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Advances in Difference Equations

, 2013:259 | Cite as

Power functions and essentials of fractional calculus on isolated time scales

  • Tomáš KiselaEmail author
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

This paper is concerned about a recently suggested axiomatic definition of power functions on a general time scale and its consequences to fractional calculus. Besides a discussion of the existence and uniqueness of such functions, we derive an efficient formula for the computation of power functions of rational orders on an arbitrary isolated time scale. It can be utilized in the introduction and evaluation of fractional sums and differences. We also deal with the Laplace transform of such fractional operators, which, apart from solving of fractional difference equations, enables a more detailed comparison of our results with those in the relevant literature. Some illustrating examples (including special fractional initial value problems) are presented as well.

MSC:26E70, 39A12, 26A33, 44A10.

Keywords

fractional calculus power functions time scales convolution Laplace transform 

1 Introduction

In the last decades, continuous fractional calculus, i.e., a calculus dealing with integrals and derivatives of noninteger orders, evolved in a respected mathematical discipline with a number of physical and technical applications (for more information we refer to monographs, e.g., [1, 2]). Its key notion can be represented by the formula for the fractional integral of a real function f
D γ a f ( t ) = a t ( t τ ) γ 1 Γ ( γ ) f ( τ ) d τ , t > a , γ > 0 , Open image in new window
(1)

where γ is an order of integration. Note that the term ‘fractional’ is used due to a tradition, because (1) allows to utilize not only rational, but also real and, in principle, even complex values of γ.

Contrary to the continuous fractional calculus, which has been studied for more than 300 years, its discrete counterpart is a rather new topic. It started by the pioneering works [3, 4], where fractional operators were introduced for the q-calculus and the difference calculus, respectively. Utilizing a current notation, corresponding fractional sums of an appropriate function f can be written as
q γ a f ( q n a ) = k = 1 n ( 1 q 1 ) q k a ( q n a q k 1 a ) q ( γ 1 ) Γ q 1 ( γ ) f ( q k a ) , n Z , γ > 0 Open image in new window
(2)
( q > 0 Open image in new window, q 1 Open image in new window, a > 0 Open image in new window) and
γ a f ( n ) = k = 1 n ( n k + 1 ) ( γ 1 ) Γ ( γ ) f ( k ) , n Z , γ > 0 . Open image in new window
(3)

For more information on definitions of discrete power functions ( ) q ( γ ) Open image in new window and ( ) ( γ ) Open image in new window we refer, e.g., to [5, 6].

In the last decade, many authors proceeded with these ideas and studied properties of corresponding fractional difference equations as well as further generalizations of this theory, e.g., to h-calculus and ( q , h ) Open image in new window-calculus (see, e.g., [5, 7, 8, 9, 10, 11] and [12, 13, 14]). In these papers, the authors often preferred the notation based on the time scales theory, which easily exposes similarities among the results derived in q-calculus, h-calculus, ( q , h ) Open image in new window-calculus and the continuous case. However, this notation was employed only formally, since there was no general time scale definition of the power function occurring in (1)-(3), therefore, the achieved results could not be generally applied to other time scales.

On this account, some ideas regarding fundamental properties which should be met by power functions on time scales were outlined in [15]. In [16], the authors introduced fractional calculus (and by extension also power functions) via the Laplace transform. However, this approach is limited to settings with the well-defined Laplace transform (see [17]) and suffers by some technical difficulties, connected to the inverse Laplace transform. Recently, in [6] and [18], the authors independently suggested an axiomatic definition of power functions on an arbitrary time scale.

It this paper, we are going to continue in this research and discuss this definition and some of its consequences. In Section 2, we recall basics of the time scales theory and present some advanced results concerning a convolution and Laplace transform on time scales. Section 3 is devoted to fractional calculus on time scales. In particular, we state the definition of power functions and fractional operators, comment on their relations to approaches in [6, 15] and [18], and point out several interesting properties. In Section 4, we show that the suggested definition is well-posed on every isolated time scale and provides an effective formula for computing values of power functions of rational orders. We also discuss a singular behaviour of negative-order power functions implied by proposed axioms. Section 5 provides a link between our axiomatic definition and the Laplace transform approach developed in [16]. We also present a few examples involving special fractional difference equations, considered on various time scales. The paper is concluded by Section 6, summarizing the obtained results and discussing some induced open problems.

2 Time scales preliminaries

A time scale Open image in new window is an arbitrary nonempty closed subset of real numbers, e.g., the reals ℝ, arithmetic sequence h Z = { n h ; n Z } Open image in new window ( h > 0 Open image in new window), geometric sequence q Z ¯ = cl { q n ; n Z } Open image in new window ( q > 0 Open image in new window, q 1 Open image in new window) or their combinations denoted as T ( q , h ) Open image in new window (see [5]). If inf T = a > Open image in new window, we denote T κ = T a Open image in new window, otherwise T κ = T Open image in new window. For any t T Open image in new window, we define the forward and backward jump operator σ ( t ) = inf { τ T ; τ > t } Open image in new window and ρ ( t ) = sup { τ T ; τ < t } Open image in new window, respectively. If ρ ( t ) < t < σ ( t ) Open image in new window for all t T Open image in new window such that inf T < t < sup T Open image in new window, we speak of an isolated time scale. We also recall the forward and backward graininess function μ ( t ) = σ ( t ) t Open image in new window and ν ( t ) = t ρ ( t ) Open image in new window, respectively. For detailed information on the time scales theory, we refer to [19, 20]. In the sequel, we utilize a notation ( a , b ) M = ( a , b ) M Open image in new window, where M is an arbitrary subset of reals (analogously for closed intervals).

