Exact solutions of some systems of fractional

Research Article Received 24 July 2014

Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.3318 MOS subject classification: 34A08; 35R11; 83C15; 35C07

Exact solutions of some systems of fractional differential-difference equations Ahmet Bekira *†, Özkan Günerb and Burcu Ayhana Communicated by M. Kirane 0

In this paper, the GG -expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential-difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential-difference equation into its differential-difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time-fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd. Keywords: fractional differential-difference equations; fractional partial differential equations; exact solutions

1. Introduction Since the study of Fermi et al. in the 1960s [1], the investigation of exact solutions of nonlinear differential-difference equations (NLDDEs) has played a very important role in the modeling of many phenomena in different fields which include condensed matter physics, plasma physics, molecular crystals, biophysics, and mechanical engineering. Their solutions are also useful in applications. In the past several decades, many effective methods for obtaining exact solutions of NLDDEs have been presented [2–9]. However, no method obeys the strength and the flexibility requirements for finding all solutions to all kinds of nonlinear NLDDEs. Baldwin et al. [10] presented an algorithm to find exact traveling wave solutions of differential-difference equations in terms of tanh 0 function. Zhang et al. [11] and Aslan [12] used the GG -expansion method to address some physically important NLDDEs. Zhang [13] and Gepreel [14] have used the Jacobi elliptic function method for constructing new and more general Jacobi elliptic solu function 0 tions of the integral discrete nonlinear Schrödinger equation. More recently, Zhang et al. [15] proposed a generalized GG -expansion method to improve and extend the works of Wang et al. [16] and Tang et al. [17] for solving variable-coefficient equations and high-dimensional equations. Fractional differential equations are generalizations of classical differential equations of integer order. In recent decades, fractional differential equations played an important role in applied physics, chemistry, biology, engineering, and finance. The exact solutions of these problems, when they exist, are very important in the understanding of the nonlinear fractional physical phenomena. There are many powerful methods for solving nonlinear fractional differential equations [18–29]. Time-fractional differential-difference equations have been the focus of many studies. The fractional derivatives in the sense of 0 modified Riemann–Liouville derivative and the GG -expansion method are employed for constructing exact solutions of nonlinear time-fractional partial differential-difference equation systems. The power of this method is presented by applying it to several examples. We will use the modified definition of the Riemann–Liouville fractional derivative. The fractional complex transform [30] is used to analytically deal with fractional differential equations. This method is extremely simple but effective for solving fractional differential equations.

a Department of Mathematics - Computer, Art-Science Faculty, Eskisehir Osmangazi University, Eskisehir, Turkey b Department of Management Information Systems, School of Applied Sciences, Dumlupınar University, Kütahya, Turkey

* Correspondence to: Ahmet Bekir, Department of Mathematics - Computer, Art-Science Faculty, Eskisehir Osmangazi University, Eskisehir, Turkey. † E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2014

A. BEKIR, Ö. GÜNER, AND B. AYHAN 0 The manuscript suggests the GG -expansion method and fractional complex transform to find exact solutions of NLDDEs with the modified Riemann–Liouville derivative by Jumarie [31]. The Jumarie’s modified Riemann–Liouville derivative of order ˛ is defined by

D˛ t f .t/

D

8 < :

R d t 1 .t .1˛/ dt 0

/˛ .f ./ f .0//d, 0 < ˛ < 1 (1)

.f .n/ .t//.˛n/ , n ˛ < n C 1, n 1.

Modified Riemann–Liouville derivative has many important properties; four of their important formulas are [32] .1 C / ˛ x , > 0, .1 C ˛/

(2)

˛ D˛ t .cf .t// D cDt f .t/, c D constant

(3)

˛ ˛ D˛ t faf .t/ C bg.t/g D aDt f .t/ C bDt g.t/,

(4)

D˛ t c D 0, c D constant,

(5)

D˛ t x D

where a and b are constant.

which are direct consequences of the equality d˛ x.t/ D .1 C ˛/dx.t/.

(6) 0 G The rest of this paper is organized as follows. In Section 2, we describe the algorithm for using the G -expansion method to solve NLDDEs. In Sections 3 and 4, to illustrate the validity and advantages of the method, we will apply it to the time-fractional Toda lattice equations and to the time-fractional relativistic Toda lattice system. In Section 5, some conclusions are given.

