International Journal of Differential Equations

Volume 2012 (2012), Article ID 780619, 11 pages

doi:10.1155/2012/780619

Research Article

Fractional Order Difference Equations

J. Jagan Mohan and G. V. S. R. Deekshitulu

Fluid Dynamics Division, School of Advanced Sciences, VIT University, Tamil Nadu, Vellore 632014, India

Received 11 May 2012; Accepted 17 October 2012

Academic Editor: Shaher Momani

Copyright © 2012 J. Jagan Mohan and G. V. S. R. Deekshitulu. 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.

Abstract.

A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations.

1. Introduction

Fractional calculus has gained importance during the past three decades due to its applicability in diverse fields of science and engineering. The notions of fractional calculus may be traced back to the works of Euler, but the idea of fractional difference is very recent.

Diaz and Osler [1] defined the fractional difference by the rather natural approach of allowing the index of differencing, in the standard expression for the th difference, to be any real or complex number. Later, Hirota [2] defined the fractional order difference operator where is any real number, using Taylor’s series. Nagai [3] adopted another definition for fractional order difference operator by modifying Hirota’s [2] definition. Recently, Deekshitulu and Mohan [4] modified the definition of Nagai [3] for in such a way that the expression for does not involve any difference operator.

The study of theory of fractional differential equations was initiated and existence and uniqueness of solutions for different types of fractional differential equations have been established recently [5]. Much of literature is not available on fractional integrodifferential equations also, though theory of integrodifferential equations [6] has been almost all developed parallel to theory of differential equations. Very little progress has been made to develop the theory of fractional order difference equations.

The main aim of this paper is to establish theorems on existence and uniqueness of solutions of various classes of fractional order difference equations. Further we define autonomous and nonautonomous fractional order difference equations and find their solutions.

2. Preliminaries

Throughout the present paper, we use the following notations: be the set of natural numbers including zero. for . Let . Then for all and , , and , that is, empty sums and products are taken to be 0 and 1, respectively. If and are in , then for this function , the backward difference operator is defined as . Now we introduce some basic definitions and results concerning nabla discrete fractional calculus.

Definition 2.1. The extended binomial coefficient , () is defined by

Definition 2.2 (see [7]). For any complex numbers and , let be defined as follows:

Remark 2.3. For any complex numbers and , when , and are neither zero nor negative integers,

for any positive integer .

In 2003, Nagai [3] gave the following definition for fractional order difference operator.

Definition 2.4. Let and be an integer such that . The difference operator of order , with step length , is defined as

The above definition of given by Nagai [3] contains operator and the term inside the summation index and hence it becomes difficult to study the properties of solution. To avoid this, Deekshitulu and Mohan [4] gave the following definition, for . Let such that .

Definition 2.5. The fractional sum operator of order is defined as

And the fractional order difference operator of order is defined as

Throughout this paper we assume that and unless until specified.

Remark 2.6. Let and , such that and ,   are constants. Then

(1). (2). (3). (4). (5) and .

3. Existence and Uniqueness of Solutions of Fractional Order Difference Equations

In the present section, we establish theorems on existence and uniqueness of solutions for various classes of fractional order difference equations.

Definition 3.1. Let be any function defined for , . Then a nonlinear difference equation of order together with an initial condition is of the form

The existence of solutions to difference equations is trivial as the solutions are expressed as recurrence relations involving the values of the unknown function at the previous arguments.

Now we consider (2.5) and replace by , we have

where for . The above recurrence relation shows the existence of solution of (3.1).

Example 3.2. If where are any two nonnegative functions then

Recently, the authors have established the following fractional order discrete Gronwall-Bellman type inequality [8].

Theorem 3.3. Let , , and be real valued nonnegative functions defined on . If, for ,

then

for .

Definition 3.4. Let and be any functions defined for , , and . Then a nonlinear difference equation of Volterra type of order together with an initial condition is of the form

From (3.2) and (3.6), we have

The above recurrence relation shows the existence of solution of (3.6). Now we establish the uniqueness of solutions for the fractional order difference equations (3.1) and (3.3).

Lemma 3.5. For ,

Proof. Consider

Theorem 3.6. Let be any function defined for , . Suppose that satisfies

where are any two nonnegative functions, and is a nonnegative constant. Then there exists unique solution for the initial value problem (3.1).

Proof. Let and be any two solutions of the initial value problem (3.1) such that . Then, using (3.2) and (3.10), we get

where be any small quantity. Let . Then

Using Discrete Fractional Gronwall’s inequality (i.e., Theorem 3.3) we get

Using Lemma 3.5, we get

Since is arbitrary, inequality (3.14) implies that tends to zero. Thus .