The symbol f ( t ) Open image in new window is the nabla derivative of f : T R Open image in new window at t T κ Open image in new window. For our purposes, it is sufficient to recall its form on isolated time scales when
f ( t ) = f ( t ) f ( ρ ( t ) ) ν ( t ) Open image in new window
and m f ( t ) = ( m 1 f ( t ) ) Open image in new window ( m Z Open image in new window, m > 1 Open image in new window). Similarly, we omit a general definition of nabla integral and specify only its isolated time scale version
a b f ( τ ) τ = τ ( a , b ] T ν ( τ ) f ( τ ) , a a f ( τ ) τ = 0 Open image in new window
(4)

for f being bounded on [ a , b ] T Open image in new window. Of course, it is possible to consider also the delta derivatives and integrals. However, the nabla version seems to be more suitable for fractional calculus as outlined, e.g., in [5, 14].

Time scale derivatives and integrals are related by several useful rules (see, e.g., [20]), e.g.,
s t ( f ( τ ) g ( τ ) ) τ = s t f ( τ ) g ( τ ) τ + s t f ( ρ ( τ ) ) g ( τ ) τ , Open image in new window
(5)
s t f ( τ ) τ = f ( t ) f ( s ) , Open image in new window
(6)
t s t y ( t , τ ) τ = s ρ ( t ) t y ( t , τ ) τ + y ( t , t ) , Open image in new window
(7)
s s t y ( τ , s ) τ = s t s y ( τ , s ) τ y ( s , ρ ( s ) ) , Open image in new window
(8)

where f, g and y are real functions such that the integrals exist and t Open image in new window represents the derivative with respect to t (an omitted index implies the derivative with respect to the first variable).

To utilize the time scales theory as a unification of continuous and discrete fractional calculus, we have to introduce reasonable extensions of standard definitions of some related notions (see, e.g., [20]). In this connection, we recall the family of monomials h ˆ m Open image in new window defined for integers m via the recursion
h ˆ 0 ( t , s ) = 1 , Open image in new window
(9)
h ˆ m ( t , s ) = s t h ˆ m 1 ( τ , s ) τ , m Z + . Open image in new window
(10)

Now, we present some advanced results, related to the Laplace transform and convolution on time scales. We start with the definition of nabla Laplace transform introduced in [17].

Definition 2.1 Let a T L Open image in new window, where T L Open image in new window is a time scale such that sup T L = Open image in new window and sup { ν ( t ) ; t T L } < Open image in new window. The Laplace transform of f : ( a , ) T L R Open image in new window is defined by
L a { f } ( z ) = a f ( τ ) e ˆ z ( a , ρ ( τ ) ) τ for  z D L ( f ) , Open image in new window

where D L ( f ) Open image in new window consists of all z C Open image in new window, for which the improper integral exists, and for which 1 z ν ( t ) 0 Open image in new window for all t ( a , ) T L Open image in new window. The symbol e ˆ z Open image in new window denotes the nabla time scale exponential function (see [20]).

Remark 2.2 Clearly, a necessary condition for existence of L a { f } ( z ) Open image in new window is lim τ f ( τ ) e z ( a , ρ ( τ ) ) = 0 Open image in new window.

By a modification of the proofs presented in [17, 21], we can prove analogues of many important properties of the Laplace transform known from continuous analysis. For our purposes, we recall
L a { h ˆ m ( , a ) } ( z ) = z m 1 , m Z 0 + , Open image in new window
(11)
L a { m f } ( z ) = z m L a { f } ( z ) j = 0 m 1 z j m j 1 f ( t ) | t = a , m Z + . Open image in new window
(12)

Regarding one of key results connected to the Laplace transform, the convolution theorem, we first have to introduce a convolution on time scales. This issue was solved in [22] for the case of the delta calculus. We adapt this approach for the nabla calculus.

Definition 2.3 Let a T Open image in new window. The shift of f : ( a , ) T R Open image in new window is defined as the solution of the shifting problem
t u ( t , ρ ( s ) ) = s u ( t , s ) , s , t T , t > s > a , u ( t , a ) = f ( t ) , t T , t > a . Open image in new window
Remark 2.4 (i) If the graininess is constant, the shifting problem has a unique solution u ( t , s ) = f ( t s ) Open image in new window.
  1. (ii)

    For all ξ T Open image in new window, the shift of h ˆ m ( t , ξ ) Open image in new window, e ˆ λ ( t , ξ ) Open image in new window ( m Z 0 + Open image in new window, λ R Open image in new window) is h ˆ m ( t , s ) Open image in new window, e ˆ λ ( t , s ) Open image in new window, respectively (see [22]).

     
Definition 2.5 Let s , t T Open image in new window be such that t s Open image in new window. The convolution of f , g : T R Open image in new window is defined by
( f g ) ( t , s ) = s t f ˆ ( t , ρ ( τ ) ) g ( τ ) τ , where  f ˆ  is the shift of  f . Open image in new window
Theorem 2.6 Let f, g be functions such that L a { f } ( z ) Open image in new window, L a { g } ( z ) Open image in new window exist. Then
L a { ( f g ) ( , a ) } ( z ) = L a { f } ( z ) L a { g } ( z ) . Open image in new window

Proof Analogous assertion was proved in [22] for the delta calculus. We utilize a similar technique with several adjustments better suiting our purposes.

