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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 230850, 6 pages
Solving Fractional Difference Equations Using the Laplace Transform Method
School of Mathematical Science, Anhui University, Hefei, Anhui 230601, China
Received 22 September 2013; Accepted 17 January 2014; Published 26 February 2014
Academic Editor: Stefan Siegmund
Copyright © 2014 Li Xiao-yan and Jiang Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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. , Holm , and Abdeljawad  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 ; 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  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.
Let us start with some definitions and preliminaries.
Definition 1 (see ). 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 ). The th fractional sum of a function , for , is defined by for , .
Definition 3 (see ). 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 ). The Laplace transform of the function on the time scale is represented by
Definition 5 (see ). 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 ). 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 ). 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 , 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
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 .
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
Since Theorem 11, we have the solution of (39):
This result coincides with the result of paper , 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 ). 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.
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)
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
- I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara, “Matrix approach to discrete fractional calculus. II. Partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137–3153, 2009.
- M. T. Holm, “The Laplace transform in discrete fractional calculus,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1591–1601, 2011.
- T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
- F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.
- C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers Mathematics With Applications, vol. 62, pp. 1251–1268, 2011.
- C. S. Goodrich, “On discrete sequential fractional boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 111–124, 2012.
- C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4740–4753, 2011.