The Riccati Sub-ODE Method For FractionalDifferential-difference Equations

BIN ZHENGShandong University of Technology

School of ScienceZhangzhou Road 12, Zibo, 255049

CHINAzhengbin2601@126.com

Abstract: In this paper, we are concerned with seeking exact solutions for fractional differential-difference equa-tions by an extended Riccati sub-ODE method. The fractional derivative is defined in the sense of the modifiedRiemann-liouville derivative. By a combination of this method and a fractional complex transformation, the it-erative relations from indices n to n ± 1 are established. As for applications, we apply this method to solve thetwo-component fractional Volterra lattice equations and the fractional m-KdV lattice equation. Some new exactsolutions for the two fractional differential-difference equations are obtained.

Key–Words: Fractional differential-difference equations; Exact solutions; Riccati sub-ODE method; Fractionalcomplex transformations; Traveling wave solutions; Nonlinear evolution equationsMSC 2010: 35Q51; 35Q53

1 Introduction

Nonlinear differential equations (NLDEs) and nonlin-ear differential-difference equations (NLDDEs) canfind their applications in many aspects of mathemati-cal physics. In the last decades, research on seekingexact solutions for NLDEs and NLDDEs has been ahot topic, and many effective methods have been pre-sented so far (see [1-28] and the references therein).Among these investigations, we notice that little atten-tion is paid to fractional differential-difference equa-tions (FDDEs).

In this paper, we extend the Riccati sub-ODEmethod to seek exact solutions for FDDEs. The frac-tional derivative is defined in the sense of modifiedRiemann-liouville derivative [29-33] as follows.

Dγt f(t) =

1

Γ(1− γ)ddt

∫ t

0(t− ξ)−γ(f(ξ)− f(0))dξ,

0 < γ < 1,(f (n)(t))(γ−n), n ≤ γ < n+ 1, n ≥ 1.

Based on a fractional complex transformation, agiven fractional differential-difference equation canbe turned into another differential-difference equationof integer order, and the iterative relations of whichfrom indices n to n ± 1 are also established. By thisapproach, we will solve two fractional differential-difference equations: the two-component fractional

Volterra lattice equations{Dγt un = un(vn − vn−1),

Dγt vn = vn(un+1 − un),

(1)

and the following fractional m-KdV lattice equation

Dγt un = (α− u2n)(un+1 − un−1), (2)

where 0 < γ ≤ 1, un = un(t), vn = vn(t), n ∈Z, and Dγ

t denotes the modified Riemann-liouvillederivative of order γ with respect to the variable t.

When γ = 1, Eqs. (1) become the known two-component Volterra lattice equations [34], while Eq.(2) becomes the m-KdV lattice equation [34].

The following properties for the modifiedRiemann-Liouville are known to us (see [30-33]):

Dγt tr =

Γ(1 + r)

Γ(1 + r − γ)tr−γ , (3)

Dγt (f(t)g(t)) = g(t)Dγ

t f(t) + f(t)Dγt g(t), (4)

Dγt f [g(t)] = f ′g[g(t)]D

γt g(t) = Dγ

gf [g(t)](g′(t))γ .

(5)

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2 Description of the extended Ric-cati sub-ODE method for frac-tional differential-difference equa-tions

The main steps of the extended Riccati sub-ODEmethod for solving fractional differential-differenceequations are summarized as follows:

Step 1. Consider a fractional differential differ-ence equation in the form

P (un+p1(x), ..., un+pk(x), ..., Dγun+p1(x), ...,

Dγun+pk(x), D2γun+p1(x), ..., D

2γun+pk(x), ...) = 0,

(6)where the dependent variable u hasM components ui,the continuous variable x has N components xj , thediscrete variable n has Q components ni, the k shiftvectors ps ∈ ZQ has Q components psj , Dkγ denotesthe collection of mixed derivative terms of order kγ,and the order of the derivatives with respect to everyxj are integer multiples of γ.

