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Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform∗ Nuno R. O. Bastos1 , Dorota Mozyrska2 and Delfim F. M. Torres3 1

Department of Mathematics, School of Technology Polytechnic Institute of Viseu 3504-510 Viseu, Portugal [email protected] 2

Faculty of Computer Science Bialystok University of Technology 15-351 Bialystok, Poland [email protected] 3

Department of Mathematics University of Aveiro 3810-193 Aveiro, Portugal delﬁ[email protected]

Abstract 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. Keywords: fractional derivatives and integrals, time scales, Laplace transform. 2010 Mathematics Subject Classification: 26A33, 26E70, 44A10.

1

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. ∗

Submitted 20-Aug-2010; accepted 11-Nov-2010. Ref: Int. J. Math. Comput. 11 (2011), no. J11, 1–9.

1

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 hα : T × T → R: h0 (t, s) = 1, Z t hα (τ, s)∆τ. hα+1 (t, s) =

(1)

s

Recursion (1) provides a definition only in the case α ∈ N0 , and there is no hope to define polynomials hα 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).

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].

2.1

Caputo and Riemann–Liouville fractional derivatives

Let f be an arbitrary integrable function. By Da−α f we denote the fractional integral of f of order α ∈ (0, 1) on [0, t], defined as Z t f (τ ) 1 −α dτ Da f (t) = Γ(α) 0 (t − τ )1−α

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]: ! [α]+1 d (Daα f ) (t) = Da−([α]−α+1) f (t) dt[α]+1 and C

Daα f

(t) =

Da−([α]−α+1) 2

d[α]+1 f dt[α]+1

!!

(t) ,

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 Daα f (t) = Daα f (t). Next proposition gives the Laplace transform of the Caputo fractional derivative. Proposition 2.1 ( [10]). Let α > 0, n be the integer such that n−1 < α ≤ n, and f a function satisfying f ∈ C n (R+ ), f (n) ∈ L1 (0, t1 ), t1 > 0, and |f (n) (t)| ≤ Beq0 t , t > t1 > 0. If the Laplace transforms L[f ](z) and L[D n f ](z) exist, and lim D k f (t) = 0 for k = 0, . . . , n − 1, t→+∞

then L

C

n−1 X D0α f (z) = z α L [f ] (z) − f (k) (0)z α−k−1 .

Remark 2.1. If α ∈ (0, 1], then L

2.2

k=0

C

D0α f (z) = z α L [f ] (z) − f (0)z α−1 .

The Laplace transform on time scales

Throughout the paper, T is an arbitrary time scale with bounded graininess, i.e., 0 < µmin ≤ µ(t) ≤ µmax for all t ∈ T. Let t0 ∈ T be fixed. We R t define functions hk (·, t0 ) : T → R, k ∈ N0 , recursively as in (1): h0 (t, t0 ) ≡ 1, hk+1 (t, t0 ) = t0 hk (τ, t0 )∆τ . Functions hk (·, t0 ) are called generalized polynomials on the time scale T. By h∆ k (t, t0 ) we denote the delta derivative of hk (t, t0 ) with respect to t. For t ∈ Tκ one has h∆ (t, t0 ) = 0 and h∆ 0 k (t, t0 ) = hk−1 (t, t0 ), k ∈ N. Example 2.2. In [5] one can find exact formulas of generalized polynomials for some particu (k) k 0 lar time scales T: if T = R, then hk (t, t0 ) = (t−tk!0 ) ; if T = Z, then hk (t, t0 ) = (t−tk!0 ) = t−t k , where (t − t0 )(0) = 1 and (t − t0 )(k) = (t − t0 )(t − t0 − 1) · · · (t − t0 − k + 1), k ∈ N. The Laplace transform on time scales is studied in [8]. Definition 2.1 ( [8]). For f : T → R, the generalized Laplace transform of f , denoted by LT [f ], is defined by Z ∞ f (t)e⊖z (σ(t), 0)∆t . LT [f ](z) = F (z) := 0

Remark 2.2. In view of Definition 2.1, the Laplace transform L of Proposition 2.1 can be written as LR . Definition 2.2 ( [8]). The function f : T → R is said to be of exponential type I if there exist constants M, c > 0 such that |f (t)| ≤ M ect . Furthermore, f is said to be of exponential type II if there exist constants M, c > 0 such that |f (t)| ≤ M ec (t, 0). The time scale exponential function ec (t, 0) is of type II while generalized polynomials hk (t, 0) are of type I. Theorem R ∞2.3 ( [8]). If f is of exponential type II with exponential constant c, then the delta integral 0 f (t)e⊖ (σ(t), 0)∆t converges absolutely for z ∈ D. From [8, Theorem 1.3] and mathematical induction one obtains the following result:

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Proposition 2.4. Let F be the generalized Laplace transform of f : T → R. Then, n

LT [f ∆ ](z) = z n F (z) −

n−1 X

k

z n−k−1 f ∆ (0) .