2. Description of the

0 G -expansion method for NLDDEs G

In this section, we would like to outline the algorithm for using the consider a system of M fractional NLDDEs in the form

0 G G

-expansion method to solve NLDDEs step by step. Let us

.r˛/ .r˛/ ˛ P unCp1 .x/ , : : : , unCpk .x/ , : : : , u˛ nCp1 .x/ , : : : , unCpk .x/ , : : : , unCp1 .x/ , : : : , unCpk .x/ D 0,

(7)

where the dependent variable un has M components ui,n , the continuous variable x has N components xj , the discrete variable n has Q components ni , the k shift vectors ps 2 ZQ , and u.r˛/ .x/ denotes the collection of modified Riemann–Liouville derivative terms of order r˛. Using a fractional complex transformation,

unCps .x/ D UnCps .n / , n D

Q X

di ni C

N X

iD1

jD1

cj x ˛ C , .s D 1, 2, : : : , k/ , .1 C ˛/ j

(8)

where the coefficients di and cj and the phase are all constants. By using the chain rule, 0

D˛ t u D t D˛ x uD 0

du ˛ D d t du ˛ x d Dx 0

,

(9)

0

0

0

where t and x are called the sigma indexes (see [33]); without loss of generality we can take t D x D l, where l is a constant. Substituting (8) with (2) and (9) into (7), we can rewrite Equation (7) as a NLDDE of integer order in the form .r/ .r/ 0 0 . / , : : : , U . / , : : : , U . / , : : : , U . / D 0. Q UnCp1 .n / , : : : , UnCpk .n / , : : : , UnCp n n n n nCp nCp nCp 1 k 1 k

(10)

Step 1: We assume the following series expansion as a solution of Equation (10): Un .n / D

m X lD0


˛l

G0 .n / G .n /

l , ˛m ¤ 0,

(11)


A. BEKIR, Ö. GÜNER, AND B. AYHAN where m and ˛i are constants to be determined later, and G .n / satisfies a second-order linear ordinary differential equation: d2 G .n / dG .n / C C G .n / D 0, dn2 dn

(12)

where and are arbitrary constants. Using the general solutions of Equation (12), we obtain the following three cases: 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ G0 .n / < G .n / ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

p

0 2 4 2

p

p 1 2 4 n CC2 cosh 2 @ p p A 2 4 2 4 C1 cosh n CC2 sinh n 2 2 p

C1 sinh

2 4 n 2

p p 1 42 42 C sin cos CC 1 n 2 n 2 2 42 @ p p A 2 42 42 C1 cos n CC2 sin n 2 2

2

, 2 4 > 0,

2

, 2 4 < 0,

0

C1 C1 n CC2

(13)

2

, 2 4 D 0,

where C1 and C2 are arbitrary constants. Step 2: We derive from (13) and put the formulas G0 n˙y D G n˙y where D ˙1 and f

p

p p .2 4/ .2 4/ f y 2 2 p .2 4/ G0 .n / 1˙ p 2 C f y 2 2 G. / 2 n 4

G0 .n / 2 C G.n / ˙

, 2

(14)

p 2 8 4 < tanh y , D 1, 2 .2 4/ p y D 2 : tan 42 y , D 1. 2

(15)

By a simple computation, we can obtain the identity nCps D n C 's , 's D ps1 d1 C ps2 d2 C : : : C psQ dQ ,

(16)

where psj is the jth component of the shift vector ps . Thus, considering trigonometric/hyperbolic function identities and using the expressions (14)–(16), we obtain

UnCps .n / D

m X

2 ˛l 4

lD0

p p .2 4/ .2 4/ f 's 2 2 p .2 4/ G0 .n / 1˙ p 2 C f 's 2 2 G. / 2 n 4

G0 .n / 2 C G.n / ˙

3l

5 2

.

(17)

Step 3: By using the homogeneous balance principle for the highest order nonlinear term(s) and the highest order partial derivative of Un .n / in Equation (10), we can easily determine the degree m of Equation (11). It should be noted that the leading terms of UnCps .n /, .ps ¤ 0/ will not affect the balancing procedure, because UnCps .n / can be interpreted as being of degree zero 0 0 . / .n / n in GG. . and we are interested in balancing the terms of GG. n/ n/ Step 4: Then, substituting Equations (11) and (17) together with (12) into Equation (10) and equating the coefficients of i G0 .n / .i D 0, 1, 2, 3, .../ to zero, we obtain a system of nonlinear algebraic equations, from which the constants ˛l , di , and G.n / cj can be explicitly determined. Finally, we substitute these values into Equation (11) and obtain various kinds of discrete exact solutions to Equation (7).