Step 2. Using a fractional complex transfor-

mation Xj =xγj

Γ(1 + γ), and letting un+ps(x) =

Un+ps(X), Eq. (6) can be turned into

P (Un+p1(X), ..., Un+pk(X), ..., U ′n+p1(X), ...,

U ′n+pk

(X), U ′′n+p1(X), ..., U ′′

n+pk(X), ...) = 0, (7)

Step 2. Using a wave transformation

Un+ps(X) = Un+ps(ξn), ξn =

Q∑i=1

dini+

N∑j=1

cjXj+ζ,

where di, cj , ζ are all constants, we can rewrite Eq.(7) in the following form

˜P (Un+p1(ξn), ..., Un+pk(ξn), ..., U

′n+p1(ξn), ...,

U ′n+pk

(ξn), ..., U(r)n+p1(ξn), ..., U

(r)n+pk

(ξn)) = 0. (8)

Step 3: Suppose the solutions of Eq. (8) can bedenoted by

Un(ξn) =

l∑i=0

aiϕi(ξn), (9)

where ai are constants to be determined later, l is apositive integer that can be determined by balancingthe highest order linear term with the nonlinear termsin Eq. (8), ϕ(ξn) satisfies the known Riccati equation:

ϕ′(ξn) = σ + ϕ2(ξn). (10)

Step 4: We present some special solutionsϕ1, ..., ϕ6 for Eq. (10):

When σ < 0:

ϕ1(ξn) = −√−σ tanh(

√−σξn + c0),

ϕ2(ξn) = −√−σ coth(

√−σξn + c0),

ϕ1,2(ξn+ps) =

ϕ1,2(ξn)−√−σ tanh(

√−σ

Q∑i=1

dipsi)

1− ϕ1,2(ξn)√−σ

tanh(√−σ

Q∑i=1

dipsi)

,

(11)where c0 is an arbitrary constant.

When σ > 0:

ϕ3(ξn) =√σ tan(

√σξn + c0),

ϕ4(ξn) = −√σ cot(

√σξn + c0),

ϕ3,4(ξn+ps) =

ϕ3,4(ξn) +√σ tan(

√σ

Q∑i=1

dipsi)

1− ϕ3,4(ξn)√σ

tan(√σ

Q∑i=1

dipsi)

,

(12)and

ϕ5(ξn) =√σ[tan(2

√σξn + c0) + | sec(2

√σξn + c0)|],

ϕ5(ξn+ps) =

ϕ(1)5 (ξn) +

√σ tan(2

√σ

Q∑i=1

dipsi)

1− ϕ(1)5 (ξn)√σ

tan(2√σ

Q∑i=1

dipsi)

+

ϕ(2)5 (ξn) sec(2

√σ

Q∑i=1

dipsi)

1− ϕ(1)5 (ξn)√σ

tan(2√σ

Q∑i=1

dipsi)

,

(13)where ϕ(1)5 (ξn) =

√σtan(2

√σξn + c0), ϕ

(2)5 (ξn) =√

σ|sec(2√σξn+c0)|, and c0 is an arbitrary constant.

When σ = 0:

ϕ6(ξn) = − 1

ξn + c0,

ϕ6(ξn+ps) =ϕ6(ξn)

1− ϕ6(ξn)

Q∑i=1

dipsi

, (14)

where c0 is an arbitrary constant.Step 5: Substituting (9) into Eq. (8), by use of

Eqs. (10)-(14), the left hand side of Eq. (8) can be

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converted into a polynomial in ϕ(ξn). Equating eachcoefficient of ϕi(ξn) to zero, yields a set of algebraicequations. Solving these equations, we can obtain thevalues of ai, di, cj .

Step 6: Substituting the values of ai into (9), andcombining with the various solutions of Eq. (10), wecan obtain a variety of exact solutions for Eq. (6).

3 Application of the extended Ric-cati sub-ODE method to the two-component Volterra lattice equa-tions

In this section, we apply the extended Riccati sub-ODE method described in Section 2 to solve the two-component Volterra lattice equations denoted by Eqs.(1).

Using a fractional complex transformation T =tγ

Γ(1 + γ), and letting un+ps(t) = Un+ps(T ), ps =

0,±1, by (3) we have Dγt T = 1, and furthermore, by

the first equality in (5), Eqs. (1) can be turned into

{U ′n(T ) = Un(T )[Vn(T )− Vn−1(T )],

V ′n = Vn(T )[Un+1(T )− Un(T )],

(15)

Letting

Un(T ) = Un(ξn), Vn(T ) = Vn(ξn),

ξn = d1n+ c1T + ζ, (16)

where d1, c1, ζ are all constants, Eqs. (15) can berewritten as the following form:

{c1U

′n(ξn) = Un(ξn)[Vn(ξn)− Vn−1(ξn−1)],

c1V′n(ξn) = Vn(ξn)[Un+1(ξn+1)− Un(ξn)],

(17)Suppose the solutions for (17) can be denoted by

Un(ξn) =

l1∑i=0

aiϕi(ξn), (18)

Vn(ξn) =

l2∑i=0

biϕi(ξn), (19)

where ϕ(ξn) satisfies Eq. (10). In Eqs. (17), by bal-ancing the order of U ′

n and UnVn, the order of V ′n and

VnUn, we obtain l1 = l2 = 1. So we have

Un(ξn) = a0 + a1ϕ(ξn). (20)

Vn(ξn) = b0 + b1ϕ(ξn). (21)

We will proceed to solve Eqs. (17) in severalcases.

Case 1: If σ < 0, and assume (10) and (11) hold,then substituting (20), (21), (10) and (11) into Eqs.(17), collecting the coefficients of ϕi1,2(ξn) and equat-ing them to zero, we obtain a series of algebra equa-tions. Solving these equations, yields

a1 = −c1, a0 = − c1√−σ

tanh(√−σd1)

,

b1 = c1, b0 = − c1√−σ

tanh(√−σd1)

,

d1 = d1, c1 = c1,

or

a1 =a0 tanh(

√−σd1)√

−σ, a0 = a0,

b1 = −a0 tanh(√−σd1)√

−σ, b0 = a0,

c1 = −a0 tanh(√−σd1)√

−σ, d1 = d1.

So we obtain the following four groups of solitarywave solutions:

un(t) = − c1√−σ

tanh(√−σd1)

+c1√−σ tanh[

√−σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

vn(t) = − c1√−σ

tanh(√−σd1)

−c1√−σ tanh[

√−σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

(22)

un(t) = − c1√−σ

tanh(√−σd1)

+c1√−σ coth[

√−σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

vn(t) = − c1√−σ

tanh(√−σd1)

−c1√−σ coth[

√−σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

(23)where d1, c1, c0 are arbitrary constants, and

un(t) = a0 − a0 tanh(√−σd1)×

tanh[√−σ(d1n− a0 tanh(

√−σd1)tγ√

−σΓ(1 + γ)+ ζ) + c0],

vn(t) = a0 + a0 tanh(√−σd1)×

tanh[√−σ(d1n− a0 tanh(

√−σd1)tγ√

−σΓ(1 + γ)+ ζ) + c0],

(24)

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un(t) = a0 − a0 tanh(√−σd1)×

coth[√−σ(d1n− a0 tanh(

√−σd1)tγ√

−σΓ(1 + γ)+ ζ) + c0],

vn(t) = a0 + a0 tanh(√−σd1)×

coth[√−σ(d1n− a0 tanh(

√−σd1)tγ√

−σΓ(1 + γ)+ ζ) + c0],

(25)where d1, c0, a0 are arbitrary constants.

In Figs 1-2, the solitary wave solutions (22) withsome special parameters are demonstrated.

Case 2: If σ > 0, and assume (10) and (12) hold,then substituting (20), (21), (10) and (12) into Eqs.(17), collecting the coefficients of ϕi3,4(ξn) and equat-ing them to zero, we obtain a series of algebra equa-tions. Solving these equations, yields

a1 = −c1, a0 = − c1√σ

tan(√σd1)

, b1 = c1,

b0 = − c1√σ

tan(√σd1)

, d1 = d1, c1 = c1,

or

a1 =a0 tan(

√σd1)√

σ, a0 = a0, b1 = −a0 tan(

√σd1)√

σ,

b0 = a0, c1 = −a0 tan(√σd1)√

σ, d1 = d1.

