Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 918529, 24 pages
doi:10.1155/2012/918529
Research Article
Fractional Difference Equations with Real Variable
Jin-Fa Cheng1 and Yu-Ming Chu2
1
2
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
School of Mathematics and Computation Science, Hunan City University, Yiyang, Hunan 413000, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 20 June 2012; Revised 14 October 2012; Accepted 25 October 2012
Academic Editor: Dumitru Bǎleanu
Copyright q 2012 J.-F. Cheng and Y.-M. Chu. 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 independently propose a new kind of the definition of fractional difference, fractional sum, and
fractional difference equation, give some basic properties of fractional difference and fractional
sum, and give some examples to demonstrate several methods of how to solve certain fractional
difference equations.
1. Introduction
Fractional calculus is an emerging field recently drawing attention from both theoretical and
applied disciplines. During the last two decades, it has been successfully applied to several
fields 1–6, and it is well known that there is a large quantity of research on what is usually
called integer-order difference equations 7, 8. However, discrete fractional calculus and
fractional difference equations represent a very new area for scientists. A pioneering work
has been done by Atici et al. 9–12, Anastassiou 13, 14, Bastos et al. 15, Abdeljawad
et al. 16–20, and Cheng 21–23, and so forth. In this paper, limited to the length of the
paper, we will introduce some of our basic works about discrete fractional calculus and
fractional difference equations. Some proofs and results of the theorems and examples in
Sections 3–5 are well proved by a more concise method. We refer to the monographer 23
for more further results. In 23 we also aim at presenting some basic properties about
discrete fractional calculus and, in a systematic manner, results including the existence and
uniqueness of solutions for the Cauchy Type and Cauchy problems, involving nonlinear
fractional difference equations, explicit solutions of linear difference equations and linear
difference system by their deduction to Volterra sum equation and by using operational
methods, applications of Z-transform, R-transform, N-transform, Adomian decomposition
method, method of undetermined coefficients, Jordan matrix theory method, and by discrete
Mittag-Leffler function and discrete Green’ function, and a theory of so-called sequential
2
Abstract and Applied Analysis
linear fractional difference equations, as well as some introduction for discrete fractional
difference variational problem, and so forth.
2. Integer-Order Difference and Sum with Real Variable
Let us start from sum and difference of the integer order. Define
t
h
xs xa xa h xa 2h · · · xt,
2.1
sa
where t a jh, j ∈ N0 {0, 1, 2, . . .}.
Definition 2.1. Let a, t be real numbers, and let h be a positive number, we call
−1
a ∇h xt
t
h
xsh
2.2
sa
one-order backward sum of xt, where t a jh, j ∈ N0 {0, 1, 2, . . .}. We call
−k−1
−k
−1
xt
a ∇h xt a ∇h a ∇h
2.3
k-order backward sum of xt, where k is a positive integer number.
Definition 2.2. Let a, t be real numbers, and let h be a positive number, we call
−1
a Δh xt
h
t−h
xsh
2.4
sa
one-order forward sum of xt, where t a jh, j ∈ N1 {1, 2, . . .}. We call
−k−1
−k
−1
xt
a Δh xt a Δh a Δh
2.5
k-order forward difference of xt, where k is a positive integer number.
Definition 2.3. Let t be a real number, and let h be a positive number, we call
∇h xt xt − xt − h
h
one-order backward difference of xt, where h is step. We call
∇kh xt ∇h ∇k−1
h xt
k-order backward difference of xt, where k is a positive integer number.
Similarly, we can define forward difference as follows.
2.6
2.7
Abstract and Applied Analysis
3
Definition 2.4. Let t be a real number, and let h be a positive number, we call
Δh xt xt h − xt
h
2.8
one-order forward difference of xt, where h is step. We call
Δkh xt Δh Δk−1
h xt
2.9
k-order forward difference of xt, where k is a positive integer number.
Theorem 2.5. The following two equalities hold:
1 ∇h a ∇−1
xt xt,
h
xt xt.
2 Δh a Δ−1
h
Definition 2.6. If k, t are real numbers, and let h be a positive number, define
tkh hk
Γt/h k
,
Γt/h
k ∈ R
2.10
rising factorial function, and set th0 1. If k is a positive integer number, then we have
tkh tt ht 2h · · · t k − 1h.
2.11
Definition 2.7. Let k, t be real numbers, and let h be a positive number, define
k
th hk
Γt/h 1
,
Γt/h 1 − k
k ∈ R
2.12
0
down factorial function, and set th 1. If k is an positive integer number, then
k
th tt − ht − 2h · · · t − k − 1h.
2.13
k
In Definitions 2.6 and 2.7, if h 1, we can simply denote thk , th as tk , tk .
