International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 1, pp. 137–153 (2009) http://campus.mst.edu/ijde A Study of the Dynamic Difference Approximations on Time Scales Qin Sheng and Anzhong Wang Baylor University CASPER and Department of Mathematics / Physics Waco, TX 76798-7328 Qin Sheng@baylor.edu Abstract Various dynamic equations have been used extensively in modeling many im- portant natural phenomena, such as the population or epidemic growth with unpre- dictable jump sizes, motion control of impulsive robot movements, and prediction of irregular option markets. Since dynamic derivatives are basic building blocks of most dynamic equations, it has been crucial to approximate the derivatives to yield computable discrete equations for numerical solutions. This motivates our investigations. This paper proposes a class of feasible approximation methods for the first and second order noncrossed dynamic derivatives. Applicable local error estimates are derived and discussed. Numerical experiments are given to illustrate our results. AMS Subject Classifications: 34A45, 39A13, 74H15, 74S20. Keywords: Dynamic derivatives, time scales, approximations, finite differences, hybrid grids, local error estimates. 1 Introduction A one-dimensional time scale T is a nonempty closed subset of the real numbers R [2,7]. We denote a = sup T,b = inf T and a, b ∈ T. Thus, T can be viewed as a closed set of real numbers superimposed over the interval [a, b] from an approximation point-of- view. Based on T, we may define the forward-jump and backward-jump functions σ, ρ for t ∈ T. We may write f σ (t)= f (σ(t)),f ρ (t)= f (ρ(t)), where f is a function defined on T. We may further define the forward-step and backward-step functions µ and η. Denote λ(t)= µ(t)/η(t) Received April 16, 2008; Accepted December 3, 2008 Communicated by Al Peterson