Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 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
Correspondence should be addressed to J. Jagan Mohan, j.jaganmohan@hotmail.com
Received 11 May 2012; Accepted 17 October 2012
Academic Editor: Shaher Momani
Copyright q 2012 J. J. 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.
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 nth difference, to
be any real or complex number. Later, Hirota 2 defined the fractional order difference
operator ∇α where α 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 0 < α < 1
in such a way that the expression for ∇α 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.
2
International Journal of Differential 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: N be the set of natural
numbers including zero. Na {a, a 1, a 2, . . .} for a ∈ Z. Let un : N0 → R. Then for
2
2
uj 0, and njn
uj 1, that is, empty sums and
all n1 , n2 ∈ N0 and n1 > n2 , njn
1
1
products are taken to be 0 and 1, respectively. If n and n 1 are in N0 , then for this function
un, the backward difference operator ∇ is defined as ∇un un − un − 1. Now we
introduce some basic definitions and results concerning nabla discrete fractional calculus.
Definition 2.1. The extended binomial coefficient na , a ∈ R, n ∈ Z is defined by
⎧
Γa 1
⎪
⎪
⎪
⎨
Γa − n 1Γn 1
a
1
⎪
n
⎪
⎪
⎩0
n > 0,
n 0,
n < 0.
2.1
Definition 2.2 see 7. For any complex numbers α and β, let αβ be defined as follows:
⎧ Γ αβ
⎪
⎪
⎪
⎪ Γα
⎪
⎪
⎨
αβ 1
⎪
⎪
⎪
0
⎪
⎪
⎪
⎩undefined
when α and α β are neither zero nor negative integers,
when α β 0,
when α 0, β is neither zero nor negative integer,
otherwise.
2.2
Remark 2.3. For any complex numbers α and β, when α, β and α β are neither zero nor
negative integers,
αβ
n
n n
k0
k
αn−k β k
2.3
for any positive integer n.
In 2003, Nagai 3 gave the following definition for fractional order difference operator.
International Journal of Differential Equations
3
Definition 2.4. Let α ∈ R and m be an integer such that m − 1 < α ≤ m. The difference operator
∇ of order α, with step length ε, is defined as
⎧
n−1
⎪
α−m
⎪
α−m
m
m−α
⎪
⎪
α > 0,
∇ ∇ un ε
−1j ∇m u n − j
⎪
⎪
j
⎪
j0
⎨
∇α un un
α 0,
⎪
⎪
⎪
n−1
⎪
α
⎪
−α
⎪
α < 0.
−1j u n − j
⎪
⎩ε
j
2.4
j0
The above definition of ∇α un given by Nagai 3 contains ∇ operator and the term
−1 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 ε m 1. Let α ∈ R such that 0 < α < 1.
j
Definition 2.5. The fractional sum operator of order α is defined as
n−1 n j α−1 n−j α−1 ∇ un u n−j u j .
j
n−j
j0
j1
−α
2.5
And the fractional order difference operator of order α is defined as
∇α un n−1 j −α
j0
j
n n−j −α−1 n−α−1
∇u n − j u j −
u0.
n−j
n−1
j1
2.6
Throughout this paper we assume that α ∈ R and 0 < α < 1 unless until specified.
Remark 2.6. Let un, vn : N0 → R and α, β ∈ R such that 0 < α, β, α β < 1 and c, d are
constants. Then
1 ∇α ∇β un ∇αβ un.
2 ∇α cun dvn c∇α un d∇α vn.
3 ∇−α ∇α un un − u0.
4 ∇α ∇−α un un.
5 ∇α u0 0 and ∇α u1 u1 − u0 ∇u1.
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.
4
International Journal of Differential Equations
Definition 3.1. Let fn, r be any function defined for n ∈ N0 , 0 ≤ r < ∞. Then a nonlinear
difference equation of order α together with an initial condition is of the form
∇α un 1 fn, un,
u0 u0 .
3.1
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 un by ∇α un, we have
∇−α ∇α un n n − j α − 1 α ∇ u j ,
n−j
j1
or un − u0 n n − j α − 1 α ∇ u j ,
n−j
j1
or un u0 n−1 n−j α−2 n−j −1
j0
or un u0 ∇ u j 1 ,
α
3.2
n−1
B n − 1, α; j f j, u j ,
j0
n−jα−1
where Bn, α; j for 0 ≤ j ≤ n. The above recurrence relation shows the existence
n−j
of solution of 3.1.
Example 3.2. If fn, un anun bn where an, bn : N0 → R are any two nonnegative functions then
un u0
n−1
n−1
n−1
1 B n − 1, α; j a j B n − 1, α; j b j
1 B n − 1, α; j ak . 3.3
j0
j0
kj1
Recently, the authors have established the following fractional order discrete Gronwall-Bellman type inequality 8.
