On Sumudu Transform Method in Discrete Fractional Calculus

and Applied Analysis 3 2. Preliminaries on Time Scales A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most wellknown examples are T R, T Z, and T q : {qn : n ∈ Z}⋃{0}, where q > 1. The forward and backward jump operators are defined by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, 2.1 respectively, where inf ∅ : supT and sup ∅ : inf T. A point t ∈ T is said to be left-dense if t > infT and ρ t t, right-dense if t < supT and σ t t, left-scattered if ρ t < t, and right-scattered if σ t > t. The graininess function μ : T → 0,∞ is defined by μ t : σ t − t. For details, see the monographs 25, 26 . The following two concepts are introduced in order to describe classes of functions that are integrable. Definition 2.1 see 25 . A function f : T → R is called regulated if its right-sided limits exist at all right-dense points in T and its left-sided limits exist at all left-dense points in T. Definition 2.2 see 25 . A function f : T → R is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T. The setT is derived from the time scaleT as follows: ifT has a left-scatteredmaximum m, then T : T − {m}. Otherwise, T : T. Definition 2.3 see 25 . A function f : T → R is said to be delta differentiable at a point t ∈ T if there exists a number fΔ t with the property that given any ε > 0, there exists a neighborhood U of t such that ∣∣[f σ t − f s ] − fΔ t σ t − s ∣∣ ≤ ε|σ t − s| ∀s ∈ U. 2.2 We will also need the following definition in order to define the exponential function on an arbitrary time scale. Definition 2.4 see 25 . A function p : T → R is called regressive provided 1 μ t p t / 0 for all t ∈ T. The set R of all regressive and rd-continuous functions forms an Abelian group under the “circle plus” addition ⊕ defined by ( p ⊕ q) t : p t q t μ t p t q t ∀t ∈ T. 2.3 The additive inverse p of p ∈ R is defined by ( p) t : − p t 1 μ t p t ∀t ∈ T. 2.4 4 Abstract and Applied Analysis Theorem 2.5 see 25 . Let p ∈ R and t0 ∈ T be a fixed point. Then the exponential function ep ·, t0 is the unique solution of the initial value problem yΔ p t y, y t0 1. 2.5 3. An Introduction to Discrete Fractional Calculus In this section, we introduce some basic definitions and a theorem concerning the discrete fractional calculus. Throughout, we consider the discrete set Na : {a, a 1, a 2, . . .}, where a ∈ R is fixed. 3.1 Definition 3.1 see 27 . Let f : Na → R and ν > 0 be given. Then the νth-order fractional sum of f is given by Δ−ν a f t : 1 Γ ν t−ν ∑ s a t − σ s ν−1f s for t ∈ Na ν. 3.2 Also, we define the trivial sum by Δ−0 a f t : f t for t ∈ Na. 3.3 Note that the fractional sum operator Δ−ν a maps functions defined on Na to functions defined on Na ν. In the above equation the term t − σ s ν−1 is the generalized falling function defined by t : Γ t 1 Γ t 1 − ν 3.4 for any t, ν ∈ R for which the right-hand side is well defined. As usual, we use the convention that division by a pole yields zero. Definition 3.2 see 27 . Let f : Na → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Riemann-Liouville fractional difference of f is given by Δaf t : Δ NΔ− N−ν a f t for t ∈ Na N−ν. 3.5 It is clear that, the fractional difference operator Δa maps functions defined on Na to functions defined on Na N−ν. As stated in the following theorem, the composition of fractional operators behaves well if the inner operator is a fractional difference. Abstract and Applied Analysis 5 Theorem 3.3 see 27 . Let f : Na → R be given and suppose ν, μ > 0 withN − 1 < ν ≤ N. Then Δa μΔ −μ a f t Δ ν−μ a f t for t ∈ Na μ N−ν. 3.6and Applied Analysis 5 Theorem 3.3 see 27 . Let f : Na → R be given and suppose ν, μ > 0 withN − 1 < ν ≤ N. Then Δa μΔ −μ a f t Δ ν−μ a f t for t ∈ Na μ N−ν. 3.6 A disadvantage of the Riemann-Liouville fractional difference operator is that when applied to a constant c, it does not yield 0. For example, for 0 < v < 1, we have Δac − c t − a −ν Γ 1 − ν . 3.7 In order to overcome this and to make the fractional difference behave like the usual difference, the Caputo fractional difference was introduced in 12 . Definition 3.4 see 12 . Let f : Na → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Caputo fractional difference of f is given by Δaf t : Δ − N−ν a Δf t for t ∈ Na N−ν. 3.8 It is clear that the Caputo fractional difference operator Δa maps functions defined on Na to functions defined on Na N−ν as well. And it follows from the definition of the Caputo fractional difference operator that Δa c 0. 3.9 4. The Discrete Sumudu Transform The following definition is a slight generalization of the one introduced by Jarad et al. 28 . Definition 4.1. The Sumudu transform of a regulated function f : Ta → R is given by