Utilizing the formulas a T a t f ( t , τ ) τ t = a T ρ ( τ ) T f ( t , τ ) t τ Open image in new window and e ˆ z ( t , s ) = e ˆ z ( t , ξ ) e z ( ξ , s ) Open image in new window (see [20]), we can write
L a { ( f g ) ( , a ) } ( z ) = a ( f g ) ( t , a ) e ˆ z ( a , ρ ( t ) ) t = a e ˆ z ( a , ρ ( t ) ) a t f ˆ ( t , ρ ( τ ) ) g ( τ ) τ t = a g ( τ ) e ˆ z ( a , ρ ( τ ) ) ρ ( τ ) e ˆ z ( ρ ( τ ) , ρ ( t ) ) f ˆ ( t , ρ ( τ ) ) t τ = a g ( τ ) e ˆ z ( a , ρ ( τ ) ) Ψ ( ρ ( τ ) ) τ , Open image in new window
(13)
where Ψ ( s ) = s e ˆ z ( s , ρ ( t ) ) f ˆ ( t , s ) t Open image in new window. We show that Ψ ( s ) Open image in new window is constant, i.e., its derivative is zero. Indeed, employing (8), (5), Definition 2.3, Remark 2.4(ii), (6) and Remark 2.2, yields
s Ψ ( s ) = s s e ˆ z ( s , ρ ( t ) ) f ˆ ( t , s ) t = s s ( e ˆ z ( s , ρ ( t ) ) f ˆ ( t , s ) ) t f ˆ ( s , ρ ( s ) ) = s ( s e ˆ z ( s , ρ ( t ) ) f ˆ ( t , ρ ( s ) ) + e ˆ z ( s , ρ ( t ) ) s f ˆ ( t , s ) ) t f ˆ ( s , ρ ( s ) ) = s ( t e ˆ z ( s , t ) f ˆ ( t , ρ ( s ) ) e ˆ z ( s , ρ ( t ) ) t f ˆ ( t , ρ ( s ) ) ) t f ˆ ( s , ρ ( s ) ) = s t ( e ˆ z ( s , t ) f ˆ ( t , ρ ( s ) ) ) t + f ˆ ( s , ρ ( s ) ) = [ e ˆ z ( s , t ) f ˆ ( t , ρ ( s ) ) ] s f ˆ ( s , ρ ( s ) ) = 0 . Open image in new window

Hence, substituting Ψ ( ρ ( τ ) ) = Ψ ( a ) Open image in new window into (13), we obtain L a { ( f g ) ( , a ) } ( z ) = L a { g } ( z ) L a { f } ( z ) Open image in new window. □

3 Foundations of fractional calculus on time scales

As indicated by (1)-(3), the issue of fractional calculus in the time scales theory is connected to the problem of defining appropriate power functions. A standard method, based on a generalization of explicit formulas for monomials to noninteger values of m, is effective only in special cases ( h Z Open image in new window, q Z ¯ Open image in new window, T ( q , h ) Open image in new window, ℝ). In general, the explicit formulas for monomials are not known, therefore, a different approach is required.

Recently, in [6] and [18], there were independently suggested quite similar axiomatic definitions of time scales power functions. We develop the one presented in [18].

Definition 3.1 Let s , t T Open image in new window, and let β , γ ( 1 , ) Open image in new window. The time scales power functions h ˆ β ( t , s ) Open image in new window are defined as a family of nonnegative functions satisfying
  1. (i)

    s t h ˆ β ( t , ρ ( τ ) ) h ˆ γ ( τ , s ) τ = h ˆ β + γ + 1 ( t , s ) Open image in new window for t s Open image in new window,

     
  2. (ii)

    h ˆ 0 ( t , s ) = 1 Open image in new window for t s Open image in new window,

     
  3. (iii)

    h ˆ β ( t , t ) = 0 Open image in new window for 0 < β < 1 Open image in new window.

     

Further, the parameter β in h ˆ β ( t , s ) Open image in new window is called the order of the power function h ˆ β ( t , s ) Open image in new window.

Compared to the definition in [6], the key underlying ideas are basically the same. However, some additional conditions are prescribed in [6]. The main difference lies in the condition h ˆ β ( t , s ) = 0 Open image in new window, whenever t s Open image in new window for all β > 1 Open image in new window. In our opinion, this assumption creates an inconsistence with the definition of time scales monomials ( h ˆ 0 ( t , s ) = 1 Open image in new window for all s , t T Open image in new window) and with the well-known power functions on T = R Open image in new window ( h ˆ β ( t , s ) = ( t s ) β Γ ( β + 1 ) Open image in new window is in general undefined for t < s Open image in new window). Thus, we prefer to leave the values of h ˆ β ( t , s ) Open image in new window undefined for t < s Open image in new window (they do not occur in fractional calculus anyway). The consequences for t = s Open image in new window will be discussed later. We point out that for t > s Open image in new window, our definition coincides with [[6], Definition 3.2] and with historically established power functions on ℝ, h Z Open image in new window, q Z ¯ Open image in new window, T ( q , h ) Open image in new window and their subintervals (see [6] and [18]). We note that condition (i) in Definition 3.1 first appeared without a deeper analysis or discussion in [15].

Remark 3.2 The well-posedness of Definition 3.1 was not discussed in [6] nor [18]. In Section 4, we show that it determines uniquely at least the power functions of rational orders on isolated time scales. In the remaining part of the current section, we assume that the power functions on corresponding time scales exist.

Lemma 3.3 Let m Z + Open image in new window, β ( 1 , ) Open image in new window, s , t T Open image in new window be such that t > s Open image in new window. Then
m h ˆ β ( t , s ) = { h ˆ β m ( t , s ) , β > m 1 , 0 , β { 0 , 1 , , m 1 } . Open image in new window

Proof Let m = 1 Open image in new window and β > 0 Open image in new window. Then, utilizing Definition 3.1(i) and (7), we can write h ˆ β ( t , s ) = s t h ˆ β 1 ( τ , s ) τ = h ˆ β 1 ( t , s ) Open image in new window. The first part of the assertion now follows by the induction principle. The second part is a consequence of h ˆ 0 ( t , s ) = 0 Open image in new window. □

Remark 3.4 Lemma 3.3 does not discuss the case β ( 1 , m 1 ] { 0 , 1 , , m 1 } Open image in new window due to an occurrence of power functions of order less than −1. Since such functions cannot be included in Definition 3.1 (e.g., due to unintegrability in real analysis), the extension of Lemma 3.3 can serve as their definition, e.g.,
h ˆ β ( t , s ) = β h ˆ β β ( t , s ) , β ( , 1 ) Z , s , t T , t σ β ( s ) , Open image in new window

where β Open image in new window is the ceiling function β = min { m Z ; m β } Open image in new window. The presented approach does not allow to introduce power functions of negative integer orders.