Theorem 3.7. Consider a nonlinear fractional difference equation of Volterra type (3.6). Suppose that and satisfies

where are any two nonnegative functions and is a nonnegative constant. Then there exists a unique solution for the initial value problem (3.6).

Proof. Let and be any two solutions of the initial value problem (3.6) such that . Then, using (3.2), we get

where be any small quantity. Let . Then

Using Discrete Fractional Gronwall’s inequality (i.e., Theorem 3.3) we get

Using Lemma 3.5 we have

Since is arbitrary, inequality (3.19) implies that tends to zero. Thus .

4. Solutions of Fractional Order Difference Equations

In this section, we define autonomous and nonautonomous fractional order difference equations and find their solutions.

Definition 4.1. Let be any real-valued function defined on . Then an autonomous difference equation of order together with an initial condition is of the form

If the function in (4.1) is replaced by a function of two variables, that is, , then we have

Equation (4.2) is called a nonautonomous difference equation of order .

Now we consider the simplest special cases of (4.1) and (4.2), namely, linear equations.

The linear autonomous difference equation of order has following general form:

where and are known constants.

The general form of linear homogeneous nonautonomous difference equation of order is given by

and the associated nonhomogeneous equation is given by

where and are real-valued functions defined for .

Now we find the solutions of (4.3), (4.4), and (4.5).

Solution of (4.4)

Using (3.2) and (4.4), we obtain

implies

Solution of (4.5)

Using (3.2) and (4.5), we obtain

implies

Solution of (4.3)

Using (4.9), we obtain

Example 4.2. Find the solution of

Solution. Using (4.10), the solution of (4.11) is

Example 4.3. Find the solution of

Solution. Using (4.9), the solution of (4.13) is

Example 4.4. Find the solution of

Solution. Using (4.7), the solution of (4.15) is

Evaluating fractional differences of various functions numerically is quite cumbersome. Now we find the numerical the solution of the following fractional order difference equation together with an initial condition

using a simple program in MATLAB for various values of .

The numerical solution for various values of and are tabulated in Table 1. The graphical representation of the numerical solution for various values of and are shown in Figure 1.

Table 1: Numerical solution of (4.17).

0.2
0.4
0.6
0.8
1.0

0
1.0000
1.0000
1.0000
1.0000
1.0000

1
2.0000
2.0000
2.0000
2.0000
2.0000

2
2.4000
2.8000
3.2000
3.6000
4.0000

3
2.6880
3.5840
4.7360
6.1920
8.0000

4
2.9245
4.3868
6.7062
10.3530
16.0000

5
3.1304
5.2221
9.2170
16.9624
32.0000

6
3.3156
6.0970
12.3917
27.3580
64.0000

7
3.4855
7.0164
16.3755
43.5659
128.0000

8
3.6437
7.9838
21.3391
68.6384
256.0000

9
3.7926
9.0020
27.4839
107.1529
512.0000

10
3.9337
10.0735
35.0464
165.9425
1024.0000

Figure 1: Numerical solution of (4.17).

References

J. B. Diaz and T. J. Osler, “Differences of fractional order,” Mathematics of Computation, vol. 28, pp. 185–202, 1974.

R. Hirota, Lectures on Difference Equations, Science-sha, Tokyo, Japan, 2000.

A. Nagai, “Discrete Mittag-Leffler function and its applications, new developments in the research of integrable systems: continuous, discrete, ultra-discrete,” RIMS Kokyuroku, no. 1302, pp. 1–20, 2003.

G. V. S. R. Deekshitulu and J. J. Mohan, “Fractional difference inequalities,” Communications in Applied Analysis, vol. 14, no. 1, pp. 89–97, 2010.

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V. Lakshmikantham and M. R. M. Rao, Theory of Integro-Differential Equations, vol. 1, Gordon and Breach Science Publishers, Lausanne, The Switzerland, 1995.

H. L. Gray and N. F. Zhang, “On a new definition of the fractional difference,” Mathematics of Computation, vol. 50, no. 182, pp. 513–529, 1988.

G. V. S. R. Deekshitulu and J. J. Mohan, “Some new fractional difference inequalities,” in Proceedings of the International Conference on Mathematical Modelling and Scientific Computation (ICMMSC '12), vol. 283 of Communications in Computer and Information Science, pp. 403–412, Springer-Verlag, Berlin, Heidelberg, 2012.