Introduction
The fractional calculus, which is as old as the usual calculus, deals with the generalization of the integration and differentiation of integer order to arbitrary order. It has recently received a lot of attention because of its interesting applications in various fields of science, such as, viscoelasticity, diffusion, neurology, control theory, and statistics, see 1-6 .
The analogous theory for discrete fractional calculus was initiated by Miller and Ross 7 , where basic approaches, definitions, and properties of the theory of fractional sums and differences were reported. Recently, a series of papers continuing this research has appeared. We refer the reader to the papers 8-12 and the references cited therein.
In the early 1990's, Watugala 13,14 introduced the Sumudu transform and applied it to solve ordinary differential equations. The fundamental properties of this transform, which are thought to be an alternative to the Laplace transform were then established in many articles 15-19 . The Sumudu transform is defined over the set of functions A : f t | ∃M, τ 1 , τ 2 > 0, f t < Me |t|/τ j , if t ∈ −1 j × 0, ∞ 1. Although the Sumudu transform of a function has a deep connection to its Laplace transform, the main advantage of the Sumudu transform is the fact that it may be used to solve problems without resorting to a new frequency domain because it preserves scales and unit properties. By these properties, the Sumudu transform may be used to solve intricate problems in engineering and applied sciences that can hardly be solved when the Laplace transform is used. Moreover, some properties of the Sumudu transform make it more advantageous than the Laplace transform. Some of these properties are i The Sumudu transform of a Heaviside step function is also a Heaviside step function in the transformed domain.
ii S{t n } u n!u n . iii vi For any real or complex number c, S{f ct } u F cu .
In particular, since constants are fixed by the Sumudu transform, choosing c 0, it gives F 0 f 0 . In dealing with physical applications, this aspect becomes a major advantage, especially in instances where keeping track of units, and dimensional factor groups of constants, is relevant. This means that in problem solving, u and G u can be treated as replicas of t and f t , respectively 20 .
Recently, an application of the Sumudu and Double Sumudu transforms to Caputofractional differential equations is given in 21 . In 22 , the authors applied the Sumudu transform to fractional differential equations.
Starting with a general definition of the Laplace transform on an arbitrary time scale, the concepts of the h-Laplace and consequently the discrete Laplace transform were specified in 23 . The theory of time scales was initiated by Hilger 24 . This theory is a tool that unifies the theories of continuous and discrete time systems. It is a subject of recent studies in many different fields in which dynamic process can be described with discrete or continuous models.
In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the discrete Sumudu transform and present some of its basic properties.
The paper is organized as follows: in Sections 2 and 3, we introduce some basic concepts concerning the calculus of time scales and discrete fractional calculus, respectively. In Section 4, we define the discrete Sumudu transform and present some of its basic properties. Section 5 is devoted to an application.

Preliminaries on Time Scales
A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most wellknown examples are T R, T Z, and T q Z : {q n : n ∈ Z} {0}, where q > 1. The forward and backward jump operators are defined by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, 2.1 respectively, where inf ∅ : sup T and sup ∅ : inf T. A point t ∈ T is said to be left-dense if t > inf T and ρ t t, right-dense if t < sup T and σ t t, left-scattered if ρ t < t, and right-scattered if σ t > t. The graininess function μ : T → 0, ∞ is defined by μ t : σ t − t. For details, see the monographs 25, 26 .
The following two concepts are introduced in order to describe classes of functions that are integrable. The set T κ is derived from the time scale T as follows: if T has a left-scattered maximum m, then T κ : T − {m}. Otherwise, T κ : T.
there exists a number f Δ t with the property that given any ε > 0, there exists a neighborhood U of t such that We will also need the following definition in order to define the exponential function on an arbitrary time scale.
The set R of all regressive and rd-continuous functions forms an Abelian group under the "circle plus" addition ⊕ defined by The additive inverse p of p ∈ R is defined by Abstract and Applied Analysis Theorem 2.5 see 25 . Let p ∈ R and t 0 ∈ T be a fixed point. Then the exponential function e p ·, t 0 is the unique solution of the initial value problem

An Introduction to Discrete Fractional Calculus
In this section, we introduce some basic definitions and a theorem concerning the discrete fractional calculus. Throughout, we consider the discrete set Definition 3.1 see 27 . Let f : N a → R and ν > 0 be given. Then the νth-order fractional sum of f is given by 3.2 Also, we define the trivial sum by Note that the fractional sum operator Δ −ν a maps functions defined on N a to functions defined on N a ν .
In the above equation the term t − σ s ν−1 is the generalized falling function defined by for any t, ν ∈ R for which the right-hand side is well defined. As usual, we use the convention that division by a pole yields zero.
Definition 3.2 see 27 . Let f : N a → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Riemann-Liouville fractional difference of f is given by It is clear that, the fractional difference operator Δ ν a maps functions defined on N a to functions defined on N a N−ν .
As stated in the following theorem, the composition of fractional operators behaves well if the inner operator is a fractional difference.