Considering the well-defined power functions, we can introduce fractional operators in the frame of the time scales theory (see [6, 15] and [12, 18]) following the continuous paradigm (see, e.g., [1, 2]).

Definition 3.5 Let γ 0 Open image in new window, α > 0 Open image in new window, a ˜ , a , b T Open image in new window be such that a ˜ a < b Open image in new window. Then for a function f : ( a ˜ , b ] T R Open image in new window we define
  1. (i)
    the fractional integral of order γ > 0 Open image in new window with the lower limit a as
    γ a f ( t ) = a t h ˆ γ 1 ( t , ρ ( τ ) ) f ( τ ) τ , t [ a , b ] T ( a ˜ , b ] T Open image in new window
     
and for γ = 0 Open image in new window, we put 0 a f ( t ) = f ( t ) Open image in new window,
  1. (ii)
    the Riemann-Liouville fractional derivative of order α with the lower limit a as
    α a f ( t ) = α a α α f ( t ) , t [ σ ( a ) , b ] T ( σ ( a ˜ ) , b ] T , Open image in new window
     
  2. (iii)
    and the Caputo fractional derivative of order α with the lower limit a ( a > a ˜ Open image in new window) as
    α a C f ( t ) = a α α α f ( t ) , t [ σ ( a ) , b ] T . Open image in new window
     
Remark 3.6 (i) For γ = 1 Open image in new window, Definition 3.5(i) reduces to a formula for the antiderivative 1 a f ( t ) = a t f ( τ ) τ Open image in new window known from the time scales theory (see [20]).
  1. (ii)

    We can see a formal agreement of Definition 3.5(i) with (1)-(3) (for detailed information see [6]).

     
Remark 3.7 (i) If a > a ˜ Open image in new window, we obtain a usual definitions of the difference calculus, q-calculus and ( q , h ) Open image in new window-calculus. In this case, it holds γ a f ( t ) | t = a = 0 Open image in new window on every time scale.
  1. (ii)

    If a = a ˜ Open image in new window, the integrated function is undefined at the lower limit of the integral. Such functions play an important role in continuous fractional calculus and, as we demonstrate in the following sections, can appear also on isolated time scales.

     

In [6] and [18], the authors proved many time scales analogues of important properties of fractional operators known from continuous analysis, e.g., the index law γ a a β f ( t ) = a ( γ + β ) f ( t ) Open image in new window ( γ , β > 0 Open image in new window). One of the key relationships of the fractional calculus is a power rule, which can be written as

Theorem 3.8 Let α R Open image in new window, β ( 1 , ) Open image in new window and s , t T Open image in new window be such that t > s Open image in new window. Then it holds
α a h ˆ β ( t , a ) = { h ˆ β α ( t , s ) , β > α 1 , 0 , β { α α , α α + 1 , , α 1 } . Open image in new window

Proof For α < 0 Open image in new window it follows by Definitions 3.5(i) and 3.1(i). For α > 0 Open image in new window, we utilize also Lemma 3.3. □

By similar arguments, we can derive an analogous formula for Caputo fractional derivative, i.e.,

Theorem 3.9 Let α > 0 Open image in new window, β ( 1 , ) Open image in new window and s , t T Open image in new window be such that t > s Open image in new window. Then it holds
α a C h ˆ β ( t , a ) = { h ˆ β α ( t , s ) , β > α 1 , 0 , β { 0 , 1 , , α 1 } . Open image in new window

4 Power functions on isolated time scales

Although the axiomatic definition of power functions has been proposed, its feasibility, i.e., the question of existence and uniqueness, has not been discussed yet. In [6], the author presents examples of such functions on ℝ, h Z Open image in new window, q Z Open image in new window or T ( q , h ) Open image in new window, but those functions are known historically, they are not originally based on the proposed axiomatic definition.

In this section, we analyze Definition 3.1 on an arbitrary isolated time scale. We constructively prove the existence and uniqueness of rational-order power functions, which enables us to present graphs of power functions on some chosen time scales for the first time. Further, we discuss some properties of negative-order power functions, which are, to the author’s knowledge, quite new in the frame of isolated time scales.