So we obtain the following periodic wave solutions:

un(t) = − c1√σ

tan(√σd1)

−c1√σ tan[

√σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

vn(t) = − c1√σ

tan(√σd1)

+c1√σ tan[

√σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

(26)

un(t) = − c1√σ

tan(√σd1)

+c1√σ cot[

√σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

vn(t) = − c1√σ

tan(√σd1)

−c1√σ cot[

√σ(d1n+ c1t

γ

Γ(1 + γ)+ ζ) + c0],

(27)where d1, c1, c0 are arbitrary constants, and

un(t) = a0 + a0 tan(√σd1)×

tan[√σ(d1n− a0 tan(

√σd1)t

γ√σΓ(1 + γ)

+ ζ) + c0],

vn(t) = a0 − a0 tan(√σd1)×

tan[√σ(d1n− a0 tan(

√σd1)t

γ√σΓ(1 + γ)

+ ζ) + c0],

(28)

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un(t) = a0 − a0 tan(√σd1)×

cot[√σ(d1n− a0 tan(

√σd1)t

γ√σΓ(1 + γ)

+ ζ) + c0],

vn(t) = a0 + a0 tan(√σd1)×

cot[√σ(d1n− a0 tan(

√σd1)t

γ√σΓ(1 + γ)

+ ζ) + c0],

(29)where d1, c0, a0 are arbitrary constants.

In Figs 3-4, the periodic wave solutions (26) withsome special parameters are demonstrated.

Case 3: If σ > 0, and assume (10) and (13) hold,then substituting (20), (21), (10) and (13) into Eqs.(17), using [ϕ

(2)5 (ξn)]

2 = σ + [ϕ(1)5 (ξn)]

2, collectingthe coefficients of [ϕ(1)5 (ξn)]

i[ϕ(2)5 (ξn)]

j and equatingthem to zero, we obtain a series of algebra equations.Solving these equations, we get that

a1 = −c1, a0 = 0, b1 = c1,

b0 = 0, d1 =π

2√σ, c1 = c1,

or

a1 = −c1, a0 = b0, b1 = c1,

b0 = b0, d1 =1

2√σarcsin(−2c1b0

√σ

b20 + c21σ), c1 = c1.

So we obtain the following trigonometric function so-

lutions:

un(t) = −c1√σ{tan[2

√σ( π

2√σn+ c1t

γ

Γ(1 + γ)+ ζ)

+c0] + | sec[2√σ( π

2√σn+ c1t

γ

Γ(1 + γ)+ ζ) + c0]|},

vn(t) = c1√σ{tan[2

√σ( π

2√σn+ c1t

γ

Γ(1 + γ)+ ζ)

+c0] + | sec[2√σ( π

2√σn+ c1t

γ

Γ(1 + γ)+ ζ) + c0]|},

(30)where c1, c0 are an arbitrary constants, and

un(t) = −c1√σ{tan[2

√σ( 1

2√σarcsin(−2c1b0

√σ

b20 + c21σ)n

+ c1tγ

Γ(1 + γ)+ ζ) + c0]

+| sec[2√σ( 1

2√σarcsin(−2c1b0

√σ

b20 + c21σ)n

+ c1tγ

Γ(1 + γ)+ ζ) + c0]|}+ b0,

vn(t) = c1√σ{tan[2

√σ( 1

2√σarcsin(−2c1b0

√σ

b20 + c21σ)n

+ c1tγ

Γ(1 + γ)+ ζ) + c0]

+| sec[2√σ( 1

2√σarcsin(−2c1b0

√σ

b20 + c21σ)n

+ c1tγ

Γ(1 + γ)+ ζ) + c0]|}+ b0,

(31)where c1, b0, c0 are an arbitrary constants.

Case 4: If σ = 0, and assume (10) and (14) hold,then substituting (20), (21), (10) and (14) into Eqs.(17), collecting the coefficients of ϕi6(ξn) and equat-ing them to zero, we obtain a series of algebra equa-tions. Solving these equations, yields

a1 = d1b0, a0 = b0, b1 = −d1b0,

b0 = b0, d1 = d1, c1 = −d1b0.

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Then we obtain the following rational solutions:un(t) =

−d1b0d1n− d1b0t

γ

Γ(1 + γ)+ ζ + c0

+ b0,

vn(t) =d1b0

d1n− d1b0tγ

Γ(1 + γ)+ ζ + c0

+ b0,(32)

where d1, b0, c0 are an arbitrary constants.

Remark 1 In [34, Eqs. (46), (47), (51), (52)], Ayhanand Bekir presented some exact solutions for the two-component Volterra lattice equations by the (G’/G)-expansion method. We note that our results (22), (23)are generalizations of [34, Eqs. (46), (47)], while(26), (27) are generalizations of [34, Eqs. (51), (52)].In fact, if we let

γ = 1, c0 = arth(C2

C1), σ =

4µ− λ2

4

or

γ = 1, c0 = arcoth(C1

C2), σ =

4µ− λ2

4,

then our results (22), (23) reduce to [34, Eq. (46),(47)]. If we let

γ = 1, c0 = arctan(−C2

C1), σ =

4µ− λ2

4

or

γ = 1, c0 = arccot(−C1

C2), σ =

4µ− λ2

4,

then our results (26), (27) reduce to [34, Eq. (51),(52)].

Remark 2 The established results by (30-32) are newexact solutions for the two-component Volterra latticeequations so far to our best knowledge.

4 Application of the extended Riccatisub-ODE method to the fractionalm-KdV lattice equation

In this section, we apply the extended Riccati sub-ODE method to solve the fractional m-KdV latticeequation denoted by Eq. (2).

Using a fractional complex transformation T =tγ

Γ(1 + γ), and letting un+ps(t) = Un+ps(T ), ps =

0,±1, by (3) we have Dγt T = 1, and furthermore, by

the first equality in (5), Eq. (2) can be turned into

U ′n(T ) = [α− U2

n(T )][Un+1(T )− Un−1(T )]. (33)

Letting

Un(T ) = Un(ξn), ξn = d1n+ c1T + ζ, (34)

where d1, c1, ζ are all constants, Eq. (33) can berewritten in the following form:

c1U′n(ξn)−(α−U2

n(ξn))(Un+1(ξn+1)−Un−1(ξn−1)) = 0.(35)

Suppose the solutions Un(ξn) for Eq. (3.3) can bedenoted by

Un(ξn) =

l∑i=0

aiϕi(ξn), (36)

where ϕ(ξn) satisfies Eq. (10). By balancing the high-est order linear term with the nonlinear terms in Eq.(35) we obtain l+1 = 2l, and then l = 1. So we have

Un(ξn) = a0 + a1ϕ(ξn). (37)

We will proceed to solve Eq. (35) in several cases.

Case 1: If σ < 0, and assume (10) and (11) hold,then substituting (37), (10) and (11) into Eq. (35),collecting the coefficients of ϕi1,2(ξn) and equatingthem to zero, we obtain a series of algebraic equa-tions. Solving these equations, yields

a1 = ±√

−ασtanh(

√−σd1), a0 = 0,

d1 = d1, c1 =2α√−σ

tanh(√−σd1).

So we obtain the following solitary wave solutions:

un(t) = ±√α tanh(

√−σd1)

tanh[√−σ(d1n+

2α√−σ

tanh(√−σd1)

tγ

Γ(1 + γ)+ζ)+c0],

(38)and

un(t) = ±√α tanh(

√−σd1)

coth[√−σ(d1n+

2α√−σ

tanh(√−σd1)

tγ

Γ(1 + γ)+ζ)+c0],

(39)

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where d1, c0 are arbitrary constants.Case 2: If σ > 0, and assume (10) and (12)

hold, then substituting (37), (10) and (12) into Eq.(35), collecting the coefficients of ϕi3,4(ξn) and equat-ing them to zero, we obtain a series of algebraic equa-tions. Solving these equations, yields

a1 = ±√α

σtan(

√σd1), a0 = 0,

d1 = d1, c1 =2α√σtan(

√σd1).

