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Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the inverse Laplace transform on time scales. Useful properties of the new fractional operators are proved.


Introduction
The fractional calculus deals with extensions of derivatives and integrals to noninteger orders. It represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering [10,13,14,16].
The fractional calculus may be approached via the theory of linear differential equations. Analogously, starting with a linear difference equation, we are led to a definition of fractional difference of an arbitrary order [3]. Our main objective is to introduce here a fractional calculus on an arbitrary time scale, using the theory of delta (or nabla) differential equations. The analysis on time scales is a fairly new area of research. It was introduced in 1988 by Stefan Hilger and his Ph.D. supervisor Bernd Aulbach [2,9]. It combines the traditional areas of continuous and discrete analysis into one theory. After the publication of three textbooks in this area [5,7,11], more and more researchers are getting involved in this fast-growing field.
Recently, two attempts have been made to provide a general definition of fractional derivative on an arbitrary time scale [1,15]. These two works address a very interesting question, but unfortunately there is a small inconsistency in the very beginning of both studies. Indeed, investigations of [1,15] are based on the following definition of generalized polynomials on time scales h α : T × T → R: Recursion (1) provides a definition only in the case α ∈ N 0 , and there is no hope to define polynomials h α for real or complex indices α with (1). Here we propose a different approach to the subject based on the Laplace transform [8].
The paper is organized as follows. In Sect. 2 we briefly review the basic notions of Riemann-Liouville and Caputo fractional integration and differentiation (Sect. 2.1) as well as necessary tools from time scales (Sect. 2.2). Our results are then given in Sect. 3: we introduce the concept of fractional integral and fractional derivative on an arbitrary time scale T (Sect. 3.1); we then prove some important properties of the fractional integrals and derivatives (Sect. 3.2).

Preliminaries
The following definitions and basic results serve essentially to fix notations. The reader is assumed to be familiar with both fractional calculus and calculus on time scales. For an introduction to fractional and time scale theories we refer to the books [10,13,14,16] and [5,7,11].

Caputo and Riemann-Liouville fractional derivatives
Let f be an arbitrary integrable function.
with Γ the well-known gamma function. For an arbitrary real number α, the Riemann-Liouville and Caputo fractional derivatives are defined, respectively, as in [12]: where [α] is the integer part of α.
The notion of Riemann-Liouville derivative is the preferred fractional derivative among mathematicians, while Caputo fractional derivative is the preferred one among engineers. If f (a) = f ′ (a) = · · · = f (n−1) (a) = 0, then both Riemann-Liouville and Caputo derivatives coincide. In particular, for α ∈ (0, 1) and f (a) = 0 one has C D α a f (t) = D α a f (t). Next proposition gives the Laplace transform of the Caputo fractional derivative.
). Let α > 0, n be the integer such that n−1 < α ≤ n, and f a function

The Laplace transform on time scales
Throughout the paper, T is an arbitrary time scale with bounded graininess, i.e., 0 < µ min ≤ µ(t) ≤ µ max for all t ∈ T. Let t 0 ∈ T be fixed. We define functions h k (·, t 0 ) : T → R, k ∈ N 0 , recursively as in (1): Example 2.2. In [5] one can find exact formulas of generalized polynomials for some particular time scales The Laplace transform on time scales is studied in [8].
Remark 2.3. If F is the Laplace transform of f : T → R, then the Laplace transform of the delta derivative of f is given by Our work is motivated by the following result: Theorem 2.5 (Inversion formula of the Laplace transform [8]). . Suppose that F is analytic in the region Re µ (z) > Re µ (c) and F (z) → 0 uniformly as |z| → ∞ in this region. Assume F has finitely many regressive poles of finite order {z 1 , z 2 , . . . , z n } andF R (z) is the transform of the functionf (t) on R that corresponds to the transform Res z=z i e z (t, 0)F (z) has transform F (z) for all z with Re(z) > c.
Remark 2.4. The inverse transform of F (z) = 1 z k+1 , k ∈ N, is h k (·, 0). Moreover, the inversion formula gives the claimed inverses for any of the elementary functions that were presented in the table of Laplace transform in [6].
Using the series e z (t, 0) = +∞ k=0 z k h k (t, 0), we see that generalized polynomials h k (t, 0) give us the difference between time scales for the inverse Laplace images.

Main Results
We begin by introducing the definition of fractional integral and fractional derivative on an arbitrary time scale T.

Fractional derivative and integral on time scales
Similarly to the classical case, the Laplace transform of a Caputo fractional derivative of order α ∈ (0, 1] is given by Simultaneously, the generalized Laplace transform on time scales gives unification and extension of the classical results. Important to us, the Laplace transform of the ∆-derivative is given by the formula L T [f ∆ ](z) = zF (z)−f (0). Our idea is to define the fractional derivative on time scales via the inverse Laplace transform formula for the complex function G(z) = z α L T [f ](z) − f (0 + )z α−1 . Furthermore, for α ∈ (n − 1, n], n ∈ N, we use a generalization of (2) to define fractional derivatives on times scales for higher orders α.

Properties
We begin with two trivial but important remarks about the fractional integral and the fractional derivative operators just introduced.
Remark 3.2. As the inverse Laplace transform is linear, we also have linearity for the new fractional integral and derivative: