A new transform method in nabla discrete fractional calculus
© Jarad et al.; licensee Springer 2012
Received: 19 July 2012
Accepted: 18 October 2012
Published: 5 November 2012
Starting from the definition of the Sumudu transform on a general nabla time scale, we define the generalized nabla discrete Sumudu transform. We obtain the nabla discrete Sumudu transform of Taylor monomials, fractional sums, and differences. We apply this transform to solve some fractional difference equations with initial value problems.
The fractional calculus deals with integrals and derivatives of arbitrary orders. Many scientists have paid a lot of attention to this calculus 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 , 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 [8–16] and the references cited therein.
In the early 1990s, Watugala [17, 18] introduced the Sumudu transform and applied it to solve ordinary differential equations. The fundamental properties of this transform, which is thought to be an alternative to the Laplace transform, were then established in many articles [19–23].
For any real or complex number c, .
Particularly, 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 can be treated as replicas of t and , respectively .
Recently, an application of the Sumudu and double Sumudu transforms to Caputo fractional differential equations is given in . In , the authors applied the Sumudu transform to fractional differential equations.
In , the authors obtained the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences and applied this transform to solve a fractional difference initial value problem.
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 . The theory of time scales was initiated by Stefan Hilger . 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 a dynamic process can be described with discrete or continuous models.
In this paper, starting from the definition of the Sumudu transform on a general nabla time scale, we define the nabla 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 nabla discrete Sumudu transform and present some of its basic properties. Section 5 is devoted to some applications.
2 Preliminaries on time scales
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 backwards graininess function is defined by . For details, see the monographs [30, 31].
The following two concepts are introduced in order to describe the classes of functions that are integrable.
Definition 2.1 
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 
A function is called ld-continuous if it is continuous at left-dense points in and its right-sided limits exist at right-dense points in .
The set is derived from the time scale as follows: If has a right-scattered minimum m, then . Otherwise, .
Definition 2.3 
We shall also need the following definition in order to define the nabla exponential function on an arbitrary time scale.
Definition 2.4 
A function is called ν-regressive provided for all .
Theorem 2.5 
We next define the nabla Taylor monomials and later generalize them for noninteger orders.
Definition 2.6 
3 An introduction to nabla discrete fractional calculus
For any function , we define the backwards difference, or nabla operator, ∇, by for . In this paper, we use the convention that . We then define higher order differences recursively by for , . In addition, we take as the identity operator.
Definition 3.1 
Also, we define the trivial sum by for .
for any for which the right-hand side is well defined. As usual, we use the convention that if and , then .
Definition 3.2 
For , we set for .
As stated in the following two theorems, the composition of fractional operators behaves well if the inner operator is a fractional difference.
Theorem 3.3 
Theorem 3.4 
with allowance for the convention for . This completes the proof. □
In order to overcome this and to make the fractional differences behave like the usual differences, we define the Caputo nabla fractional difference operator, different from the definitions given in  and , as follows.
For , we set for .
4 The discrete nabla Sumudu transform
where is fixed, is an unbounded time scale with infimum a, and is the set of all nonzero complex constants u for which is ν-regressive and the integral converges.
for each for which the series converges. For the convergence of the Sumudu transform, we need the following definition.
Definition 4.2 
The proof of the following lemma is obvious.
If , then the series in the definition of the nabla Sumudu converges for all u such that and diverges elsewhere;
If , the series converges for all u;
If , the series diverges everywhere, except perhaps when .
The following lemma of uniform convergence can be proved easily.
Lemma 4.6 If , the series in the definition of Sumudu is convergent for all u such that .
Lemma 4.7 (Initial value)
Proof The proof can be done by taking term by term limit as . □
Lemma 4.8 (Final value)
Theorem 4.9 (Uniqueness theorem)
Let be a function. Then if and only if , .
Since the series converges uniformly, taking the limit of the left side as , we get .
Corollary 4.10 Let be such that and exist and , then , .
Proof By the linearity it is obvious. □
Below we state the definition of the Taylor monomials which are very useful for applying the Sumudu transform in discrete fractional calculus.
Definition 4.11 
The following lemma is crucial for finding the Sumudu transform of fractional order nabla Taylor monomials.
Lemma 4.12 
where the last equality follows from the generalized binomial theorem.
The proof is now complete. □
Definition 4.14 
Theorem 4.15 
for all such that . □
provided (4.6) holds. □
We can now generalize this result for an arbitrary positive integer n as follows.
Proof The proof follows from Theorem 4.18 by applying induction. □
The following theorem presents the Sumudu transform of the nabla fractional difference of a function.
Proof The proof can be done by replacing f by in Theorem 4.19 and then using Theorem 4.17. □
In the following theorem, the Sumudu transform of the Caputo nabla fractional difference is given.
where , and .
In the following theorem, we establish the Sumudu transform of the Mittag-Leffler function.
In this section, we will illustrate the possible use of the discrete Sumudu transform by applying it to solve some initial value problems.
where and .
where and .
The authors would like to thank the reviewers for their valuable comments.
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