Lemma 4.1 Let β ( 1 , ) Open image in new window, s , t T κ Open image in new window be such that t > s Open image in new window. Then h ˆ β Open image in new window solves the shifting problem, i.e.,
t h ˆ β ( t , ρ ( s ) ) = s h ˆ β ( t , s ) . Open image in new window
Proof For β Z 0 + Open image in new window, the result is known (see Remark 2.4(ii)). Thus, for β ( 1 , ) Z 0 + Open image in new window, we can write
0 = t h ˆ β + 1 ( t , ρ ( s ) ) + s h ˆ β + 1 ( t , s ) = t ρ ( s ) t h ˆ β ( t , ρ ( τ ) ) h ˆ β β ( τ , ρ ( s ) ) τ + s s t h ˆ β ( t , ρ ( τ ) ) h ˆ β β ( τ , s ) τ = ρ ( s ) ρ ( t ) t h ˆ β ( t , ρ ( τ ) ) h ˆ β β ( τ , ρ ( s ) ) τ + h ˆ β ( t , ρ ( t ) ) h ˆ β β ( t , ρ ( s ) ) + s t h ˆ β ( t , ρ ( τ ) ) s h ˆ β β ( τ , s ) τ h ˆ β ( t , ρ ( s ) ) h ˆ β β ( s , ρ ( s ) ) , Open image in new window
(14)
where we employ (7) and (8). Utilizing (6) and (5), we have
h ˆ β ( t , ρ ( t ) ) h ˆ β β ( t , ρ ( s ) ) h ˆ β ( t , ρ ( s ) ) h ˆ β β ( s , ρ ( s ) ) = s t τ ( h ˆ β ( t , ρ ( τ ) ) h ˆ β β ( τ , ρ ( s ) ) ) τ = s t τ h ˆ β ( t , ρ ( τ ) ) h ˆ β β ( ρ ( τ ) , ρ ( s ) ) τ + s t h ˆ β ( t , ρ ( τ ) ) τ h ˆ β β ( τ , ρ ( s ) ) τ = ρ ( s ) ρ ( t ) τ h ˆ β ( t , τ ) h ˆ β β ( τ , ρ ( s ) ) τ + s t h ˆ β ( t , ρ ( τ ) ) τ h ˆ β β ( τ , ρ ( s ) ) τ . Open image in new window
Substituting this into (14), we get
t h ˆ β + 1 ( t , ρ ( s ) ) + s h ˆ β + 1 ( t , s ) = ρ ( s ) ρ ( t ) ( t h ˆ β ( t , ρ ( τ ) ) + τ h ˆ β ( t , τ ) ) h ˆ β β ( τ , ρ ( s ) ) τ + s t h ˆ β ( t , ρ ( τ ) ) ( s h ˆ β β ( τ , s ) + τ h ˆ β β ( τ , ρ ( s ) ) ) τ . Open image in new window
Now, using (7) and (8), we evaluate the derivatives of h ˆ β β Open image in new window, i.e.,
s h ˆ β β ( τ , s ) = s s τ h ˆ β β 1 ( τ , ρ ( τ 1 ) ) τ 1 = h ˆ β β 1 ( τ , ρ ( s ) ) , τ h ˆ β β ( τ , ρ ( s ) ) = τ ρ ( s ) τ h ˆ β β 1 ( τ 1 , ρ ( s ) ) τ 1 = h ˆ β β 1 ( τ , ρ ( s ) ) . Open image in new window
Thus, we arrive at
t h ˆ β + 1 ( t , ρ ( s ) ) + s h ˆ β + 1 ( t , s ) = ρ ( s ) ρ ( t ) ( t h ˆ β ( t , ρ ( τ ) ) + τ h ˆ β ( t , τ ) ) h ˆ β β ( τ , ρ ( s ) ) τ = 0 , Open image in new window

which implies the assertion. □

Remark 4.2 Lemma 4.1 enables us to write Definition 3.1(i) via the convolution. Indeed, setting f ( t ) = h ˆ β ( t , s ) Open image in new window and g ( t ) = h ˆ γ ( t , s ) Open image in new window, Definition 2.5 yields
( h ˆ β h ˆ γ ) ( t , s ) = h ˆ β + γ + 1 ( t , s ) , t s , β , γ > 1 . Open image in new window

Theorem 4.3 Let Open image in new window be an isolated time scale, and let r ( 1 , ) Q Open image in new window. Then Definition  3.1 determines uniquely the power function h ˆ r ( t , s ) Open image in new window for all s , t T Open image in new window such that t > s Open image in new window.

Proof We prove this assertion constructively. Due to Definition 3.1(ii), the power functions for r Z 0 + Open image in new window reduce to monomials uniquely given by (9), (10). Now, let r = u w + 1 Open image in new window ( u Z Open image in new window, w Z + Open image in new window, u w Open image in new window) be the order of the power function and a , t T Open image in new window be such that t > a Open image in new window. Subsequent application of Definition 3.1(i) yields
Setting t = σ n ( a ) Open image in new window, we expand the left-hand side of the equation (see Definition 2.5 and (4)) and get
i 1 = n k + 1 n ν ( σ i 1 ( a ) ) h ˆ r ( σ n ( a ) , σ i 1 1 ( a ) ) i 2 = n k + 1 i 1 ν ( σ i 2 ( a ) ) h ˆ r ( σ i 1 ( a ) , σ i 2 1 ( a ) ) i w = n k + 1 i w 1 ν ( σ i w ( a ) ) h ˆ r ( σ i w 1 ( a ) , σ i w 1 ( a ) ) h ˆ r ( σ i w ( a ) , σ n k ( a ) ) = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) . Open image in new window
It turns out that the key parameter in solving of this equation is k. For k = 1 Open image in new window, we obtain a simple form ν w ( σ n ( a ) ) ( h ˆ r ( σ n ( a ) , σ n 1 ( a ) ) ) w + 1 = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) Open image in new window. Utilizing the property of monomials h ˆ m ( t , ρ ( t ) ) = ν m ( t ) Open image in new window ( m Z 0 + Open image in new window), we can write
h ˆ r ( t , ρ ( t ) ) = ν r ( t ) , t T κ . Open image in new window
(15)

Note, that only positive values of h ˆ r ( t , ρ ( t ) ) Open image in new window are allowed even for w being odd. Indeed, if we consider h ˆ r ( t , s ) = ( h ˆ r 1 2 h ˆ r 1 2 ) ( t , s ) Open image in new window for s = ρ ( t ) Open image in new window, we get h ˆ r ( t , ρ ( t ) ) = ν ( t ) ( h ˆ r 1 2 ( t , ρ ( t ) ) ) 2 > 0 Open image in new window.