Definition 2.8. For any k, γ ∈ R, h > 0, we define
Γ k
γ
γ
,
k
Γ γ Γk 1
⎡ ⎤
γ
γ
γ⎢ ⎥
h ⎣ ⎦.
k
k h
h
2.14
4
Abstract and Applied Analysis
If k ∈ N, h 1, then it is easy to see that
γ γ 1 ··· γ k − 1
γ
.
k
k!
2.15
If we let t/h t, or t th, then we clearly have the following.
Theorem 2.9. Assume that k ∈ R, h > 0, t/h t; then
k
tkh hk tk ;
th hk tk .
2.16
Theorem 2.10. Let k ∈ R, h > 0, then, following equality holds:
lim tk
h→0 h
k
lim th tk .
2.17
h→0
3. Fractional Sum and Difference with Real Variable
−γ
Before giving the definitions of fractional sum a ∇h xt, γ > 0, let us revisit the calculation of
the sum of the integer order. By Definition 2.1, we have
−1
a ∇h xt
h
t
xsh,
then
−2
a ∇h xt
−1
−1
a ∇h a ∇h xt
t a jh, j ∈ N0 ,
3.1
sa
t
h
sa
−1
a ∇h xsh
h
t
s
h
h
sa
xrh
ra
t
t
t
t
t−r
h
2
xr h t − r h1h xrh,
h h h xr h h
h
ra sr
ra
ra
3.2
t
t
s
t
−
r
h
−2
−2
−3
−1
2
xrh
a ∇h xt a ∇h a ∇h xt h
h
a ∇h xsh h
h
h
sa
sa ra
t t
1 t − r 2h
t−r
h
h3 2
xr t − r hh xrh . . . .
h
h
2
h
h
2! ra
ra
2
By recursive, it is not hard to obtain
−m
a ∇h xt
t hm
t−s
h
t − s 2h
t − s m − 1h
···
xs
h
h
h
h
m − 1! sa
t
t
m−1
1
1
m−1
t − ρh s h xsh ,
t − s hh xsh h
h
Γm sa
Γm sa
where ρh s s − h.
3.3
Abstract and Applied Analysis
5
Obviously, the right side of formula 3.3 is also meaningful for all real m > 0, so we
define fractional sum as follows.
Definition 3.1. Let γ > 0, a ∈ R, h > 0, t a kh, k ∈ N0 , we call
−γ
a ∇h xt
1
Γ γ
γ−1
t − ρh s h xsh
t
h
sa
3.4
γ order fractional sum of xt.
For any positive number order fractional difference, we take the following.
Definition 3.2. Let μ > 0, and assume that m − 1 < μ < m, where m denotes a positive integer.
Define
μ
a ∇h xt
∇m
h
−m−μ
xt
a ∇h
3.5
as μ order R-L type backward fractional difference. Meantime, define
C μ
a ∇h xt
−m−μ
a ∇h
∇m
h xt
3.6
as μ order Caputo type backward fractional difference.
If we start from Definition 2.2,
−1
a Δh xt
h
t−h
xsh,
3.7
t a jh, j ∈ N1 ,
sa
completely in a similar way, we get positive integer m-order forward sum
−m
a Δh xt
1
Γm
t−mh
t −
h
sa
m−1
xsh
σh sh
,
3.8
where σh s s h.
The right side of 3.8 is meaningful for all real m > 0, so we can define forward
fractional sum as follows.
Definition 3.3. Let γ > 0, a ∈ R, h > 0, t a γh kh, k ∈ N0 , define
−γ
a Δh xt
⎡
⎤
t−γh
1 ⎣ γ−1
t − σh sh xsh⎦
h
Γν sa
as γ order fractional sum of xt, where σh s s h.
3.9
6
Abstract and Applied Analysis
Definition 3.4. Let μ > 0, and assume that m − 1 < μ < m, where m denotes a positive integer.
Define
μ
a Δh xt
Δm
h
−m−μ
xt
a Δh
3.10
as μ order R-L type forward fractional difference. Meantime, define
C μ
a Δh xt
−m−μ
a Δh
Δm
h xt
3.11
as μ order Caputo type forward fractional difference.
In Definitions 3.1–3.4, if step h 1, it is a kind of important situation. At this time, we
−γ
−γ
μ
μ
simply denote a ∇h , a Δh ; ∇h , Δh as a ∇−γ , a Δ−γ ; ∇μ , Δμ . When h 1, backward fractional
sum is defined as follows.
Definition 3.5. Let γ > 0, and define
t
γ−1
1 −γ
∇
xt
xs
t − ρs
a
Γ γ sa
3.12
as γ order fractional sum of xt, where t a mod 1, ρs s − 1.
For any positive number order fractional difference, we can take the following way.
Definition 3.6. Let μ > 0 and assume that m − 1 < μ < m, where m denotes a positive integer.
Define
μ
a ∇ xt
∇m
−m−μ
xt
a∇
3.13
as μ order R-L type backward fractional difference. Meantime, define
C μ
a ∇ xt
a∇
−m−μ
∇m xt
3.14
as μ order Caputo type backward fractional difference.