(2)

k=0

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 LT [f ∆ ](z) = zF (z) − f (0) . 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 Reµ (z) > Reµ (c) and F (z) → 0 uniformly as |z| → ∞ in this region. Assume F has finitely many regressive poles of finite order {z1 , z2 , . . . , zn } and F˜R (z) is the transform of the function f˜(t) on R that corresponds to the transform F (z) = FT (z) of f (t) on T. If Z

c+i∞

c−i∞

then f (t) =

˜ FR (z) |dz| < ∞ ,

n X

Resz=zi ez (t, 0)F (z)

i=1

has transform F (z) for all z with Re(z) > c. 1 , k ∈ N, is hk (·, 0). Moreover, the inversion Remark 2.4. The inverse transform of F (z) = z k+1 formula gives the claimed inverses for any of the elementary functions that were presented in the table of Laplace transform in [6]. P k Using the series ez (t, 0) = +∞ k=0 z hk (t, 0), we see that generalized polynomials hk (t, 0) give us the difference between time scales for the inverse Laplace images.

3

Main Results

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

3.1

Fractional derivative and integral on time scales

Similarly to the classical case, the Laplace transform of a Caputo fractional derivative of C α order α ∈ (0, 1] is given by L D0+ f (z) = z α L [f ] (z) − f (0+ )z α−1 . 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 LT [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 LT [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.

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Definition 3.1 (Fractional integral on time scales). Let α > 0, T be a time scale, and f : T → R. The fractional integral of f of order α on the time scale T, denoted by ITα f , is defined by −1 F (z) α (t) . IT f (t) = LT zα Definition 3.2 (Fractional derivative on time scales). Let T be a time scale, F (z) = LT [f ](z), and α ∈ (n − 1, n], n ∈ N. The fractional derivative of function f of order α on the time scale T, denoted by f (α) , is defined by # " n−1 X k z α F (z) − (3) f (α) (t) = L−1 f ∆ (0+ )z α−k−1 (t) . T k=0

Remark 3.1. For α ∈ (0, 1] we have

3.2

Properties

α f (α) (t) = L−1 z F (z) − f (0+ )z α−1 (t) . T

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: ITα (af + bg)(t) = aITα f (t) + bITα g(t), (af + bg)(α) (t) = af (α) (t) + bg(α) (t) . Remark 3.3. Since h0 (t) ≡ 1 for any time scale T, from the definition of Laplace transform (α) (α) and the fractional derivative we conclude that h0 (t) = 0. For α > 1 one has h1 (t) = 0. We now prove several important properties of the fractional integrals and fractional derivatives on arbitrary time scales. Proposition 3.1. Let α ∈ (n − 1, n], n ∈ N. If k ≤ n − 1, then (α)

hk (t, 0) = 0 . Proof. From (3) it follows that " " # # n−1 k α α X X z z i (α) α−i−1 hk (t, 0) = L−1 (t) = L−1 h∆ hk−i (0)z α−i−1 (t) − − k (0)z T T z k+1 z k+1 i=0 i=0 α z − z α−k−1 (t) = 0 . = L−1 T z k+1 Proposition 3.2. Let α ∈ (n − 1, n], n ∈ N. If k ≥ n, then 1 (α) −1 hk (t, 0) = LT (t) . z k+1−α 5

Proof. From (3) we have " # # " n−1 n−1 α α X X z z i (α) α−i−1 (t) = L−1 h∆ hk−i (0)z α−i−1 (t) hk (t) = L−1 − − k (0)z T T z k+1 z k+1 i=0 i=0 α z 1 = L−1 (t) = L−1 (t) . T T z k+1 z k+1−α Proposition 3.3. Let α ∈ (n − 1, n], n ∈ N. If c(t) ≡ m, m ∈ R, then c(α) (t) = 0. (α)

Proof. From the linearity of the inverse Laplace transform and the fact that h0 (t, 0) = 0, it (α) follows that c(α) (t) = (m × 1)(α) = (m × h0 (t, 0))(α) = m × h0 (t, 0) = m × 0 = 0. Proposition 3.4. Let α, β > 0. Then, ITβ (ITα f ) (t) = ITα+β f (t) . Proof. By definition we have h i i h −α−β −1 −β α s F (z) (t) = ITα+β f (t) . f ] (t) = L z L [I ITβ (ITα f ) (t) = L−1 T T T T Proposition 3.5. Let α, β ∈ (0, 1]. (a) If α + β ≤ 1, then h i (β) β−1 (α) z f (0) (t) . (t) = f (α+β) (t) − L−1 f (α) T (b) If 1 < α + β ≤ 2, then h i (β) h i −1 α+β−2 ∆ β−1 (α) z f (0) (t) . z f (0) (t) + L f (α) (t) = f (α+β) (t) − L−1 T T (c) If β ∈ (0, 1], then f∆ Proof. (a) Let α + β ≤ 1. Then