3. Time-fractional Toda lattice equations We next consider the two coupled time-fractional PDDE through the well-known Toda lattice equations [34]: @˛ un D un .vn vn1 / , @t˛ ˛ @ vn D vn .unC1 un / , @t˛

(18)

where 0 < ˛ 1, un D u .n, t/, vn D v .n, t/, and n 2 Z. Using a fractional complex transformation, Copyright © 2014 John Wiley & Sons, Ltd.


A. BEKIR, Ö. GÜNER, AND B. AYHAN un˙1 D vn˙1 D

t˛ C , .1 C ˛/ t˛ C , Vn˙1 .n / , n D d1 n C c1 .1 C ˛/

Un˙1 .n / , n D d1 n C c1

(19)

and substituting (19) with (2) and (9) into (18), Equation (18) turns into c1 Un0 Un .Vn Vn1 / D 0

(20)

c1 Vn0 Vn .UnC1 Un / D 0.

In this case, ps D 1, 's D d1 , and prime denotes a derivative with respect to n . By using the homogeneous balance principle, we find the balancing term for Un and Vn . Let m be the balancing term of Un and k be the balancing term of Vn . Here the terms UnC1 and Vn1 do not affect the balance: 0 m X G .n / l Un .n / D ˛l , ˛m ¤ 0, (21) G .n / lD0

UnCps .n / D

m X

2 ˛l 4

lD0

p p .2 4/ .2 4/ f 's 2 2 p .2 4/ G0 .n / 1˙ p 2 C f 's 2 2 G. / 2 n 4

G0 .n / 2 C G.n / ˙

Vn .n / D

k X

ˇl

lD0

VnCps .n / D

k X lD0

2 ˇl 4

G0 .n / G .n /

3l

5 2

,

(22)

l , ˇk ¤ 0,

p p .2 4/ .2 4/ f 's 2 2 p .2 4/ G0 .n / f 's 1˙ p 2 C 2 2 G. / 2 n 4

G0 .n / 2 C G.n / ˙

(23) 3l

5 2

.

(24)

In the first equation of Equation system (20), from the highest order nonlinear term Un Vn and the highest order partial derivative of Un , we find the balancing term of Vn as follows: Un0 Un Vn ) m C 1 D m C k ) k D 1.

(25)

In the second equation of Equation system (20), from the highest order nonlinear term Un Vn and the highest order partial derivative of Vn , we find the balancing term of Un as follows: Vn0 Un Vn ) k C 1 D m C k ) m D 1.

(26)

Thus, substituting these balancing terms into (21)–(24), we obtain Un D ˛0 C ˛1 2 6 UnC1 .n / D ˛0 C ˛1 4

0 p

.2 4/ 2

B @

p 1 .2 4/ d1 2 C . / p A 0 . / 2 4/ . G n 2 1C p C f d1 G.n / 2 2 4 2

p

2 2 4

Vn D ˇ0 C ˇ1 2 6 Vn1 .n / D ˇ0 C ˇ1 4

0 p

.2 4/ 2

B @

G0 .n / , ˛1 ¤ 0, G .n /

G0 .n / 2 C G.n /

3

C f

7 5, 2

G0 .n / , ˇ1 ¤ 0 G .n /

1 p .2 4/ d1 2 C . / A p .2 4/ G0 .n / 2 p 1 C f d1 G.n / 2 2 4 2

p

(27)

2 2 4

G0 .n / 2 C G.n /

(28)

(29) 3

f

7 5, 2

(30)

for the traveling wave solutions (20). Substituting (27)–(30) along with (12) into Equation (20), clearing the denominator of Equation 0

i

.n / and setting the coefficients of GG. to zero, we derive a system of nonlinear algebraic equations for ˛0 , ˛1 , ˇ0 , ˇ1 , d1 , and c1 . Solving n/ this algebraic system by the use of Maple, we obtain hyperbolic and trigonometric function solutions for Equation (20) as follows:



A. BEKIR, Ö. GÜNER, AND B. AYHAN In the case when 2 4 > 0, p

Let tanh

2 4 d1 2

D for simplicity, and from the first equation of system (20), we find coefficients of

G0 .n / G.n / G0 .n / G.n / G0 .n / G.n / G0 .n / G.n /

0 1 2 3


0 1 2 3

G0 .n / G.n /

i

.0 i 3/:

p : ˛1 c1 2 4 2ˇ1 ˛0 D 0, : ˛1 c1

p

2 4 2˛1 c1 2ˇ1 ˛1 2ˇ1 ˛0 D 0,

: 3 ˛1 c1 C ˛1 c1

p

(31) .2

4/ 2ˇ1 ˛0 2ˇ1 ˛1 D 0,

: 2˛1 .c1 C ˇ1 / D 0.

