1. 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 by

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).(iii).(iv).(v).(vi).

In particular, since constants are fixed by the Sumudu transform, choosing , it gives .

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 Caputo-fractional 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.

2. Preliminaries on Time Scales

A time scale is an arbitrary nonempty closed subset of the real numbers . The most well-known examples are , , and , where . The forward and backward jump operators are defined by respectively, where and . A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if . The graininess function is defined by . 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 is called regulated if its right-sided limits exist at all right-dense points in and its left-sided limits exist at all left-dense points in .

Definition 2.2 (see [25]). A function is called rd-continuous if it is continuous at right-dense points in and its left-sided limits exist at left-dense points in .

The set is derived from the time scale as follows: if has a left-scattered maximum , then . Otherwise, .

Definition 2.3 (see [25]). A function is said to be delta differentiable at a point if there exists a number with the property that given any , there exists a neighborhood of such that

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 is called regressive provided for all .

The set of all regressive and rd-continuous functions forms an Abelian group under the “circle plus” addition defined by

The additive inverse of is defined by

Theorem 2.5 (see [25]). Let and be a fixed point. Then the exponential function is the unique solution of the initial value problem

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

Definition 3.1 (see [27]). Let and be given. Then the -order fractional sum of is given by Also, we define the trivial sum by Note that the fractional sum operator maps functions defined on to functions defined on .

In the above equation the term is the generalized falling function defined by for any 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 and be given, and let be chosen such that . Then the -order Riemann-Liouville fractional difference of is given by

It is clear that, the fractional difference operator maps functions defined on to functions defined on .

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 be given and suppose with . Then

A disadvantage of the Riemann-Liouville fractional difference operator is that when applied to a constant , it does not yield 0. For example, for , 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 and be given, and let be chosen such that . Then the -order Caputo fractional difference of is given by

It is clear that the Caputo fractional difference operator maps functions defined on to functions defined on as well. And it follows from the definition of the Caputo fractional difference operator that

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 is given by where is fixed, is an unbounded time scale with infimum and is the set of all nonzero complex constants for which is regressive and the integral converges.

In the special case, when , every function is regulated and its discrete Sumudu transform can be written as for each for which the series converges. For the convergence of the Sumudu transform, we need the following definition.

Definition 4.2 (see [27]). A function is of exponential order () if there exists a constant such that

The following lemma can be proved similarly as in Lemma 12 in [27].

Lemma 4.3. Suppose is of exponential order . Then

The following lemma relates the shifted Sumudu transform to the original.

Lemma 4.4. Let and and are of exponential order . Then for all such that ,

Proof. For all such that , we have

Taylor monomials are very useful for applying the Sumudu transform in discrete fractional calculus.

Definition 4.5 (see [27]). For each , define the -Taylor monomial to be

Lemma 4.6. Let and such that . Then for all such that , one has

Proof. By the general binomial formula for such that , where as in [27], it follows from (4.10) and where that for and .
And since , we have for all such that ,

Definition 4.7 (see [27]). Define the convolution of two functions by

Lemma 4.8. Let be of exponential order . Then for all such that ,

Proof. Since the substitution yields for all such that .

Theorem 4.9. Suppose is of exponential order and let with . Then for all such that ,

Proof. First note that the shift formula (4.5) implies that for all such that , taking zeros of into account. Furthermore, by (4.9), (4.15), and (4.16), Then we obtain

Theorem 4.10. Suppose is of exponential order and let with . Then for all such that ,

Proof. Let , and be as in the statement of the theorem. We already know from Theorem 3.8 in [28] that (4.24) holds when , that is, If , then and hence it follows from (3.6), (4.19), and (4.25) that

In the following theorem the Sumudu transform of the Caputo fractional difference operator is presented.

Theorem 4.11. Suppose is of exponential order and let with . Then for all such that ,

Proof. Let , and be as in the statement of the theorem. We already know from (4.25) that , (4.27) holds. If , then and hence it follows from (4.19) and (4.25) that

Lemma 4.12. Let be given. For any and with , one has

Proof. Let f, v, N, and p be given as in the statement of the lemma. Then

Corollary 4.13. Suppose is of exponential order , with and . Then for all such that ,

Proof. The proof follows from (4.25), (4.27), and (4.29).

5. 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.

Example 5.1. Suppose is of exponential order and let with . The unique solution to the fractional initial value problem is given by where for .

Proof. Since is of exponential order , then exists for all such that . So, applying the Sumudu transform to both sides of the fractional difference equation in (5.1), we have for all such that , Then from (4.24), it follows 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 , since for .
Consequently, we have Since Sumudu transform is a one-to-one operator (see [28, Theorem 3.6]), we conclude that for , where (see [27, Theorem 11]).

Example 5.2. Consider the initial value problem (5.1) with the Riemann-Liouville fractional difference replaced by the Caputo fractional difference.
Applying the Sumudu transform to both sides of the difference equation, we get for all such that ,
Then from (4.27), it follows By (4.20), we have Since from [28], we have hence