Then we have the following periodic wave solutions:

un(t) = ±√α tan(

√σd1)

tan[√σ(d1n+

2α√σtan(

√σd1)

tγ

Γ(1 + γ)+ζ)+c0], (40)

andun(t) = ±

√α tan(

√σd1)

cot[√σ(d1n+

2α√σtan(

√σd1)

tγ

Γ(1 + γ)+ζ)+c0], (41)

where d1, c0 are arbitrary constants.In [34, Eqs. (32) and (36)], Ayhan and Bekir pre-

sented some exact solutions for m-KdV lattice equa-tion by the (G’/G)-expansion method as follows:

un = ±√α tanh(

√λ2 − 4µ

2d1)×

(C1 sinh(

√λ2 − 4µ

2ξn) + C2 cosh(

√λ2 − 4µ

2ξn)

C1 cosh(

√λ2 − 4µ

2ξn) + C2 sinh(

√λ2 − 4µ

2ξn))

),

(42)

where ξn = d1n+4α√λ2 − 4µ

tanh(

√λ2−4µ2 d1)t+ζ,

and

un = ±√α tan(

√4µ− λ2

2d1)×

(−C1 sin(

√4µ− λ2

2ξn) + C2 cos(

√4µ− λ2

2ξn)

C1 cos(

√4µ− λ2

2ξn) + C2 sin(

√4µ− λ2

2ξn))

),

(43)

where ξn = d1n+ 4α√4µ− λ2

tan(

√4µ−λ22 d1)t+ ζ.

We note that our results (38) and (40) are solu-tions of more general forms than Eqs. (42) and (43).In fact, if we let γ = 1, c0 = arth(C2

C1), σ =

4µ− λ2

4 or γ = 1, c0 = arcoth(C1C2

), σ =4µ− λ2

4 ,then our result (38) reduces to (42). If we let γ =

1, c0 = arctan(−C2C1

), σ =4µ− λ2

4 or γ = 1, c0 =

arccot(−C1C2

), σ =4µ− λ2

4 , then our result (40) re-duces to (43).

Case 3: If σ > 0, and assume (10) and (13) hold,then substituting (37), (10) and (13) into Eq. (35), us-ing [ϕ

(2)5 (ξn)]

2 = σ+ [ϕ(1)5 (ξn)]

2, collecting the coef-ficients of [ϕ(1)5 (ξn)]

i[ϕ(2)5 (ξn)]

j and equating them tozero, we obtain a series of algebraic equations. Solv-ing these equations, we get three families of values asfollows:

a1 = ±√α

σ, a0 = 0,

d1 = − π

4√σ, c1 = − 2α√

σ,

a1 = ±√α

σ, a0 = 0,

d1 =π

4√σ, c1 =

2α√σ,

or

a1 = ±

√2α− α sin2(2

√σd1)− 2α cos(2

√σd1)

√σ sin(2

√σd1)

,

d1 = d1, a0 = 0, c1 = −2α(cos(2√σd1)− 1)√

σ sin(2√σd1)

So we obtain the following trigonometric function so-lutions:

un(t) = ±√α{tan[2

√σ(

π

4√σn+

2αtγ√σΓ(1 + γ)

+ ζ)

+c0]+ | sec[2√σ(

π

4√σn+

2αtγ√σΓ(1 + γ)

+ζ)+c0]|},

(44)

un(t) = ±√α{tan[2

√σ(− π

4√σn− 2αtγ√

σΓ(1 + γ)+ ζ)

+c0]+| sec[2√σ(− π

4√σn− 2αtγ√

σΓ(1 + γ)+ζ)+c0]|},

(45)where c0 is an arbitrary constant, and

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E-ISSN: 2224-2880 198 Volume 13, 2014

un(t) = ±

√2α− α sin2(2

√σd1)− 2α cos(2

√σd1)

sin(2√σd1)

{tan[2√σ(d1n− 2α(cos(2

√σd1)− 1)tγ√

σ sin(2√σd1)Γ(1 + γ)

+ ζ) + c0]

+| sec[2√σ(d1n−

2α(cos(2√σd1)− 1)tγ√

σ sin(2√σd1)Γ(1 + γ)

+ ζ)+ c0]|},

(46)where d1, c0 are arbitrary constants.

Case 4: If σ = 0, and assume (10) and (14) hold,then substituting (37), (10) and (14) into Eq. (35), col-lecting the coefficients of ϕi6(ξn) and equating them tozero, we obtain a series of algebraic equations. Solv-ing these equations, yields

a1 = ±√αd1, a0 = 0, d1 = d1, c1 = 2d1α.

Then we obtain the following rational solution:

un(t) =±√αd1

d1n+2d1αt

γ

Γ(1 + γ)+ ζ + c0

, (47)

where d1, c0 are arbitrary constants.