From the second equation of system (20), we find coefficients of

G0 .n / G.n /

i

.0 i 3/:

p : ˇ1 c1 2 4 C 2ˇ0 ˛1 D 0, p

: ˇ1 c1

2 4 C 2ˇ1 c1 C 2ˇ1 ˛1 C 2˛1 ˇ0 D 0, (32)

p : 3ˇ1 c1 ˇ1 c1 .2 4/ C 2ˇ0 ˛1 C 2ˇ1 ˛1 D 0, : 2ˇ1 .c1 ˛1 / D 0.

Solving the algebraic systems (31) with (32) for ˛0 , ˛1 , d1 , and c1 by the use of Maple, we obtain ˛0 D

p p 2 4 c1 2 4 coth d1 2 2

ˇ0 D

c1

, ˛1 D c1 ,

p p 2 4 d1 C 2 4 coth 2 2

(33) , ˇ1 D c1 ,

d1 D d1 , c1 D c1 , D 1. Substituting these values into (27) and (29), we have the hyperbolic function solutions as follows: 0 p p 1 p 2 C1 sinh 224 n CC2 cosh 224 n p 2 4 @ 4 p p A, un,1 .t/ D c1 d1 C coth 2 2 2 4 2 4 C1 cosh

p

vn,1 .t/ D c1

0 2 4 2

@coth

p

2 4 d1 2

n CC2 sinh

2

C1 sinh

2 4 n 2

(34)

n

p 1 2 4 n CC2 cosh 2 p p A, 2 4 2 4 C1 cosh n CC2 sinh n 2 2 p

C

2

(35)

˛

t where n D d1 n C c1 .1C˛/ C , C1 , and C2 are arbitrary constants. If we take, in particular, 2 4 D 4 and C1 D 0 in (27) and (29), the solutions of Equation (18) become t˛ un,2 D c1 coth .d1 / C coth.d1 n C c1 .1C˛/ C / ,

t˛ C / ; vn,2 D c1 coth .d1 / C coth.d1 n C c1 .1C˛/

(36)

and if we take 2 4 D 4 and C2 D 0 in (27) and (29), the solutions of Equation (18) become t˛ un,3 D c1 coth .d1 / C tanh.d1 n C c1 .1C˛/ C / , vn,3

t˛ D c1 coth .d1 / C tanh.d1 n C c1 .1C˛/ C / .

(37)

In the case when 2 4 < 0, Copyright © 2014 John Wiley & Sons, Ltd.


A. BEKIR, Ö. GÜNER, AND B. AYHAN Let tan

p

42 d1 2



0 1 2 3


0 1 2 3

G0 .n / G.n /

i

.0 i 3/:

p : ˛1 c1 4 2 2ˇ1 ˛0 D 0, : ˛1 c1

p

4 2 2˛1 c1 2ˇ1 ˛1 2ˇ1 ˛0 D 0, (38)

p

: 3 ˛1 c1 C ˛1 c1 4

2

2ˇ1 ˛0 2ˇ1 ˛1 D 0,

: 2˛1 .c1 C ˇ1 / D 0.


G0 .n / G.n /

i

.0 i 3/:

p : ˇ1 c1 4 2 C 2ˇ0 ˛1 D 0, : ˇ1 c1

p

4 2 C 2ˇ1 c1 C 2ˇ1 ˛1 C 2˛1 ˇ0 D 0, (39)

p : 3ˇ1 c1 ˇ1 c1 4 2 C 2ˇ0 ˛1 C 2ˇ1 ˛1 D 0, : 2ˇ1 .c1 ˛1 / D 0.