We can define forward fractional sum as follows.
Definition 3.7. Let γ > 0, and define
t−γ
1 −γ
Δ
xt
t − σsγ−1 xs
a
Γ γ sa
as γ order forward fractional sum of xt, where t − γ a mod 1, σs s 1.
3.15
Abstract and Applied Analysis
7
Definition 3.8. Let μ > 0, and assume that m − 1 < μ < m, where m denotes a positive integer.
Define
μ
a Δ xt
Δm
−m−μ
xt
aΔ
3.16
as μ order R-L type forward fractional difference. Meantime, define
−m−μ
Δm xt
3.17
γ−1
1 γ
t − ρs
,
t−s
Γ γ
1
γ
t − σsγ−1 .
t−γ −s
Γ γ
3.18
C μ
a Δ xt
aΔ
as μ order Caputo type forward fractional difference.
By Definition 2.8, it is easy to calculate
By Theorem 2.9 we have
⎡
γ−1
t − s hh
Γ γ
γ−1
hh
t − s −
Γ γ
hγ−1
t − s/h 1
Γ γ
γ−1
⎢
hγ−1 ⎣
⎡
hγ−1
⎤
γ
⎥
,
t − s⎦
h
γ
⎤
3.19
t − s/h − 1γ−1
⎥
⎢
hγ−1 ⎣
⎦.
t−s
Γ γ
−γ
h
Therefore, if we adopt Definition 2.8, then Definitions 3.1, 3.3, 3.5, and 3.7 can be
rewritten as follows.
Definition 3.9. Assume that γ > 0, let a ∈ R, h > 0, t a kh, k ∈ N0 , and define
−γ
a ∇h xt t γ
xs
h
t−s h
sa
3.20
as γ order backward fractional sum of xt.
Definition 3.10. Assume that γ > 0, let a ∈ R, h > 0, t a γh kh, k ∈ N0 , and define
−γ
a Δh xt
as γ order forward fractional sum of xt.
t−γh
h
sa
γ
xs
t − s − γh h
3.21
8
Abstract and Applied Analysis
Definition 3.11. Assume that γ > 0, t, a ∈ R, and t a mod 1, and define
t γ
xs
a ∇ xt t−s
sa
−γ
3.22
as γ order backward fractional sum of xt.
Definition 3.12. Assume that γ > 0, t, a ∈ R, and t − γ a mod 1, and define
−γ
aΔ
xt t−γ sa
γ
xs
t−γ −s
3.23
as γ order forward fractional sum of xt.
Set a/h a, t/h t, or a ah, t th, and set xt xth yt; then by Theorem 2.9
and Definitions 3.1–3.4, one obtains the following.
Theorem 3.13. For any γ, μ > 0, the following equalities hold:
1
−γ
a ∇h xt
2
μ
a ∇h xt
−γ
hγ a ∇−γ yt; a Δh xt hγ a Δ−γ yt,
μ
h−μ a ∇μ yt; a Δh xt h−μ a Δμ yt,
−μ C μ
−μ C μ
C
3 C
a ∇h xt h a ∇ yt; a Δh xt h a Δ yt.
μ
μ
−γ
From Theorem 3.13 we can see, by stretching t th, the functions a ∇h xt and
with common step h, can be convert into the functions a ∇−γ yt and a ∇μ yt with
step h 1, respectively. In essence, nothing arises much different, but the latter is more
convenient in research.
In view of Definitions 3.1–3.4 and Theorem 2.10, if we let h → 0, then we can obtain
the following.
μ
a ∇h xt,
Corollary 3.14. Assume that xt is integrable, then:
t
−γ
−γ
−γ
1 limh → 0 a ∇h xt limh → 0 a ∇h xt 1/Γγ a t − sγ−1 xtds Dt xt,
μ
−m−μ
μ
2 limh → 0 a ∇h xt limh → 0 a ∇h xt Dm a Dt
μ
−m−μ
μ
C
m
3 limh → 0 C
a ∇h xt limh → 0 a ∇h xt D a Dt
μ
xt a Dt xt,
μ
xt C
a Dt xt.
4. Some Basic Properties
We sometimes only list some basic results here, for more detailed results and their proofs can
been seen in monographer 23.
Theorem 4.1. Assume that the following function is well defined; then
γ
γ
γ−1
γ−1
1 ∇h th γth , Δh th γth
2 t γ
γhth
γ
1
th ,
t −
γ
γhth
,
γ
1
th
, γ ∈ R,
Abstract and Applied Analysis
9
αγ
αγ
3 If 0 < γ < 1, then th ≤ tαh γ , th
α
β
4 th
β
α
β
t βαh th , th
α
≥ th γ ,
α β
t − βh th ,
γ
γ
γ
γ
γ
γ
γ
γ
5 Let 0 < t ≤ r, if γ > 0, then th ≤ rh , th ≤ rh ; If γ < 0, then th ≥ rh , th ≥ rh .
Theorem 4.2. Let 0 ≤ m − 1 < γ ≤ m, m ∈ N, where xt is defined in Nh,a {a, a h, a 2h, . . .},
then
−γ
−γ
1 a ∇h xt a Δh xt γh, t ∈ Nh,a ,
2 a ∇γ xt a Δγ xt − γh, t ∈ Nh,m
a .