Now let k > 1 Open image in new window. For the sake of brevity we introduce the symbol φ n k = h ˆ r ( σ n ( a ) , σ n k ( a ) ) Open image in new window. By shifting all indices by n k + 1 Open image in new window and rearranging the sums, we get
i 1 = 0 k 1 i 2 = 0 i 1 i w = 0 i w 1 ( = 1 w ν ( σ n k + 1 + i ( a ) ) ) ( = 1 w 1 φ n k + 1 + i 1 + i i + 1 ) φ n k i 1 φ n k + 1 + i w 1 + i w = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) . Open image in new window
Further, we perform the transformation i 1 k 1 i 1 Open image in new window, i 2 k 1 i 1 i 2 Open image in new window etc. to obtain
i 1 = 0 k 1 i 2 = 0 k 1 i 1 i w = 0 k 1 p = 1 w 1 i p ( = 1 w ν ( σ n p = 1 i p ( a ) ) φ n p = 1 1 i p 1 + i ) φ n p = 1 w i p k p = 1 w i p = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) . Open image in new window
For an easier work, we introduce the multiindex notation, namely,
i = ( i 1 , i 2 , , i w ) , | i | = p = 1 i p , { 1 , 2 , , w } , | i | = | i | w and i ¯ = max { 1 , , w } i . Open image in new window
Then we can write shortly
0 | i | k 1 φ n | i | k | i | = 1 w ν ( σ n | i | ( a ) ) φ n | i | 1 i + 1 = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) . Open image in new window
Now, we extract the summands involving φ n k Open image in new window, which are characterized by | i | = 0 Open image in new window or i ¯ = k 1 Open image in new window, i.e.,
φ n k ( ν w ( σ n ( a ) ) ( φ n 1 ) w + φ n k + 1 1 = 1 w ν w + 1 ( σ n k + 1 ( a ) ) ν 1 ( σ n ( a ) ) ( φ n k + 1 1 ) w ( φ n 1 ) 1 ) + 0 < | i | k 1 i ¯ k 1 φ n | i | k | i | = 1 w ν ( σ n | i | ( a ) ) φ n | i | 1 i + 1 = h ˆ u + w ( σ n ( a ) , σ n k ( a ) ) . Open image in new window
Evaluating the terms φ 1 Open image in new window by (15), we can express h ˆ r ( t , ρ k ( t ) ) = φ n k Open image in new window ( t = σ n ( a ) Open image in new window) and arrive at recursion
h ˆ r ( t , ρ k ( t ) ) = h ˆ u + w ( t , ρ k ( t ) ) 0 < | i | k 1 i ¯ k 1 h ˆ r ( ρ | i | ( t ) , ρ k ( t ) ) = 1 w ν ( ρ | i | ( t ) ) h ˆ r ( ρ | i | 1 ( t ) , ρ | i | + 1 ( t ) ) = 0 w ν ( r + 1 ) ( t ) ν ( r + 1 ) ( w ) ( ρ k 1 ( t ) ) , k > 1 , Open image in new window
(16)

which, supplied with (15), determines uniquely h ˆ r ( t , s ) Open image in new window for all r ( 1 , ) Q Open image in new window, s , t T Open image in new window such that t > s Open image in new window. □

Remark 4.4 As a byproduct of the proof of Theorem 4.3, we have obtained the recursive formula (16). Hence, employing (15), (16), there can be created an effective algorithm evaluating numerically the power functions of rational orders on a given isolated time scale. Note that for special isolated time scales such that ν ( t ) ν ( s ) Open image in new window for all s , t T κ Open image in new window, Lemma 4.1 yields a more simple formula
h ˆ r ( t , s ) = ν ( t ) h ˆ r ( t , σ ( s ) ) ν ( σ ( s ) ) h ˆ r ( ρ ( t ) , s ) ν ( t ) ν ( σ ( s ) ) . Open image in new window

Remark 4.5 Definition 3.1 prescribes nonnegative values of h ˆ β ( t , s ) Open image in new window. Although so far known power functions fulfilling (15), (16) have this property (i.e., power functions on ℝ, h Z Open image in new window, q Z ¯ Open image in new window and T ( q , h ) Open image in new window, as well as the ones obtained numerically), a general proof of nonnegativity of solutions of (15), (16) is still an open problem. However, since (15), (16) represent the unique functions satisfying Definition 3.1, an occurrence of a negative value of a power function on some time scale would imply a reevaluation of the nonnegativity condition.

Theorem 4.6 Let Open image in new window be an isolated time scale, let t T Open image in new window, and let r ( 1 , ) Q Open image in new window. Then
  1. (i)

    h ˆ r ( t , t ) = 0 Open image in new window for r > 0 Open image in new window,

     
  2. (ii)

    h ˆ r ( t , t ) = 1 Open image in new window for r = 0 Open image in new window,

     
  3. (iii)

    the value of h ˆ r ( t , t ) Open image in new window is unbounded for 1 < r < 0 Open image in new window.

     

Proof First, let r = 0 Open image in new window. Then h ˆ 0 ( t , t ) = 1 Open image in new window, as prescribed by Definition 3.1(ii). Let r > 0 Open image in new window. For 0 < r < 1 Open image in new window, the zero value is given by Definition 3.1(iii). For r 1 Open image in new window, the assertion follows by subsequent application of the relation h ˆ r ( t , t ) = t t h ˆ r 1 ( τ , t ) τ = 0 Open image in new window, where the zero value is implied by (4). Finally, let 1 < r < 0 Open image in new window. Since r 1 > 1 Open image in new window, we can write by Definition 3.1(i), (ii) t t h ˆ r 1 ( t , ρ ( τ ) ) h ˆ r ( τ , t ) τ = h ˆ 0 ( t , t ) = 1 Open image in new window. Considering (4) and (15), we conclude that the value of h ˆ r ( t , t ) Open image in new window is unbounded. □

Remark 4.7 (i) Theorem 4.6 corresponds with the continuous case, where power functions ( t s ) r Γ ( r + 1 ) Open image in new window, r ( 1 , 0 ) Open image in new window are unbounded in a neighborhood of t = s Open image in new window. It is known that in the continuous case, we can utilize an improper integration to calculate t t ( τ t ) r Γ ( r + 1 ) d τ = 0 Open image in new window. On isolated time scales, it follows by Definition 3.1(iii).
  1. (ii)

    The power functions of order r ( 1 , 0 ) Q Open image in new window represent a new class of ‘singular’ functions on time scales. Besides continuous calculus, this phenomenon appeared, e.g., in [9], where the singular functions were introduced on q Z ¯ Open image in new window with a singularity at the cluster point.

     
  2. (iii)

    Considering Remark 3.4, power functions of orders r ( 2 , 1 ) Open image in new window have unbounded values h ˆ r ( t , t ) Open image in new window and h ˆ r ( t , ρ ( t ) ) Open image in new window. It implies that such functions are not integrable, which agrees with the continuous analysis.