Solving the algebraic systems (38) with (39) for ˛0 , ˛1 , d1 , and c1 by the use of Maple, we obtain

˛0 D

ˇ0 D

p p 42 c1 42 cot d1 2 2

, ˛1 D c1 ,

p p 42 c1 C 42 cot d1 2 2

(40) , ˇ1 D c1 ,

d1 D d1 , c1 D c1 , D 1. Substituting these values into (27) and (29), we have the trigonometric function solutions of Equation (18) as follows:

un,4 .t/ D

p 2 c1 4 2

p

vn,4 .t/ D c1

0 @ cot 0

42 2

@cot

p

p

42 d1 2

42 d1 2

C

C

p p 1 42 42 C1 sin n CC2 cos n 2 2 p p A, 42 42 C1 cos n CC2 sin n 2 2 p p 1 42 42 C1 sin n CC2 cos n 2 2 p p A, 42 42 C1 cos n CC2 sin n 2 2

(41)

(42)

˛

t where n D d1 n C c1 .1C˛/ C , C1 , and C2 are arbitrary constants. Similarly, if we take 4 2 D 4 and C1 D 0 in (41) and (42), the solutions of Equation (18) become

t˛ un,5 .t/ D c1 cot .d1 / C cot.d1 n C c1 .1C˛/ C / , t˛ C ; vn,5 .t/ D c1 cot .d1 / C cot d1 n C c1 .1C˛/

(43)

and if we take 4 2 D 4 and C2 D 0 in (41) and (42), the solutions of Equation (18) become t˛ un,6 .t/ D c1 cot .d1 / C tan.d1 n C c1 .1C˛/ C / , t˛ C . vn,6 .t/ D c1 cot .d1 / C tan d1 n C c1 .1C˛/ Copyright © 2014 John Wiley & Sons, Ltd.

(44)


A. BEKIR, Ö. GÜNER, AND B. AYHAN

4. Time-fractional relativistic Toda lattice system In this section, we will consider the time-fractional relativistic Toda lattice system [35]: Dt un D .1 C ˛un / .vn vn1 / , (45) Dt un D vn .unC1 un C ˛vnC1 ˛vn1 / , where 0 < 1, un D u .n, t/, vn D v .n, t/, and n 2 Z. Using a fractional complex transformation,

t un˙1 D Un˙1 .n / , n D d1 n C c1 .1C C , /

(46)

t C , vn˙1 D Vn˙1 .n / , n D d1 n C c1 .1C /

and substituting (46) with (2) and (9) into (45), Equation (45) turns into c1 Un0 .1 C ˛Un / .Vn Vn1 / D 0, (47) c1 Vn0 Vn .UnC1 Un C ˛VnC1 ˛Vn1 / D 0. In this case, ps D 1, 's D d1 , and prime denotes derivative with respect to n . By using the homogeneous balance principle for the highest order nonlinear term and the highest order partial derivative of Un and Vn in Equation (47), we find the balancing terms. Let m be the balancing term of Un and k be the balancing term of Vn . Here the terms UnC1 and Un1 do not affect the balance. In the first equation of Equation system (47), from the highest order nonlinear term Un Vn and the highest order partial derivative of Un , we find the balancing term of Vn as follows: Un0 Un Vn ) m C 1 D m C k ) k D 1.

(48)

In the second equation of Equation system (47), from the highest order nonlinear term Un Vn and the highest order partial derivative of Vn , we find the balancing term of Un as follows: Vn0 Un Vn ) k C 1 D m C k ) m D 1.

(49)

Thus, substituting these balancing terms into (48)–(49), we obtain Un D ˛0 C ˛1 2 6 UnC1 .n / D ˛0 C ˛1 4

0 p

.2 4/ 2

B @

p 1 .2 4/ d1 2 C . / p A .2 4/ G0 .n / 1C p 2 C f d 1 G.n / 2 2 4 2

p

2 2 4

Vn D ˇ0 C ˇ1 2 6 Vn1 .n / D ˇ0 C ˇ1 4

0 p

.2 4/ 2

B @

G0 .n / , ˛1 ¤ 0, G .n /

G0 .n / 2 C G.n /

3

C f

7 5, 2

G0 .n / , ˇ1 ¤ 0, G .n /

1 p .2 4/ d1 2 C . / p A .2 4/ G0 .n / 2 p f d1 1 C G.n / 2 2 4 2

p

(50)

2 2 4

G0 .n / 2 C G.n /

(51)