Theorem 4.3. Let 0 ≤ m − 1 < γ ≤ m, m ∈ N, xt is defined in Nh,a {a, a h, a 2h, . . .}, then
−γ
−γ
1 a Δh xt a ∇h xt − γh, t ∈ Nh,a
γ ,
γ
γ
2 a Δh xt a ∇h xt γh, t ∈ Nh,a−γ
m .
Theorem 4.4. For any real γ, the following equality holds:
−γ
−γ
−γ
−γ
γ−1
1 a ∇h ∇h xt ∇h a ∇h xt − t − a − 1h /Γγxa − h,
γ−1
2 a Δh Δh xt Δh a Δh xt − t − ah
/Γγxa.
Theorem 4.5. For any real γ and p > 0, the following equality holds:
p−1
−γ p
p
−γ
γ−p
k
/Γγ k − p 1∇kh xa − h,
1 a ∇h ∇h xt ∇h a ∇h xt − k0 t − a 1h
p−1
−γ p
p
−γ
γ−p
k
2 a Δh Δh xt Δh a Δh xt − k0 t − ah
/Γγ k − p 1Δkh xa.
Theorem 4.6. Let p, γ > 0, then
p
−γ
−γ−p
p
−γ
−γ−p
1 ∇h a ∇h xt a ∇h
2 Δh a Δh xt a Δh
xt,
xt.
In the previous theorems, we only need to consider the simplest case h 1, but actually
the methods of proof and conclusions can also be extended for general step h > 0. In fact, we
only need do a stretching transformation and then make use of Theorem 2.9.
Next, we discusses fractional sum transform such as: Z transform, N transform, R
transform, and some properties of these transforms.
Definition 4.7. Let ft be defined in N0 {0, 1, 2, . . .}, we call
ft ∞
ftz−t
4.1
t0
is a Z transform of ft, denote it by Zft.
Definition 4.8. Let ft be defined in Nt0 {t0 , t0 1, t0 2, . . .}, t0 ∈ R, and define N transform
as follows:
∞
Nt0 ft s 1 − st−1 ft.
tt0
If the domain of the function ft is N1 , then we use the notation Nft.
4.2
10
Abstract and Applied Analysis
If we set t − t0 n ∈ N0 , define
{t }
{t }
fn 0 fn t0 ft,
{t }
f0 0
0
fn − 1 t0 ft − 1, . . . ,
fn−1
f0 t0 ft0 .
4.3
Then, ft0 , ft0 1, . . . , ft, . . . can be regarded as a sequence
{t0 }
f0
{t0 }
, f1
{t }
4.4
, . . . fn 0 , . . . .
Under this definition, N transform can be simply rewritten as
∞
N0 ft s 1 − st−1 fs
tt0
∞
1 − sn
t0 −1 fn t0 4.5
n0
1 − st0 −1
∞
{t }
1 − sn fn 0 .
n0
Set z 1/1 − s, then we have
∞
{t }
N0 ft s z1−t0 fn 0 z−n z1−t0 Fz,
4.6
n0
{t }
where Fz is Z transform of sequence fn 0 .
If t0 1, then
N ft Fz,
z
1
.
1−s
4.7
Theorem 4.9. For any γ ∈ R \ {. . . , −2, −1, 0}, then
1 Nt
γ−1
s Γγ/sγ , |1 − s| < 1,
2 Nt
γ−1
α−t s αγ−1 Γγ/s α − 1γ , |1 − s| < α.
Proof. 1 Making use of 4.7, we get
N
tγ−1
Γ γ
Fz,
4.8
Abstract and Applied Analysis
11
{1}
where Fz is Z transform of sequence fn
{1}
fn
fn 1,
n 1γ−1
γ
fn 1 .
n
Γ γ
4.9
Since see 21–23
F
1
z − 1 −γ
γ
γ,
n
z
s
hence
N
tγ−1
Γ γ
1
,
sγ
|z| > 1, |1 − s| < 1,
|1 − s| < 1.
4.10
4.11
2 It is only to use
∞
∞ 1
s α − 1 t−1 γ−1
t ,
1−
1 − st−1 tγ−1 α−t α t1
α
t1
4.12
then the proof of 2 follows from the proof of 1.