(β)

(t) = f (β+1) (t) .

i i (β) h h β−1 (α) β (α) (z) − z f (0) (t) z L f (t) = L−1 f (α) T h i h i −1 β−1 (α) α+β α+β−1 z f (0) (t) z F (z) − z f (0) (t) − L = L−1 T T h i z β−1 f (α) (0) (t) . = f (α+β) (t) − L−1 T 6

(b) For 1 < α + β ≤ 2 we have h i −1 α+β α+β−1 α+β−2 ∆ (α+β) L [f ] (z) − z f (0) − z f (0) (t) . f (t) = LT z Hence,

h i h i (β) −1 β−1 (α) α+β−2 ∆ z f (0) (t) . z f (0) (t) − L (t) = f (α+β) (t) + L−1 f (α) T T

(c) The intended relation follows from (b) by choosing α = 1. (α) (β) (t). However, the equality holds in particular cases: (t) 6= f (β) In general f (α) (α) (β) (t). (t) = f (β) Proposition 3.6. If α + β ≤ 1 and f (0) = 0, then f (α) Proof. It follows from item (a) of Proposition 3.5.

k

k

Proposition 3.7. Let α ∈ (n − 1, n], n ∈ N, and lim f ∆ (t) = f ∆ (0+ ) exist, k = 0, . . . , n − t→0+

1. The following equality holds:

n−1 X k f ∆ (0+ )hk (t) . ITα f (α) (t) = f (t) − k=0

Proof. Let F (z) = LT [f ] (z). Then, ITα

# " n−1 i i h h α−k−1 X k −1 −1 ∆ + z (α) −α (α) f (0 ) (t) (z) (t) = LT F (z) − (t) = LT z LT f f zα k=0 n−1 n−1 X k X k 1 −1 ∆ + = f (t) − f (0 )LT (t) = f (t) − f ∆ (0+ )hk (t) . z k+1 k=0

k=0

Proposition 3.8. Let α ∈ (n − 1, n], n ∈ N, and F (z) = LT [f ] (z). If lim F (z) = 0 and z→∞

F (z) α−k z→∞ z

lim

= 0, k ∈ {1, . . . , n}, then (ITα f )(α) (t) = f (t). k

Proof. Firstly let us notice that lim (ITα f )∆ (t) = 0 for k ∈ {0, . . . , n − 1}. For that we check t→0+

that lim zL [ITα f ] (z) = lim z Fz(z) α = 0. Nextly let us assume that it holds for i = 0, . . . , k − 1 z→∞ z→∞ and let us calculate ! k−1 i h X k−1−i α ∆i k F (z) α ∆k z (IT ) (0) (z) = lim z z lim zL (IT f ) − z→∞ z→∞ zα i=0 F (z) k F (z) = lim α−k−1 = 0 . = lim z z α z→∞ z z→∞ z Then we easily conclude that (ITα f )(α)

(t) =

L−1 T

F (z) zα α z

7

(t) = f (t) .

The convolution of two functions f : T → R and g : T × T → R on time scales, where g is rd-continuous with respect to the first variable, is defined in [4, 8]: Z t f (τ )g(t, σ(τ ))∆τ . (f ∗ g) (t) = 0

As function g we can consider, e.g., ec (t, t0 ) or hk (t, t0 ). Proposition 3.9. Let t0 ∈ T. If α ∈ (0, 1), then α α (f ∗ g(·, t0 ))(α) (t) = f ( 2 ) ∗ g( 2 ) (·, t0 ) (t) = f (α) ∗ g(·, t0 ) (t) , where we assume the existence of the involved derivatives.

Proof. From the convolution theorem for the generalized Laplace transform [8, Theorem 2.1], i h LT (f ∗ g(·, t0 ))(α) (z) = z α F (z)G(z) .

Hence,

h α h α i i −1 −1 α 2 F (z) (t)L 2 G(z) (t) z [z F (z)G(z)] (t) = L (f ∗ g(·, t0 ))(α) (t) = L−1 z T T T = f (α/2) ∗ g(α/2) (·, t0 ) (t) .

Equivalently, we can write

−1 −1 α (α) α [G(z)] (t) = f ∗ g(·, t ) (t) . [z F (z)] (t)L [z F (z)G(z)] (t) = L L−1 0 T T T

Acknowledgements This work is part of the first author’s Ph.D. project carried out at the University of Aveiro. The financial support of the Polytechnic Institute of Viseu and The Portuguese Foundation for Science and Technology (FCT), through the “Programa de apoio `a forma¸c˜ao avan¸cada de docentes do Ensino Superior Polit´ecnico”, Ph.D. fellowship SFRH/PROTEC/49730/2009, is here gratefully acknowledged. The second author was supported by BUT grant S/WI/1/08; the third author by the R&D unit CIDMA via FCT.

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