(52) 3

f

7 5, 2

(53)

for the traveling wave solutions of Equation (47). Substituting (50)–(53) along with (12) into Equation system (47), clearing the denom 0

i

.n / inator, and setting the coefficients of GG. to zero, we derive a system of nonlinear algebraic equations for ˛0 , ˛1 , ˇ0 , ˇ1 , d1 , and c1 . n/ Solving this algebraic system by the use of Maple, we obtain solutions for Equation system (47) as follows: In the case when 2 4 > 0,



A. BEKIR, Ö. GÜNER, AND B. AYHAN Let tanh

p

2 4 d1 2



0 1 2 3

G0 .n / G.n /

G0 .n / G.n /

G0 .n / G.n /

G0 .n / G.n / G0 .n / G.n /

0

1

2

3 4

G0 .n / G.n /

i

.0 i 3/:

p : ˛1 c1 2 4 2ˇ1 .1 C ˛0 ˛/ D 0, : ˛1 c1 .2 C 2 C / 2ˇ1 .˛˛0 C ˛˛1 C / D 0, : ˛1 c1

p

(54)

.2 4/ 3 2ˇ1 ˛˛1 2ˇ1 .1 C ˛˛0 / D 0,

: 2˛1 .c1 C ˇ1 ˛/ D 0.

From the second equation of system (47), we find the coefficients of

G0 .n / G.n /

i

.0 i 4/:

p : ˇ1 c1 .4 C 2 2 2 / C 2ˇ0 ˛1 2 4 ˛1 p C2˛ˇ1 2 4 D 0, : ˇ1 c1 ..1 2 /2 4.1 C 2 // C 2˛1 2 .ˇ0 2 C 2ˇ0 C ˇ1 / p 2 2 4.ˇ1 C ˇ0 /.˛1 C 2˛ˇ1 / D 0, : 2 4ˇ1 c1 5ˇ1 c1 2 C 2ˇ1 ˛1 2 C 6ˇ0 ˛1 C 4ˇ1 ˛1 C ˇ1 c1 2 4 p 2 2 4 ˇ0 ˛1 C ˇ1 ˛1 C 2˛ˇ12 C 2ˇ0 ˛ˇ1 D 0, : 2 2 .4ˇ1 c1 2ˇ0 ˛1 3˛1 ˇ1 / 2

p

(55)

2 4.˛1 ˇ1 C 2˛ˇ12 / D 0,

: 4ˇ1 2 .c1 ˛1 / D 0.


˛0 D

ˇ1 ˛

p 2 4 2 4 coth d1 2 2

ˇ0 D

p

ˇ1

p p 2 4 d1 2 4 coth 2 2

1 , ˛

˛1 D ˇ1 ˛, (56)

, ˇ1 D ˇ1 ,

d1 D d1 , c1 D ˇ1 ˛, D 1. Substituting these values into Equations (50) and (52), we have the hyperbolic solutions of Equation (45): p

un,1 .t/ D ˛ˇ1

2

p

vn,1 .t/ D ˇ1

0 2 4

@coth 0

2 4 2

@coth

p

2 4 2

p

2 4 2

p 1 2 4 n CC2 cosh 2 p p A 2 4 2 4 C1 cosh n CC2 sinh n 2 2 p

C1 sinh

C1 sinh

d1

d1 C

2 4 n 2

p 1 2 4 n CC2 cosh 2 p p A 2 4 2 4 C1 cosh n CC2 sinh n 2 2 p

2 4 n 2

(57)

(58)

t C , C1 , and C2 are arbitrary constants. where n D d1 n C c1 .1C / If we take, in particular, 2 4 D 4 and C1 D 0 in (57) and (58), the solutions of Equation (45) become

t C / , un,2 D ˛ˇ1 coth .d1 / coth.d1 n C c1 .1C / t C / ; vn,2 D ˇ1 coth .d1 / C coth.d1 n C c1 .1C / Copyright © 2014 John Wiley & Sons, Ltd.

(59)


A. BEKIR, Ö. GÜNER, AND B. AYHAN and if we take 2 4 D 4 and C2 D 0 in (57) and (58), the solutions of Equation (45) become t un,3 D ˛ˇ1 coth .d1 / tanh.d1 n C c1 .1C C / , / vn,3 2 In the case when 4 < 0, p

Let tan

42 d1 2



0 1 2 3

: .˛1 c1

p

G0 .n / G.n /

G0 .n / G.n /

G0 .n / G.n /

G0 .n / G.n / G0 .n / G.n /

G0 .n / G.n /

: ˛1 c1

p

4

2

(61)

3 2ˇ1 ˛˛1 2ˇ1 .1 C ˛˛0 / D 0,

: 2˛1 .c1 C ˇ1 ˛/ D 0.