Theorem 4.10. Let ft and gt be defined in Na , and define convolution of ft, gt as follows:
t
h t − ρs gs.
h ∗ g a t 4.13
t
γ−1
1 gs a ∇−γ gt.
h ∗ g a t t − ρs
Γ γ sa
4.14
sa
For ht tγ−1 /Γγ, then
Theorem 4.11. Let f, g be defined in Na , then
Na f ∗ g N1 f N a g .
4.15
Theorem 4.12. For any real γ, one has
Na a ∇−γ ft s−γ Na ft .
4.16
Theorem 4.13. For 0 < γ ≤ 1, one has
Na
1 a ∇−γ ft sγ Na ft s − 1 − sα−1 fa,
where f is defined in Na .
4.17
12
Abstract and Applied Analysis
Theorem 4.14. Let μ ∈ R \ {. . . , −2, −1, 0}, γ > 0, then
−γ
1∇
tμ
Γ μ
1
tμ
γ
.
Γ μ
γ 1
4.18
Theorem 4.15. Let f be a real function, μ, γ > 0, then
−μ
−γ
a∇
a ∇ ft
a ∇−μ
γ ft a ∇−μ a ∇−γ ft .
4.19
Definition 4.16. Let ft be defined in Nt0 , and define R transform as follows:
Rt0
∞
ft tt0
1
s
1
t
1
4.20
ft.
In Definition 4.16, if we set t − t0 n ∈ N0 , and define:
{t }
{t }
fn 0 fn t0 ft,
{t0 }
f0
0
fn − 1 t0 ft − 1, . . . ,
fn−1
f0 t0 ft0 ,
4.21
then, ft0 , ft0 1, . . . , ft, . . . can be regarded as a sequence
{t0 }
f0
{t0 }
, f1
{t }
4.22
, . . . fn 0 , . . . .
Under this definition, R transform can be simply rewritten as
t
1
∞ 1
Rt0 ft s ft
s
1
tt0
n
t0 1
∞ 1
fn t0 s
1
n0
1
s
1
t0 1 ∞ n0
1
1
s
n
4.23
{t }
fn 0 .
Set z 1 s, then
∞
{t }
Rt0 ft s z−1−t0 fn 0 z−n z−1−t0 Fz,
n0
{t }
where Fz is a Z transform of sequence fn 0 .
4.24
Abstract and Applied Analysis
13
Theorem 4.17. For any γ ∈ R \ {. . . , −2, −1, 0}, then
1 Rγ−1 tγ−1 s Γγ/sγ ,
2 Rγ−1 tγ−1 αt s αγ−1 Γγ/s 1 − αγ .
Proof. 1 let t0 γ − 1, then
Rγ−1
tγ−1
Γ γ
z−γ Fz,
{γ−1}
where Fz is a Z transform of sequence fn
{γ−1}
fn
4.25
. Since
γ−1 n
γ −1
γ
f n
γ −1 ,
n
Γ γ
4.26
and see 22, 23
F
z − 1 −γ
γ
,
n
z
4.27
hence
Rγ−1
tγ−1
Γ γ
z − 1−γ s−γ ,
|1 s| < 1,
4.28
or
Rγ−1
tγ−1
Γ γ
1
,
sγ
|1 s| < 1.
4.29
2 The proof of 2 follows from the proof of 1.
Definition 4.18. Define convolution of ht and gt as follows:
t−γ
ht − σsgs.
h ∗ g t 4.30
sa
If ht tγ−1 /Γγ, then
t−γ
γ−1
1 gs a Δ−γ gt.
h ∗ g a t t − ρs
Γ γ sa
4.31
14
Abstract and Applied Analysis
Theorem 4.19. For any γ ∈ R \ {. . . , −2, −1, 0}, then
Rγ
a h ∗ g Rγ−1 hRa g .
4.32
Theorem 4.20. Let μ > 0, m − 1 < μ ≤ m ∈ N1 , and let ft be defined in Nμ−m {μ − m, μ − m 1, . . .}, then
R0 Δ ft s s Rμ−m ft s −
μ
μ
m−1
m−k−1
s
Δ
k−m
μ
k0
ft
.
4.33
t0
Theorem 4.21. Let μ ∈ R \ {. . . , −2, −1, 0}, γ > 0, then
−γ
Δ
tμ
Γ μ
1
tμ
γ
.
Γ μ
γ 1
4.34
Theorem 4.22. Let f be a real function, μ, γ > 0, then for all t μ γ mod 1, one has
Δ−γ Δ−μ ft Δ−μ
γ ft Δ−μ Δ−γ ft .
4.35
5. The Solution of the Fractional Difference Equations with
Real Variable
In this section, we give examples to demonstrate the solving method of fractional difference
equations and reveal the inner relationship between fractional differential equations and
fractional differential equations.
Theorem 5.1. Let μ ∈ R, γ ∈ R, then
1 ∇γ tμ μγ tμ−γ , Δγ tμ μγ tμ−γ ,
2 Δγ tμ μγ t γμ−γ , ∇γ tμ μγ t − γμ−γ .
