Jan Cieśliński
Standard and Nonstandard Exponential Functions on Time Scales, Including Discrete and q-Calculus Cases
A time scale is a closed domain on which we can define functions and dynamical systems. The extreme cases are differential equations (the continuum case) and difference equations (the discrete case). Considering an arbitrary time scale allows for a unified treatment of the continuous and discrete cases, q-analogues, and all intermediate cases, in particular fractal ones.
Within time-scale calculus, the term exponential function usually denotes the delta exponential function or, less frequently, the nabla exponential function (both are related to well-known q-analogues). Unfortunately, these standard exponential functions fail to map the imaginary axis onto the unit circle, leading to rather poor properties of the associated trigonometric functions which are typically neither periodic nor bounded. We present alternative classes of exponential functions leading to trigonometric functions with significantly better properties, including the Pythagorean identity. These constructions are related to symmetric Padé approximants. The simplest, yet most important, example is the Cayley exponential function.
An important motivation for this approach comes from the relationship between dynamical systems on discrete time scales and numerical integration methods. In this setting, first-order delta and nabla equations correspond to the explicit and implicit Euler schemes. respectively, while other numerical schemes applied to linear equations can produce other types of discrete exponential functions.
In the continuous setting, exact solutions of many equations are naturally expressed in terms of exponential or trigonometric functions. We present analogous examples of exact solutions for equations on time scales, including linear equations with constant coefficients, the logistic equation and the sine-Gordon equation. In the last case both the delta exponential function and the Cayley exponential function prove to be useful in deriving the exact soliton solution.
