Solving Fractional Difference Equations Using the Laplace Transform Method

Abstract

We discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.

1. Introduction

The study of continuous fractional calculus and equations has seen tremendous growth over the past few decades involving many aspects [1–4], such as initial value problem (IVP), boundary value problems (BVP), and stability of fractional equations. Compared with the continuous fractional calculus and fractional order differential equations, we can see that the research about the discrete fractional calculus and fractional difference equations has seen slower progress, but in recent years, a number of papers have appeared, and the study of the discrete fractional calculus and fractional difference equations has been arising. For example, Podlubny et al. [5], Holm [6], and Abdeljawad [7] have explored the definitions of fractional sum and difference operators and obtained many of their properties. Also, Atici and Eloe considered discrete fractional IVPs in paper [8]; moreover, discrete fractional BVPs were discussed in papers by Goodrich [9–11].

We know that the Laplace transform method has played an important role in solving basic problems of differential equations. Holm [6] developed properties of the Laplace transform in a discrete and applied the Laplace transform to solve a fractional initial value problem, which can be described as In this paper, we will discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffler functions and use the Laplace transform method to solve another kind of discrete fractional IVPs.

2. Preliminaries

Let us start with some definitions and preliminaries.

Definition 1 (see [8]). The generalized falling function is defined by where denotes the special gamma function and whenever .
Here are some of the properties of the above fractional function:(i);(ii);(iii);(iv).

Definition 2 (see [4]). The th fractional sum of a function , for , is defined by for , .

Definition 3 (see [7]). The -order Caputo left fractional difference is defined by where , . If , then .

In this paper, we will mainly discuss the problems involving the Caputo left fractional difference.

Definition 4 (see [6]). The Laplace transform of the function on the time scale is represented by

Definition 5 (see [6]). One says that a function is of exponential order , if there exists a constant such that Via a geometric series, it is straightforward to show that if is of exponential order , then

Let be given and suppose and are of exponential order . Then for ,

Definition 6 (see [6]). For , define the convolution of and by By a standard convolution on sums, it is understood that .

3. The Laplace Transform of Caputo Fractional Difference

Lemma 7 (see [6]). Suppose is of exponential order and let be given with . Then both and converge for all , and

Theorem 8. Suppose is of exponential order and let be given. Then for each fixed , is of exponential order and converge for all .

Proof. Consider the relationship between Caputo fractional difference and Riemann-Liouville difference and in Lemma 7 we have since for sufficiently large when , will eventually grow larger than the function . So we get Choose ; there exists an small enough so that , and Lemma 7 tells us that is of exponential order , so it follows from (7) that is well defined.

Theorem 9. Suppose is of exponential order and let be given with . Then for

Proof. Since , then Because and we get

4. The Laplace Transform of Discrete Mittag-Leffler Function

In paper [8], the discrete Mittag-Leffler function is introduced as the following form.

Definition 10. For any constant and with , the discrete Mittag-Leffler functions are defined by For , it is written that

Theorem 11. We assume and in (20); then for any fixed And so exists for .

Proof. For , we have Moreover, for sufficiently large we get then, We know that Now, it is easy to see that for sufficiently large , where is a constant and , so the function converges and exists for .
We will discuss the Laplace transform of the discrete Mittag-Leffler function .

Theorem 12. Let , ; then one gets

Proof. When , we have From Lemma 7 we get by (10), we conclude that For Definition 1 implies that So, we get Recalling (28), we have Using (33), (27) can be rewritten as When , the result is , which coincided with integer order.
With this in mind, let us discuss the Laplace transform of the Mittag-Leffler function ; we will use this result in the following section.

Theorem 13. Letting , , then one has

Proof. We recall that By property (iii) of the generalized falling function, we get Then

5. Laplace Transform Method for Solving Fractional Difference Equation with Caputo Fractional Difference

In this section, we first consider the following Caputo fractional difference equations: where , and .

The solution of (39) is given by Atici and Eloe in [8] using the method of successive approximation; we will give the solution of (39) by the method of Laplace transform.

Theorem 14. Equation (39) has its solution given by

Proof. Both sides of (39) carried out Laplace transform; we get Equations (6) and (9) and Theorem 9 imply that Then Since Theorem 11, we have the solution of (39):
This result coincides with the result of paper [7], which obtains the solution of (40) using the method of successive approximation.
Next, we consider the solution of Caputo nonhomogeneous difference equation where , , .
The following standard rule for composing the Laplace transform with the convolution is necessary for solving the fractional initial value problems (45).

Lemma 15 (see [6]). Let be of exponential order . Then

Theorem 16. Let of be exponential order ; then (45) has its solution given by

Proof. Using Laplace transform on both sides of (45), we obtain because ; that is, ; similar to the above discussion, it is easy to obtain the following: Then we obtain Carrying out Laplace inverse transform of both sides of (45), according to (10), (28), (33), and (35), we have Letting , formula (45) yields which is the expression of the Caputo nonhomogeneous difference equation (45).
In our future research work, we will consider the solution of fractional difference equations (39) and (45) in general situation: , .

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This research had been supported by the National Nature Science Foundation of China (no. 11371027), Starting Research Fund for Doctors of Anhui University (no. 023033190249), National Natural Science Foundation of China, Tian Yuan Special Foundation (no. 11326115), and the Special Research Fund for the Doctoral Program of the Ministry of Education of China (no. 20123401120001)