G0 .n / G.n /

i

.0 i 4/:

p : ˇ1 c1 .4 C 2 2 2 / C 2ˇ0 ˛1 4 2 ˛1 p C2˛ˇ1 4 2 D 0,

1

: ˇ1 c1 .1 2 /2 4.1 C 2 / C 2˛1 2 ˇ0 2 C 2ˇ0 C ˇ1 p 2 4 2 .ˇ1 C ˇ0 / .˛1 C 2˛ˇ1 / D 0,

2

: 2 4ˇ1 c1 5ˇ1 c1 2 C 2ˇ1 ˛1 2 C 6ˇ0 ˛1 C 4ˇ1 ˛1 C ˇ1 c1 2 4 p 2 4 2 ˇ0 ˛1 C ˇ1 ˛1 C 2˛ˇ12 C 2ˇ0 ˛ˇ1 D 0,

4

.0 i 3/:

: ˛1 c1 .2 C 2 C / 2ˇ1 .˛˛0 C ˛˛1 / D 0,

0

3

i

4 2 2ˇ1 .1 C ˛0 ˛// D 0,


(60)

t D ˇ1 coth .d1 / C tanh.d1 n C c1 .1C C / . /

: 2 2 .4ˇ1 c1 2ˇ0 ˛1 3˛1 ˇ1 / 2

p

(62)

4 2 .˛1 ˇ1 C 2˛ˇ12 / D 0,

: 4ˇ1 2 .c1 ˛1 / D 0.


˛0 D

ˇ0 D

ˇ1 ˛

p p 42 42 cot d1 2 2

p p 42 ˇ1 42 cot d1 2 2

1 , ˛

˛1 D ˇ1 ˛, (63)

, ˇ1 D ˇ1 ,

d1 D d1 , c1 D ˇ1 ˛, D 1. Substituting these values into Equations (50) and (52), we have the trigonometric solutions of Equation (45): p

un,4 .t/ D ˛ˇ1

2

p

vn,4 .t/ D ˇ1

0 42

@cot 0

42 2

@cot

p

42 2

p

42 2

d1

d1 C

p p 1 42 42 C1 sin n CC2 cos n 2 2 p p A 42 42 C1 cos n CC2 sin n 2 2 p p 1 42 42 C1 sin n CC2 cos n 2 2 p p A 42 42 C1 cos n CC2 sin n 2 2

(64)

(65)

t where n D d1 n C c1 .1C C , C1 , and C2 are arbitrary constants. /



A. BEKIR, Ö. GÜNER, AND B. AYHAN Similarly, if we take 4 2 D 4 and C1 D 0 in (64) and (65), the solutions of Equation (45) become t un,5 .t/ D ˛ˇ1 cot .d1 / cot.d1 n C c1 .1C C / , / t C / ; vn,5 .t/ D ˇ1 cot .d1 / C cot.d1 n C c1 .1C /

(66)

and if we take 4 2 D 4 and C2 D 0 in (64) and (65), the solutions of Equation (45) become t un,6 .t/ D ˛ˇ1 cot .d1 / tan.d1 n C c1 .1C C / , / t C / . vn,6 .t/ D ˇ1 cot .d1 / C tan.d1 n C c1 .1C /

(67)

5. Conclusion In this work, we have used the .G0 =G/-expansion method for solving fractional differential-difference equations in the sense of modified Riemann–Liouville derivative and applied it to find exact solutions of time-fractional Toda lattice equations and to the time-fractional relativistic Toda lattice system. As a result, some generalized exact solutions for them have been successfully found. Hyperbolic function and trigonometric function solutions with parameters are obtained, from which some known solutions including kink-type solitary wave solution are recovered by setting the parameters as special values. The performance of this method is reliable and simple and gives many new exact solutions. Furthermore, the fractional complex transform is extremely simple but effective for solving fractional differential equations, and this transformation is very important; it ensures that fractional differential-difference equation systems can be turned into differential-difference equation systems of integer order. We deduce that this method can also be applied to other nonlinear fractional differential-difference equation systems.

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