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International Journal of Differential Equations
Volume 2012 (2012), Article ID 780619, 11 pages
Fractional Order Difference Equations
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.
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.
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  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  defined the fractional order difference operator where is any real number, using Taylor’s series. Nagai  adopted another definition for fractional order difference operator by modifying Hirota’s  definition. Recently, Deekshitulu and Mohan  modified the definition of Nagai  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 . Much of literature is not available on fractional integrodifferential equations also, though theory of integrodifferential equations  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.
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 ). 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  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  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  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.
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 .
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 ,
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 .
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 .
Example 4.2. Find the solution of
Example 4.3. Find the solution of
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 .
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