Proof. 1 The proof of 1 directly follows from Theorem 4.1 and Theorem 4.2.
2 By Theorem 4.2 and 1, we have
μ
μ−γ
Δγ tμ ∇γ t γ μγ t γ
,
μ
μ−γ
∇γ tμ Δγ t − γ
μγ t − γ
.
5.1
Example 5.2. Consider Euler type fractional difference equations
t2α Δ2α xt atα Δα xt bxt 0,
0 < α < 1.
5.2
Abstract and Applied Analysis
15
Set xt tγ , and take it into previous equation, we get
t2α γ 2α t 2αγ−2α atα γ α t αγ−α btγ 0.
5.3
By Theorem 4.1 4, we obtain
γ 2α tγ aγ α tγ btγ 0,
5.4
γ 2α aγ α b 0.
5.5
and get indicator equation
Therefore, we can transform Euler type fractional difference equations into its indicator equation.
Example 5.3. Consider initial value problem of homogeneous linear γ order 0 < γ ≤ 1
fractional difference equation with constant coefficient
∇γ yt a∇0 yt 0, t ∈ N0 ,
∇γ−1 t
a0 .
5.6
t−1
Note that ∇γ−1 yt is defined in N−1 {−1, 0, 1, 2, . . .}, since
γ−1
ft
−1 ∇
t−1
t
−γ
1
t − ρs ys
Γ 1 − γ s−1
1−γ
y−1 y−1.
Γ 1−γ
5.7
Therefore, initial problem of 5.6 is equivalent to initial problem
∇γ yt a∇0 yt 0,
t ∈ N,
y−1 a0 .
5.8
The solution of initial problem of 5.6 is equivalent to the solution of sum equations
yt t
γ−1
t 1γ−1
ys.
t − ρs
a0 a
Γ γ
s0
5.9
16
Abstract and Applied Analysis
We use approximation method to solve these sum equations. Set
y0 t t 1γ−1
a0 ,
Γ γ
t
γ−1
a ym t y0 t ym−1 s
t − ρs
Γ γ s0
y0 t a∇−γ ym−1 t,
5.10
m 1, 2, . . . .
Applying power law Theorem 4.22, we get
−γ
y1 t y0 t a∇ y0 t a0
t 1γ−1
t 12γ−1
a
Γ γ
Γ 2γ
.
5.11
Applying power law repeatedly, and by recursion, we obtain
ym t a0
m
ai tiγ
γ−1
,
i0 Γ i 1γ
m 0, 1, 2, . . . .
5.12
Let m → ∞, then
∞ i
∞
a t 1iγ
γ−1
i iγ γ
yt a0
.
a0 a
t
i0 Γ i 1γ
i0
5.13
Example 5.4. Let γ 1/q, q ∈ N, we call
∇γ yt − a∇0 yt 0,
t ∈ N0 ,
5.14
the fractional difference equation of order 1, q.
In order to solve this equation, we need to introduce some special functions.
Definition 5.5. Define function
Λ t, γ, λ a ∇−γ λt ,
γ ∈ R,
where t a mod 1. Sometimes denote it Λγ, λ or Λt, γ, λ; a.
In view of Theorems 4.2 and 4.3, we can establish the following theorem.
Theorem 5.6. Assume the following function is well defined; then
1 Λt, γ, λ 1 − 1/λΛt, γ 1, λ t − a 1γ /Γγ 1,
2 ∇Λt, γ 1, λ Λt, γ, λ,
3 ∇p Λt, γ t, λ Λt, γ, λ, where p 0, 1, 2, . . .,
5.15
Abstract and Applied Analysis
17
4 ∇μ Λt, γ, λ Λt, γ − μ, λ, where p − 1 < μ ≤ p,
5 ∇−μ Λt, γ, λ Λt, γ μ, λ.
Now we will use the method of undetermined coefficients to solve Example 5.4. By
Theorem 5.6, we notice that
∇γ Λt, 0, λ Λ t, −γ, λ ,
∇γ Λ t, −γ, λ Λ t, −2γ, λ ,
..
.
∇ Λ t, − q − 2 γ, λ Λ t, − q − 1 γ, λ ,
1
Λt, 0, λ.
∇γ Λ t, − q − 1 γ, λ Λt, −1, λ 1 −
λ
γ
5.16
The significance of these applications is that if we apply the operator ∇γ to
Λt, 0, λ, Λ t, −γ, λ , . . . , Λ t, − q − 1 γ, λ ,
5.17
then we get a cyclic permutation of the same functions. That is, no new functions are
introduced. Therefore, we will choose a linear combination of these functions as a candidate
for a solution of 5.14. Say
yt b0 Λt, 0, λ b1 Λ t, −γ, λ
. . . bq−2 Λ t, − q − 2 γ, λ bq−1 Λ t, − q − 1 γ, λ .