Remark 3 Our results (44)-(47) have not been re-ported by other authors so far to our best knowledge.

5 Conclusions

The Riccati sub-ODE method is extended to seekexact solutions for fractional differential-differenceequations. By this method, we solved the two-component fractional Volterra lattice equations andthe fractional m-KdV lattice equation successfully,and as a result, some generalized exact solutions in-cluding solitary wave solutions, periodic wave solu-tions and rational function solutions for them havebeen found with the aid of mathematical software. Be-ing concise and effective, we note that this approachcan be applied to solve other fractional differential-difference equations.

References:

[1] M. L. Wang and X. Z. Li, Applications of F-expansion to periodic wave solutions for a newHamiltonian amplitude equation, Chaos, Soli-tons and Fractals, 24, 2005, pp. 1257-1268.

[2] J. H. He and X. H. Wu, Exp-function method fornonlinear wave equations, Chaos, Solitons andFractals, 30, 2006, pp. 700-708.

[3] M. L. Wang, X. Z. Li and J. L. Zhang, The(G’/G)-expansion method and travelling wavesolutions of nonlinear evolution equations inmathematical physics, Phys. Lett. A, 372, 2008,pp. 417-423.

[4] E. M. E. Zayed and K. A. Gepreel, The mod-ified (G’/G)-expansion method and its applica-tions to construct exact solutions for nonlinearPDEs, WSEAS Transactions on Mathematics,10(8), 2011, pp. 270-278.

[5] E. M. E. Zayed, A further improved (G’/G)-expansion method and the extended tanh-method for finding exact solutions of nonlin-ear PDEs, WSEAS Transactions on Mathemat-ics, 10(2), 2011, pp. 56-64.

[6] E. M. E. Zayed and M. Abdelaziz, Exact trav-eling wave solutions of nonlinear variable co-efficients evolution equations with forced termsusing the generalized (G’/G)-expansion method,WSEAS Transactions on Mathematics, 10(3),2011, pp. 115-124.

[7] Q. H. Feng and B. Zheng, Traveling WaveSolutions for the Fifth-Order Sawada-KoteraEquation and the General Gardner Equationby (G’/G)-Expansion Method, WSEAS Transac-tions on Mathematics, 9(3), 2010, pp. 171-180.

[8] Y. Wang, Variable-coefficient Simplest EquationMethod For Solving Nonlinear Evolution Equa-tions In Mathematical Physics, WSEAS Transac-tions on Mathematics, 12(5), 2013, pp. 512-520.

[9] E. M. E. Zayed and S. Al-Joudi, An Improved(G’/G)-expansion Method for Solving NonlinearPDEs in Mathematical Physics, ICNAAM 2010,AIP. Conf. Proc., Vol. 1281, 2010, pp. 2220-2224.

[10] Q. H. Feng and B. Zheng, Traveling WaveSolutions for the Fifth-Order Kdv Equationand the BBM Equation by (G’/G)-ExpansionMethod, WSEAS Transactions on Mathematics,9(3), 2010, pp. 201-210.

[11] B. Zheng, Application Of A GeneralizedBernoulli Sub-ODE Method For Finding Trav-eling Solutions Of Some Nonlinear Equations,WSEAS Transactions on Mathematics, 11(7),2012, pp. 634-642.

[12] C. C. Kong, D. Wang, L. N. Song andH. Q. Zhang, New exact solutions to MKDV-Burgers equation and (2+1)-dimensional disper-sive long wave equation via extended Riccatiequation method, Chaos, Solitons and Fractals,39, 2009, pp. 697-706.

WSEAS TRANSACTIONS on MATHEMATICS Bin Zheng

E-ISSN: 2224-2880 199 Volume 13, 2014

[13] M. Eslami, A. Neyrame and M. Ebrahimi, Ex-plicit solutions of nonlinear (2+1)-dimensionaldispersive long wave equation, J. King SaudUniver.-Sci. , 24, 2012, pp. 69-71.

[14] I. Aslan, Discrete exact solutions to some non-linear differential-difference equations via the(G’/G)-expansion method, Appl. Math. Comput.,215, 2009, pp. 3140-3147.

