Research Article Fractional Difference Equations with

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, [email protected] Received 20 June 2012; Revised 14 October 2012; Accepted 25 October 2012 Academic Editor: Dumitru Bˇaleanu 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

xs h

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  xs h

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 h t 2h · · · t k − 1 h .

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 − h t − 2h · · · t − k − 1 h .

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  xs h,

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 xs h

h

t s   h

h sa

 xr h

ra

   t t t t     t−r h 2 xr  h t − r h 1h xr h,  h h h xr  h h h ra sr ra ra 3.2   t t s      t − r

h −2 −2 −3 −1 2 xr h a ∇h xt  a ∇h a ∇h xt  h h a ∇h xs h  h h h sa sa ra       t  t 1  t − r 2h t−r h h3  2 xr   t − r h h xr h . . . . 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 − 1 h  ··· xs h h h h m − 1 ! sa     t t   m−1  1 1 m−1  t − ρh s h xs h , t − s h h xs h  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 xs h

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  xs h,

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 xs h σh s h

,

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 s h xs h⎦ 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 − σs γ−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 − σs γ−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 h h   Γ γ γ−1 h h

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 − 1 γ−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 − s γ−1 xt ds  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 γh th



γ 1 th ,

t −

γ γh th

, 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 β αh th , th

α

≥ th γ ,

α β

 t − β h 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 − 1 h /Γγ xa − h , γ−1

2 a Δh Δh xt  Δh a Δh xt − t − a h

/Γγ xa .

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 1 h p−1 −γ p p −γ γ−p k 2 a Δh Δh xt  Δh a Δh xt − k0 t − a h /Γγ k − p 1 Δkh 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 

∞  ft z−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 − s t−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 − s t−1 fs tt0 ∞  1 − s n t0 −1 fn t0



4.5

n0

 1 − s t0 −1

∞ 

{t }

1 − s n 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 − 1 γ , |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 1 γ−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 − s t−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 − s α−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 − α γ . 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 − σs gs . 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 h Ra 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 γ μ−γ , ∇γ tμ  μγ t − γ μ−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 2α γ−2α atα γ α t α γ−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 1 γ−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 1 γ−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 1 γ−1 t 1 2γ−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 1 γ

m  0, 1, 2, . . . .

5.12

Let m → ∞, then   ∞ i ∞   a t 1 iγ γ−1 i iγ γ yt  a0 .    a0 a t i0 Γ i 1 γ 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/λ Λt, γ 1, λ t − a 1 γ /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 − s t−1 ft t1



∞ 

5.30

1 − s t−1 ft − 1 − s −1 f0 .

t0

Taking them into previos equation, we get     sγ N0 ft − 1 − a 1 − s −1 f0 − aN0 ft  0,

5.31

and we have   N0 ft  1 − a y0

1 1 − s sγ − a q−1 q−1−k kγ a s k0  1 − a y0  q−1 1 − s sγ − a k0 aq−1−k skγ q−1

 1 − a y0

k0

aq−1−k skγ

5.32

.

1 − s s − 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 − s λ , 2 N0 Λt, −kγ, λ  N0 ∇kγ λt  1/1 − s · skγ /1 − 1 − s λ · 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 − a y0

 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 1 γ−1 fr a Γ γ r0



n 1  −γ {a} {a}   n − r 1 γ−1 fr  0 ∇n fn . Γ γ r0

6.3

And it is easy to prove that 

 μ>0 .

μ μ {a} a ∇t ft 0 ∇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.

References 1 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, Russia, 1987. 2 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. 3 I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.

24

Abstract and Applied Analysis

4 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. 5 K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. 6 Z.-X. Zheng, “On the developments applications of fractional equations,” Journal of Xuzhou Normal University, vol. 26, no. 2, pp. 1–10, 2008. 7 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, Marcel Dekker, New York, NY, USA, 2000. 8 M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2003. 9 F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007. 10 F. M. Atici and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 3, pp. 1–12, 2009. 11 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. 12 F. M. Atici and S. S¸engul, ¨ “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1–9, 2010. 13 G. A. Anastassiou, “Discrete fractional calculus and inequalities,” http://arxiv.org/abs/0911.3370 . 14 G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562–571, 2010. 15 N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513–524, 2011. 16 T. Abdeljawad, “On riemann and caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. 17 T. Abdeljawad, “Principles of delta and nabla fractional differences,” http://arxiv.org/abs/1112 .5795. 18 F. Jarad, T. Abdeljawad, D. Baleanu, and K. Bic¸en, “On the stability of some discrete fractional nonautonomous systems,” Abstract and Applied Analysis, vol. 2012, Article ID 476581, 9 pages, 2012. 19 T. Abdeljawad and D. Baleanu, “Caputo q-fractional initial value problems and a q-analogue mittagleffler function,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4682–4688, 2011. 20 T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 2011. 21 J.-F. Cheng and Y.-M. Chu, “On the fractional difference equations of order 2, q ,” Abstract and Applied Analysis, vol. 2011, Article ID 497259, 16 pages, 2011. 22 J.-F. Cheng, “Solutions of fractional difference equations of order k, q ,” Acta Mathematicae Applicatae Sinica, vol. 34, no. 2, pp. 313–330, 2011. 23 J.-F. Cheng, Theory of Fractional Difference Equations, Xiamen University Press, 2011.

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