     
Corollary 4.8 Let γ > 0 Open image in new window, r ( 1 , 0 ) Q Open image in new window, a T Open image in new window and Open image in new window be an isolated time scale. Then it holds
  1. (i)

    γ a h ˆ r ( t , a ) | t = a = 0 Open image in new window for γ > r Open image in new window,

     
  2. (ii)

    γ a h ˆ r ( t , a ) | t = a = 1 Open image in new window for γ = r Open image in new window,

     
  3. (iii)

    γ a h ˆ r ( t , a ) | t = a Open image in new window is unbounded for γ < r Open image in new window.

     

Proof The assertion follows by Theorems 3.8 and 4.6. □

To illustrate and clarify the notion of the power functions on time scales, we conclude this section by the following examples.

Example 4.9 It can be proved that Definition 3.1 (and also recursive formulas (15), (16)) is satisfied by the power functions h ˆ r ( t , s ) Open image in new window, historically established on ℝ, h Z Open image in new window, q Z ¯ Open image in new window and T ( q , h ) Open image in new window provided t > s Open image in new window (this matter was discussed, e.g., in [6]). We demonstrate this on the case T = Z Open image in new window, when Definition 3.1 yields
h ˆ r ( t , s ) = { ( k + r 1 k 1 ) , k > 0 , 0 , k = 0 , undefined , k < 0 , where  r ( 0 , ) Q , t = σ k ( s ) . Open image in new window
(17)
Indeed, properties (17)2 and (17)3 are directly implied by Theorem 4.6(i) and Definition 3.1, respectively. For r Z + Open image in new window, the property (17)1 is reduced into the classical result known from the theory of time scales monomials. To confirm the validity of (17)1 also for rational values of r, we employ the induction principle. Applying (15), we can easily verify the case k = 1 Open image in new window. Now, denote r = u w + 1 Open image in new window ( u , w Z + Open image in new window) and assume that (17)1 holds for k 1 Open image in new window. Thus, (16) yields
h ˆ u w + 1 ( n , n k ) = 1 w + 1 ( ( k + u + w 1 k 1 ) 0 < | i | k 1 i ¯ k 1 ( k | i | + u w + 1 1 k | i | 1 ) = 1 w ( | i | | i | 1 + u w + 1 | i | | i | 1 ) ) . Open image in new window
Treating the occurring summation 0 < | i | k 1 i ¯ k 1 Open image in new window is in general technically quite difficult. Hence, for the sake of simplicity, we perform the calculations only for w = 2 Open image in new window. We get
h ˆ u 3 ( n , n k ) = 1 3 ( ( k + u 1 k 1 ) i 1 = 1 k 2 ( i 1 + u 3 i 1 ) i 2 = 0 k 1 i 1 ( k i 1 i 2 1 + u 3 k i 1 i 2 1 ) ( i 2 + u 3 i 2 ) i 2 = 1 k 2 ( k i 2 1 + u 3 k i 2 1 ) ( i 2 + u 3 i 2 ) ) = 1 3 ( ( k + u 1 k 1 ) i 1 = 1 k 2 ( i 1 + u 3 i 1 ) ( k i 1 1 + u 3 k i 1 1 ) ( k 1 + 2 u 3 k 1 ) + 2 ( k 1 + u 3 k 1 ) ) = 1 3 ( ( k + u 1 k 1 ) ( k 1 + 3 u 3 k 1 ) + ( k 1 + 2 u 3 k 1 ) ( k 1 + 2 u 3 k 1 ) + 3 ( k 1 + u 3 k 1 ) ) = ( k + u 3 1 k 1 ) , Open image in new window

where we utilize the well-known Vandermonde’s convolution [[23], formula (6:5:5)] twice. Note that similar technique can be utilized also for power functions on h Z Open image in new window, q Z ¯ Open image in new window and T ( q , h ) Open image in new window provided the corresponding version of Vandermonde’s convolution is employed.

Example 4.10 We consider isolated time scales
T 1 = Z = { t n ; t n = n , n Z } , T 2 = { t n ; t n = 0.2 n 2 + n 2 , n Z } , where  ξ = max { m Z ; m ξ } , T 3 = { t n ; t n = n  for  n 4  and  t n = 4 + 0.2 ( n 4 )  for  n 5 , n Z } , T 4 = { t n ; t n  are randomly generated points such that  0 t n 5 , n Z } , T 5 = { t n ; t n = 0.1 n  for  n 10  and  t n = 2.5 + 0.3 ( n 11 )  for  n 11 , n Z } . Open image in new window
For these time scales, we depict the power functions of orders 1 2 Open image in new window and 1 3 Open image in new window. For a better comparison, we plot at Figures 1-4 also their differences with respect to power function in ℝ. We can observe a trend of tending to the values of continuous power functions with decreasing values of graininess and increasing t.

5 Laplace transform

The Laplace transform is a tool of a key importance regarding solving and analyzing of fractional differential equations in the continuous (see, e.g., [1, 2]), as well as in the discrete case (see, e.g., [7] and [14]). In this section, we deal with the Laplace transform of the power functions on isolated time scales T L Open image in new window (see Definition 2.1), which implies also the Laplace transforms of fractional operators. We demonstrate a usage of the Laplace transform method for solving of initial value problems by a simple example.

Theorem 5.1 Let a T L Open image in new window, and let r ( 1 , ) Q Open image in new window. Then it holds
L a { h ˆ r ( , a ) } ( z ) = z r 1 . Open image in new window
Proof Let r = u w + 1 Open image in new window ( u Z Open image in new window, w Z + Open image in new window, u w Open image in new window). It is known that L a { h ˆ u + w ( , a ) } ( z ) = z u w 1 Open image in new window (see (11)). Assume that g is a function such that L a { g } ( z ) = z r 1 Open image in new window. Then Theorem 2.6 implies
On the other hand, a subsequent application of Definition 3.1(i) (see Remark 4.2) yields
Theorem 4.3 guarantees the uniqueness of h ˆ r Open image in new window, therefore, it follows
h ˆ r ( t , a ) = g ( t ) L a { h ˆ r ( , a ) } ( z ) = z r 1 , Open image in new window

which concludes the proof. □

Now, we can write the time scales analogues of some relations of continuous fractional calculus.