Theorem 3.3. Let un, an, and bn be real valued nonnegative functions defined on N0 . If, for
0 < α < 1,
un ≤ u0 n−1
B n − 1, α; j a j u j b j
3.4
j0
then
un ≤ u0
n−1
j0
for n ∈ N0 .
n−1
n−1
1 B n − 1, α; j a j B n − 1, α; j b j
1 B n − 1, α; j ak 3.5
j0
kj1
International Journal of Differential Equations
5
Definition 3.4. Let fn, r and gn, m, r be any functions defined for n, m ∈ N0 , m ≤ n and
0 ≤ r < ∞. Then a nonlinear difference equation of Volterra type of order α together with an
initial condition is of the form
∇α un 1 fn, un n−1
gn, m, um,
u0 u0 .
3.6
m0
From 3.2 and 3.6, we have
j−1
n−1
un u0 B n − 1, α; j f j, v j g j, m, vm .
3.7
m0
j0
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 n ∈ N0 ,
n
nα
B n, α; j .
n
j0
3.8
Proof. Consider
Γ n−j α
j0 ΓαΓ n − j 1
n
n
B n, α; j j0
n 1
n
1j αn−j using Definition 2.2
j
Γn 1 j0
3.9
1
1 αn using Remark 2.3
Γn 1
Γn α 1
1
nα
.
n
Γn 1 Γα 1
Theorem 3.6. Let fn, r be any function defined for n ∈ N0 , 0 ≤ r < ∞. Suppose that f satisfies
fn, vn − fn, wn ≤ L|vn − wn|,
3.10
where vn, wn : N0 → R are any two nonnegative functions, and L is a nonnegative constant.
Then there exists unique solution for the initial value problem 3.1.
6
International Journal of Differential Equations
Proof. Let vn and wn be any two solutions of the initial value problem 3.1 such that
v0 w0. Then, using 3.2 and 3.10, we get
n |vn 1 − wn 1| B n, α; j f j, v j − f j, w j j0
≤
n
B n, α; j f j, v j − f j, w j j0
n
≤ L B n, α; j v j − w j 3.11
j0
< L
n
B n, α; j v j − w j ,
j0
where > 0 be any small quantity. Let yn |vn − wn|. Then
yn 1 < L
n
B n, α; j y j .
3.12
j0
Using Discrete Fractional Gronwall’s inequality i.e., Theorem 3.3 we get
⎤
⎡
n−1
n−1
1 LB n − 1, α; j < exp⎣L B n − 1, α; j ⎦.
yn < j0
3.13
j0
Using Lemma 3.5, we get
nα−1
yn < exp L
.
n−1
3.14
Since is arbitrary, inequality 3.14 implies that yn tends to zero. Thus vn wn.
Theorem 3.7. Consider a nonlinear fractional difference equation of Volterra type 3.6. Suppose that
f and g satisfies
fn, vn − fn, wn ≤ L|vn − wn|,
n−1 gn, m, vm − gn, m, wm ≤ L2 |vn − wn|,
3.15
m0
where vn, wn : N0 → R are any two nonnegative functions and L is a nonnegative constant.
Then there exists a unique solution for the initial value problem 3.6.
International Journal of Differential Equations
7
Proof. Let vn and wn be any two solutions of the initial value problem 3.6 such that
v0 w0. Then, using 3.2, we get
n |vn 1 − wn 1| B n, α; j f j, v j − f j, w j
j0
j−1
n
B n, α; j
g j, m, vm − g j, m, wm m0
j0
≤
n
B n, α; j f j, v j − f j, w j j0
j−1
n
g j, m, vm − g j, m, wm B n, α; j
m0
j0
≤L
n
n
B n, α; j v j − w j L2 B n, α; j v j − w j j0
j0
n
≤ L L2
B n, α; j v j − w j j0
n
< L L2
B n, α; j v j − w j ,
j0
3.16
where > 0 be any small quantity. Let yn |vn − wn|. Then
n
yn 1 < L L2
B n, α; j y j .
3.17
j0
Using Discrete Fractional Gronwall’s inequality i.e., Theorem 3.3 we get
yn < ⎡
⎤
n−1
L L2 B n − 1, α; j < exp⎣ L L2
B n − 1, α; j ⎦.
n−1 3.18
j0
j0
Using Lemma 3.5 we have
yn < exp
L L2
n α − 1
.
n−1
Since is arbitrary, inequality 3.19 implies that yn tends to zero. Thus vn wn.
3.19
8
International Journal of Differential Equations
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 gr be any real-valued function defined on R. Then an autonomous difference equation of order α together with an initial condition is of the form
∇α un 1 gun,
u0 u0 .
4.1
If the function g in 4.1 is replaced by a function f of two variables, that is, f : N0 × R → R,
then we have
∇α un 1 fn, un,
u0 u0 .
4.2
Equation 4.2 is called a nonautonomous difference equation of order α.
