 10.3934/dcdsb.2010.13.229 [11] Jian-Hua Zheng. Dynamics of hyperbolic meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2273-2298. doi: 10.3934/dcds.2015.35.2273 [12] Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293 [13] Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101 [14] Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653 [15] Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439 [16] Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803 [17] Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161 [18] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631 [19] Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055 [20] Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over ${\mathbb F}_{p}$ with ${q}$ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020018

Impact Factor: