First Order Linear Equations On Time Scales

Abstract

Unification and extension of the concepts of difference calculus with that of differential calculus led to the development of time scale calculus, by which we can recover, as special cases the former and the latter concepts, plus more. We consider First Order Linear Dynamic Equations (F.O.L.D.Es) on an arbitrary time scale. These F.O.L.D.Es have as special cases the ordinary differential equations, difference equations and many others e.g q-difference equations. We begin by taking a step back and explain what a time scale is, covering the main theorems and special properties of functions on a time scale. We then introduce the exponential function for a time scale including its properties, and use these properties to derive solutions for F.O.L.D.Es with constant coefficients. Several examples and applications, among them an insect population model, and the logistics equation are considered.