5.18
Then
∇γ yt b0 Λ t, −γ, λ b1 Λ t, −2γ, λ
1
. . . bq−2 Λ t, − q − 1 γ, λ bq−1 1 −
Λt, 0, λ.
λ
5.19
Taking yt, ∇γ yt into the left side of 5.14, we obtain
1
∇γ yt − ayt bq−1 1 −
− ab0 Λt, 0, λ
λ
b0 − ab1 Λ t, −γ, λ · · · bq−2 − abq−1 Λ t, − q − 1 γ, λ .
5.20
In order to make the right side equate zero, set
bk cα−k ,
k 1, 2, . . . , q − 1 .
5.21
18
Abstract and Applied Analysis
Then
bk−1 − abk c α−k
1 − aα−k cα−k α − a.
5.22
If we let α be a root of the indicial equation
P x x − a 0,
5.23
or α a, then we have
k 1, 2, . . . , q − 1 .
bk−1 − abk ca−k P a 0
5.24
Since we also need
1 −q
1
0 bq−1 1 −
− ab0 ac 1 −
a −1 ,
λ
λ
5.25
so let us set
1
1−
aq ,
λ
λ
1
.
1 − αq
5.26
Since c is an arbitrary number, set c aq−1 , then
5.27
bk aq−1−k .
Therefore, we obtain a solution of fractional difference of order 1, q as
yt q−1
bk Λ t, −kγ, λ
k0
q−1
k0
aq−1−k Λ t, −kγ,
1
1 − aq
5.28
λa t.
The fractional difference equation of order 1, q in Example 5.4 can be solved by the
method of N0 transform. Make N1 transform to the following equation:
∇γ yt − a∇0 yt 0.
5.29
Abstract and Applied Analysis
We have
19
sγ N0 ft − 1 − s−1 f0 aN1 ft 0,
∞
N1 ft 1 − st−1 ft
t1
∞
5.30
1 − st−1 ft − 1 − s−1 f0.
t0
Taking them into previos equation, we get
sγ N0 ft − 1 − a1 − s−1 f0 − aN0 ft 0,
5.31
and we have
N0 ft 1 − ay0
1
1 − ssγ − a
q−1 q−1−k kγ
a
s
k0
1 − ay0
q−1
1 − ssγ − a k0 aq−1−k skγ
q−1
1 − ay0
k0
aq−1−k skγ
5.32
.
1 − ss − aq In 23, we have the following
Theorem 5.7. The following equality holds:
1 N0 Λt, 0, λ N0 λt 1/1 − s · 1/1 − 1 − sλ,
2 N0 Λt, −kγ, λ N0 ∇kγ λt 1/1 − s · skγ /1 − 1 − sλ · k 1, 2, . . . , q − 1.
Set λ 1/1 − aq , then
N0 Λ t, 0,
N0 Λ t, −kγ,
1
1 − aq
1
1 − aq
1 − aq
1
·
,
1 − s s − aq
skγ 1 − aq
·
.
1 − s s − aq
k 1, 2, . . . , q − 1 .
5.33
By Theorem 5.7 and 5.33, we know that
yt 1 − ay0
q−1
aq−1−k Λ t, −kγ,
k0
1
1 − aq
5.34
is a solution of 5.14. Except a constant, the solution yt is the same as the solution 5.28, where
yt λa t,
which is solved by the method of undetermined coefficients before.
5.35
20
Abstract and Applied Analysis
6. Relationship between the Fractional Difference Equations and
the Fractional Differential Equations
In this section, we only give an example to demonstrate the relationship between integers
order difference equations and integral order differential equation.
Let us recall the definition of fractional sum when step h 1
−γ
a ∇t ft
t
γ−1
1 fs,
t − ρs
Γ γ sa
6.1
where t ∈ Na {a, a 1, a 2, . . .}. If we set
t − a n ∈ N0 ,
{a}
fr
fr a fs,
s − a r ∈ N0 ,
{a}
fn
fn a ft,
6.2
then
t
n
a
γ−1
γ−1
1 1 fs fs
t − ρs
n a − ρs
Γ γ sa
Γ γ sa
n
1 n a − r a 1γ−1 fr a
Γ γ r0
n
1 −γ {a}
{a}
n − r 1γ−1 fr 0 ∇n fn .
Γ γ r0
6.3
And it is easy to prove that
μ>0 .
μ
μ {a}
a ∇t ft0 ∇n fn ,
6.4
Therefore, we have the following.
{a}
Theorem 6.1. Let t ∈ Na , and set t − a n ∈ N0 , fn
−γ
a ∇t ft
−γ
{a}
0 ∇n fn ;
μ
a ∇t ft
fn a ft, then
μ {a}
0 ∇n fn ,
μ, γ > 0 .