Corollary 5.2 Let γ , α ( 0 , ) Q Open image in new window, a T L Open image in new window and f : T L R Open image in new window be such that L a { f } ( z ) Open image in new window exists. Then it holds
  1. (i)

    L a { a γ f } ( z ) = z γ L a { f } ( z ) Open image in new window,

     
  2. (ii)

    L a { a α f } ( z ) = z α L a { f } ( z ) j = 0 α 1 z j a α j 1 f ( t ) | t = a Open image in new window,

     
  3. (iii)

    L a { a C α f } ( z ) = z α L a { f } ( z ) j = 0 α 1 z α α + j a α j 1 f ( t ) | t = a Open image in new window.

     

Proof All three formulas are direct consequences of Theorem 5.1, supplied with (11), (12), Theorems 2.6 and 5.1. The proofs are analogous to the continuous paradigm (see, e.g., [2]). □

Remark 5.3 In [16], the authors introduce the fractional operators via the Laplace transform in the frame of the delta calculus. Note that this idea is transferable also to the nabla case. In this sense, Corollary 5.2 shows that Definition 3.1 is consistent for special time scales T L Open image in new window with the approach utilized in [16].

Corollary 5.2 enables us to utilize the Laplace transform method for an effective solving of some initial value problems. We point out that such results are not only symbolical, but due to (15), (16), they have a solid meaning, and it is possible to plot the solutions on every isolated time scale. We demonstrate this method by the following example.

Example 5.4 Let α ( 0 , 1 ] Q Open image in new window, a T L Open image in new window. Consider the pair of initial value problems with Riemann-Liouville and Caputo fractional operators
α a y ( t ) = 4 5 , t > a , a C α y ( t ) = 4 5 , t > a , α 1 a y ( t ) | t = a = y α 1 , y ( a ) = y 0 , Open image in new window
where y α 1 , y 0 R Open image in new window. Note that the initial condition α 1 a y ( t ) | t = a = y α 1 Open image in new window can be interpreted via Corollary 4.8 (see also [14]). Utilizing the Laplace transform, namely Corollary 5.2 and (11), we arrive at
L a { y RL } ( z ) = 4 5 1 z 1 + α + y α 1 1 z α , L a { y C } ( z ) = 4 5 1 z 1 + α + y 0 1 z . Open image in new window
Applying Theorem 5.1, we obtain the solutions
y RL ( t ) = 4 5 h ˆ α ( t , a ) + y α 1 h ˆ α 1 ( t , a ) , y C ( t ) = 4 5 h ˆ α ( t , a ) + y 0 h ˆ 0 ( t , a ) . Open image in new window
We point out that we would get the same results also for T = R Open image in new window. In Figures 5 and 6, we plot the solutions on time scales T 1 Open image in new window, T 2 Open image in new window and T 5 Open image in new window introduced in Example 4.10.
Figure 5

y RL ( t ) Open image in new window for α = 1 3 Open image in new window , y 2 3 = 3 4 Open image in new window .

Figure 6

y C ( t ) Open image in new window for α = 1 3 Open image in new window , y 0 = 3 4 Open image in new window .

6 Conclusions and open problems

Fractional calculus on time scales is a new topic providing many directions for further research. In this paper, we have discussed relations of the axiomatic approach, represented by Definitions 3.1 and 3.5 to the work of the other authors, namely, [6, 16]. The key assertion, i.e., Theorem 4.3 concerning existence and uniqueness of power functions, was derived under the assumption of rational orders and isolated time scales. This result has many important consequences, e.g., Theorem 5.1, therefore, its extension for time scales, involving dense points or for irrational orders of power functions, seems to be very desirable.

Regarding the values of the power functions, we have introduced the recursive formulas (15), (16). Besides the question of the nonnegativity (see Remark 4.5), it remains an open problem if a corresponding relation exists also in a closed form. Considering the power functions of negative orders, we have also mentioned a notion of singular time scales functions, a matter which is quite unexplored so far (see Remark 4.7).

A solving of more advanced fractional differential equations on time scales requires an introduction of analogues of various special functions utilized in the continuous fractional calculus, e.g., a dynamic Mittag-Leffler function E η , β λ ( t , s ) = j = 0 λ j h ˆ η j + β 1 ( t , s ) Open image in new window (see, e.g., [2, 10] and [12]). A question of convergence of the related infinite series, as well as many other topics such as stability analysis, is closely connected to an issue of asymptotic properties of the general dynamic power functions. We conjecture that the estimate
ν max r ( t ) ( n + r 1 n 1 ) h ˆ r ( t , s ) ν min r ( t ) ( n + r 1 n 1 ) , r ( 1 , 0 ) Q , t = σ n ( s ) Open image in new window

holds, where ν max ( t ) = max { ν ( τ ) ; τ ( s , t ] T } Open image in new window and ν min ( t ) = min { ν ( τ ) ; τ ( s , t ] T } Open image in new window. However, its general proof is not available so far, therefore, it remains in a form of hypothesis.

Author’s contributions

TK is the only author of this paper. The author read and approved the final manuscript.

Notes

Acknowledgements

The research was supported by the project CZ.1.07/2.3.00/30.0039 of Brno University of Technology.

Supplementary material

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Authors’ original file for figure 1
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Authors’ original file for figure 2
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Authors’ original file for figure 3
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Authors’ original file for figure 4
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Authors’ original file for figure 5
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Authors’ original file for figure 6

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© Kisela; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institute of MathematicsBrno University of TechnologyBrnoCzech Republic

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