Now we consider the simplest special cases of 4.1 and 4.2, namely, linear equations.
The linear autonomous difference equation of order α has following general form:
∇α un 1 aun b,
u0 u0 ,
4.3
where a /
0 and b are known constants.
The general form of linear homogeneous nonautonomous difference equation of order
α is given by
∇α un 1 anun,
u0 u0
4.4
and the associated nonhomogeneous equation is given by
∇α un 1 anun bn,
u0 u0 ,
4.5
where an / 0 and bn are real-valued functions defined for n ∈ N0 .
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
un u0 n−1
B n − 1, α; j a j u j
j0
4.6
International Journal of Differential Equations
9
implies
un u0
n−1
1 B n − 1, α; j a j .
4.7
jn0
Solution of 4.5
Using 3.2 and 4.5, we obtain
un u0 n−1
B n − 1, α; j a j u j b j
4.8
j0
implies
un u0
n−1
n−1
n−1
B n − 1, α; j b j
1 B n − 1, α; j a j 1 Bn − 1, α; kak.
jn0
jn0
kj1
4.9
Solution of 4.3
Using 4.9, we obtain
un u0
n−1
n−1
n−1
1 aB n − 1, α; j b B n − 1, α; j
1 Bn − 1, α; ka
j0
j0
u0
n−1
j0
u0
n−1
kj1
n−1
n−1
b aB n − 1, α; j
1 aB n − 1, α; j 1 aBn − 1, α; k
a j0
kj1
1 aB n − 1, α; j
j0
⎤
⎡
n−1
b ⎣
1 aB n − 1, α; j − 1 ⎦
a j0
4.10
n−1
b
b u0 1 aB n − 1, α; j − .
a j0
a
Example 4.2. Find the solution of
u0 0.
4.11
⎞⎤
3
⎣1 ⎝n − j − 2 ⎠⎦ − 1.
un n−j −1
j0
4.12
∇1/2 un 1 un 1,
Solution. Using 4.10, the solution of 4.11 is
n−1
⎡
⎛
10
International Journal of Differential Equations
Table 1: Numerical solution of 4.17.
n
0
1
2
3
4
5
6
7
8
9
10
0.2
1.0000
2.0000
2.4000
2.6880
2.9245
3.1304
3.3156
3.4855
3.6437
3.7926
3.9337
α
0.6
1.0000
2.0000
3.2000
4.7360
6.7062
9.2170
12.3917
16.3755
21.3391
27.4839
35.0464
0.4
1.0000
2.0000
2.8000
3.5840
4.3868
5.2221
6.0970
7.0164
7.9838
9.0020
10.0735
0.8
1.0000
2.0000
3.6000
6.1920
10.3530
16.9624
27.3580
43.5659
68.6384
107.1529
165.9425
1.0
1.0000
2.0000
4.0000
8.0000
16.0000
32.0000
64.0000
128.0000
256.0000
512.0000
1024.0000
Example 4.3. Find the solution of
∇1/2 un 1 un n,
u0 0.
4.13
Solution. Using 4.9, the solution of 4.13 is
⎛
⎛
⎞
⎞⎤
⎡
3
3 n−1
n−j − ⎠
⎣1 ⎝n − k − 2 ⎠⎦.
un j⎝
2
n − j − 1 kj1
n−k−1
j0
n−1
4.14
Example 4.4. Find the solution of
u0 1.
4.15
⎞⎤
3
⎣1 ⎝n − j − 2 ⎠⎦.
un n−j −1
j0
4.16
∇1/2 un 1 un,
Solution. Using 4.7, the solution of 4.15 is
n−1
⎡
⎛
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
∇α un 1 un,
u0 1
4.17
using a simple program in MATLAB for various values of α.
The numerical solution for various values of n and α are tabulated in Table 1. The
graphical representation of the numerical solution for various values of n and α are shown in
Figure 1.
International Journal of Differential Equations
11
104
u(n)
103
102
101
100
0
1
2
3
4
5
6
7
8
9
10
n
α = 0.2
α = 0.4
α = 0.6
α = 0.8
α=1
Figure 1: Numerical solution of 4.17.
References
1 J. B. Diaz and T. J. Osler, “Differences of fractional order,” Mathematics of Computation, vol. 28, pp.
185–202, 1974.
2 R. Hirota, Lectures on Difference Equations, Science-sha, Tokyo, Japan, 2000.
3 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.
4 G. V. S. R. Deekshitulu and J. J. Mohan, “Fractional difference inequalities,” Communications in Applied
Analysis, vol. 14, no. 1, pp. 89–97, 2010.
5 V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, CSP, Cambridge, UK,
2009.
6 V. Lakshmikantham and M. R. M. Rao, Theory of Integro-Differential Equations, vol. 1, Gordon and Breach
Science Publishers, Lausanne, The Switzerland, 1995.
7 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.
8 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.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014