[15] X. B. Hu and W. X. Ma, Application of Hirota’sbilinear formalism to the Toeplitz lattice somespecial soliton-like solutions, Phys. Lett. A, 293,2002, pp. 161-165.

[16] B. Tang, Y. N. He, L. L. Wei and S. L. Wang,Variable-coefficient discrete (G’/G)-expansionmethod for nonlinear differential-differenceequations, Phys. Lett. A , 375, 2011, pp. 3355-3361.

[17] W. Zhen, Discrete tanh method for non-linear difference-differential equations, Com-put. Phys. Commun., 180, 2009, pp. 1104-1108.

[18] I. Aslan, A discrete generalization of theextended simplest equation method, Com-mun. Nonlinear Sci. Numer. Simul., 15, 2010,pp. 1967-1973.

[19] C. Q. Dai, X. Cen and S. S. Wu, Exact travel-ling wave solutions of the discrete sine-Gordonequation obtained via the exp-function method,Nonlinear Anal., 70, 2009, pp. 58-63.

[20] C. S. Liu, Exponential function rational expan-sion method for nonlinear differential-differenceequations, Chaos, Solitons and Fractals, 40,2009, pp. 708-716.

[21] S. Zhang, L. Dong, J. M. Ba and Y. N. Sun,The (G’/G)-expansion method for nonlineardifferential-difference equations, Phys. Lett. A,373, 2009, pp. 905-910.

[22] B. Ayhan and A. Bekir, The (G’/G)-expansionmethod for the nonlinear lattice equations, Com-mun. Nonlinear Sci. Numer. Simulat., 17, 2012,pp. 3490-3498.

[23] H. Xin, The exponential function rational ex-pansion method and exact solutions to nonlinearlattice equations system, Appl. Math. Comput.,217, 2010, pp. 1561-1565.

[24] K. A. Gepreel and A. R. Shehata, Rational Ja-cobi elliptic solutions for nonlinear differential-difference lattice equations, Appl. Math. Lett.,25, 2012, pp. 1173-1178.

[25] W. H. Huang and Y. L. Liu, Jacobi elliptic func-tion solutions of the Ablowitz-Ladik discretenonlinear Schrodinger system, Chaos, Solitonsand Fractals, 40, 2009, pp. 786-792.

[26] C. B. Wen, Traveling Wave Solutions For TwoNonlinear Lattice Equations By An ExtendedRiccati Sub-equation Method, WSEAS Transac-tions on Mathematics, 11(12), 2012, pp. 1085-1093.

[27] C. B. Wen and B. Zheng, A New Fractional Sub-equation Method For Fractional Partial Differen-tial Equations, WSEAS Transactions on Mathe-matics, 12(5), 2013, pp. 564-571.

[28] Q. H. Feng and B. Zheng, Traveling Wave So-lutions For The Variant Boussinseq EquationAnd The (2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) System By (G’/G)-expansionmethod, WSEAS Transactions on Mathematics,9(3), 2010, pp. 191-200.

[29] G. Jumarie, Modified Riemann-Liouville deriva-tive and fractional Taylor series of non-differentiable functions further results, Com-put. Math. Appl., 51, 2006, pp. 1367-1376.

[30] B. Lu, Backlund transformation of fractionalRiccati equation and its applications to non-linear fractional partial differential equations,Phys. Lett. A, 376, 2012, pp. 2045-2048.

[31] S. M. Guo, L. Q. Mei, Y. Li and Y. F. Sun,The improved fractional sub-equation methodand its applications to the space-time fractionaldifferential equations in fluid mechanics, Phys.Lett. A, 376, 2012, pp. 407-411.

[32] S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlin-ear fractional PDEs, Phys. Lett. A, 375, 2011,pp. 1069-1073.

[33] B. Zheng, (G’/G)-Expansion Method for Solv-ing Fractional Partial Differential Equations inthe Theory of Mathematical Physics, Com-mun. Theor. Phys. (Beijing, China), 58, 2012,pp. 623-630.

[34] B. Ayhan and A. Bekir, The (G’/G)-expansionmethod for the nonlinear lattice equations, Com-mun. Nonlinear Sci. Numer. Simulat., 17, 2012,pp. 3490-3498.

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E-ISSN: 2224-2880 200 Volume 13, 2014