Theorem 3.3 see 27 .
Let f : N a → R be given and suppose ν, A disadvantage of the Riemann-Liouville fractional difference operator is that when applied to a constant c, it does not yield 0. For example, for 0 < v < 1, we have In order to overcome this and to make the fractional difference behave like the usual difference, the Caputo fractional difference was introduced in 12 .
Definition 3.4 see 12 . Let f : N a → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Caputo fractional difference of f is given by It is clear that the Caputo fractional difference operator C Δ ν a maps functions defined on N a to functions defined on N a N−ν as well. And it follows from the definition of the Caputo fractional difference operator that C Δ ν a c 0. 3.9

The Discrete Sumudu Transform
The following definition is a slight generalization of the one introduced by Jarad et al. 28 .
Definition 4.1. The Sumudu transform of a regulated function f : T a → R is given by where a ∈ R is fixed, T a is an unbounded time scale with infimum a and D{f} is the set of all nonzero complex constants u for which 1/u is regressive and the integral converges.
In the special case, when T a N a , every function f : N a → R is regulated and its discrete Sumudu transform can be written as for each u ∈ C \ {−1, 0} for which the series converges. For the convergence of the Sumudu transform, we need the following definition. The following lemma can be proved similarly as in Lemma 12 in 27 . Then The following lemma relates the shifted Sumudu transform to the original. Proof. For all u ∈ C \ {−1, 0} such that | u 1 /u| > r, we have

4.7
Taylor monomials are very useful for applying the Sumudu transform in discrete fractional calculus. Proof. By the general binomial formula as in 27 , it follows from 4.10 and Abstract and Applied Analysis where k ∈ N 0 that for ν ∈ R and |y| < 1.

4.22
Then we obtain

4.23
Theorem 4.10. Suppose f : N a → R is of exponential order r ≥ 1 and let ν > 0 with N −1 < ν ≤ N. Then for all u ∈ C \ {−1, 0} such that | u 1 /u| > r, Proof. Let f, r, ν, and N be as in the statement of the theorem. We already know from Theorem 3.8 in 28 that 4.24 holds when ν N, that is,

4.26
In the following theorem the Sumudu transform of the Caputo fractional difference operator is presented.
Proof. Let f, r, ν, and N be as in the statement of the theorem. We already know from 4.25 that v N, 4.27 holds. If N − 1 < ν < N, then 0 < N − ν < 1 and hence it follows from 4.19 and 4.25 that

4.28
Lemma 4.12. Let f : N a → R be given. For any p ∈ N 0 and ν > 0 with N − 1 < ν ≤ N, one has Proof. Let f, v, N, and p be given as in the statement of the lemma. Then

Applications
In this section, we will illustrate the possible use of the discrete Sumudu transform by applying it to solve some initial value problems. The following initial value problem was solved in Theorem 23 in 27 by using the Laplace transforms.
The unique solution to the fractional initial value problem is given by Proof. Since f is of exponential order r, then S a {f} u exists for all u ∈ C \ {−1, 0} such that | u 1 /u| > r. So, applying the Sumudu transform to both sides of the fractional difference equation in 5.1 , we have for all u ∈ C \ {−1, 0} such that | u 1 /u| > r, Then from 4.24 , it follows Abstract and Applied Analysis 13 and hence By 4.20 , we have Considering the terms in the summation, by using the shifting formula 4.5 , we see that for each k ∈ {0, 1, . . . , N − 1}, Consequently, we have

5.10
Since Sumudu transform is a one-to-one operator see 28, Theorem 3.6 , we conclude that for t ∈ N a ν−N ,

5.13
Applying the Sumudu transform to both sides of the difference equation, we get for all u ∈ C \ {−1, 0} such that | u 1 /u| > r,

5.16
Since from 28 , we have S 0 {t n } u n!u n , n ∈ N 0 , 5.17

5.18
Remark 5.3. The initial value problem 5.1 can also be solved by using Proposition 15 in 12 .
Example 5.4. Consider the initial value problem

5.19
where 0 < ν ≤ 1. Applying the Sumudu transform to both sides of the equation and using 4.31 and 4.27 , respectively, we get Hence we get Since from 28 , we have S 0 1 λ t u 1 1 − λu for 1 λ u u 1 < 1, 5.22 then y t A 0 − A 1 A 1 2 t−a−ν 1 .