6.5
Example 6.2. 1 Set γ 1/q, q ∈ N, n ∈ N, and solve the fractional difference equation of
order 1, q,
∇γ xn − αxn 0.
6.6
2 Let t ∈ R, and solve the equation
∇γ xt − αxt 0.
6.7
Abstract and Applied Analysis
21
3 Let h ∈ R
, t ∈ R, and solve the equation
γ
6.8
∇h xt − αxt 0.
4 If we let h → 0, we ask whether the limit solution of 6.8 is equivalent to that of
the following fractional differential equation? Consider
Dγ xt − αxt 0,
t ∈ R.
6.9
Solution 1. 1 By a result in Chapter 7 of book 23, the solution of 6.6 is
q−1
q−k−1
α
Λn −kγ,
xn λα n k0
1
1 − αq
n .
6.10
2 Set t − t0 n ∈ N0 , and define
{t }
{t }
xn 0 xn t0 xt,
0
xn − 1 t0 xt − 1, . . . ,
xn−1
{t }
x0 0 x0 t0 xt0 .
6.11
Hence, we can regard the following xt0 , xt0 1, . . . , xt, . . . as a sequence
{t }
{t }
{t }
6.12
x0 0 , x1 0 , . . . xn 0 , . . . .
Under this definition, 6.7 is actually equivalent to the following integer variable difference
equation:
{t }
{t }
6.13
∇γ xn 0 − αxn 0 0.
By 1, we know that its solution is
{t }
xn 0 q−1
αq−k−1 Λn −kγ,
k0
1
1 − αq
q−1
q−k−1
α
Λ t, −kγ,
k0
n 1
1 − αq
6.14
t .
That is
xt q−1
k0
q−k−1
α
Λ t, −kγ,
1
1 − αq
t λα t.
6.15
22
Abstract and Applied Analysis
3 Set t sh, xt xsh ys, then 6.8 is equivalent to
h−γ ∇γ ys − αys 0.
6.16
By 2, we obtain that the solution of 6.16 is
xt ys λαhγ s
q−1
αhγ q−k−1 Λ s, −kγ,
k0
Since
Λ s, −kγ,
s 1
1 − αhγ q
1
1 − αhγ q
h Λh t, −kγ,
kγ
s 6.17
.
1
1 − αhγ q
t/h ,
6.18
hence we have
q−1
γ q−k−1 kγ
xt h Λh t, −kγ,
αh k0
1
1 − αhγ q
t/h .
6.19
4 Let h → 0, and since
h Λh t, −kγ,
kγ
1
1 − αq h
t/h 1
1 − αq h
h
kγ
t/h
kγ
∇h
q
−→ eα ,
1
1 − αq h
t/h
kγ αq
−→ D e
6.20
E −kγ, α .
q
We then obtain
xt eα t q−1
αq−k−1 E −kγ, αq ,
6.21
k0
and this is exactly the solution of 6.9. See Chapter 5 in monographer 2.
Remark 6.3. If we take γ 1/2, q 2, then the followong occurs.
1 The solution of 6.19 reduces to
xt α
t
t
1
1
1/2
∇
1 − α2
1 − α2
1
1
1
,
F
t,
−
,
αF t, 0,
2 1 − α2
1 − α2
6.22
and this result is consistent with the solution 5.28 or 5.34 in Example 5.4 in Section 5.
Abstract and Applied Analysis
23
2 The solution 6.21 reduces to
t/h
t/h
1
1
1/2 1/2
h
∇
h
1 − αq h
1 − αq h
t/h
t/h 1
1
1/2
1/2
α
.
h
∇h
1 − αq h
1 − αq h
xt αh1/2
6.23
Let h → 0, then
α
1
1 − αq h
t/h
∇1/2
h
1
1 − αq h
t/h 6.24
tend to
q
q
αeα t D1/2 eα t eα t.
6.25
The results perfectly coincide with the monographer 2.
From Theorem 6.1, we see that if we take t as a, a
1, a
2, . . ., it is only a sequence with
step 1, but the initial time is not zero but a. If we make a translation variable transformation,
set t n a, n ∈ N0 , then we can change the definition of fractional sum and fractional
difference with real variable into the definition of fractional sum and difference with integer
variable. But, no doubt, it will be more convenient for us to study fractional sum and
difference with integer variable.
7. Conclusion
This work reveals some results in discrete fractional calculus and fractional h-difference
equations. This study also provides a reference for researchers in this area. First, this
paper gives the definition of the fractional h-difference from the difference of integer order.
Then some integral transforms are proposed, that is, Z transform, N transform, and R
transform. These integral transforms are applied to linear fractional h-difference equations,
and approximate solutions are obtained. At last, the study explains the relationship between
the fractional difference equations and the fractional differential equations.
Acknowledgment
The authors wish to thank the referee for the very careful reading of the manuscript and
fruitful comments and suggestions.
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