Exact solutions of some systems of fractional differential-difference equations

$Page 1: Exact solutions of some systems of fractional differential-difference equations$
Research Article

Received 24 July 2014 Published online in Wiley Online Library

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

Exact solutions of some systems of fractionaldifferential-difference equations

Ahmet Bekira*†, Özkan Günerb and Burcu Ayhana

Communicated by M. Kirane

In this paper, the�

G0

G

�-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 trans-form is proposed to convert a fractional partial differential-difference equation into its differential-difference equationof integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time-fractional Toda latticeequations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exactsolutions 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 (NLD-DEs) 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 pastseveral 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

function. Zhang et al. [11] and Aslan [12] used the�

G0

G

�-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 function solu-

tions of the integral discrete nonlinear Schrödinger equation. More recently, Zhang et al. [15] proposed a generalized�

G0

G

�-expansion

method to improve and extend the works of Wang et al. [16] and Tang et al. [17] for solving variable-coefficient equations andhigh-dimensional equations.

Fractional differential equations are generalizations of classical differential equations of integer order. In recent decades, fractionaldifferential equations played an important role in applied physics, chemistry, biology, engineering, and finance. The exact solutions ofthese problems, when they exist, are very important in the understanding of the nonlinear fractional physical phenomena. There aremany 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

modified Riemann–Liouville derivative and the�

G0

G

�-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 exam-ples. We will use the modified definition of the Riemann–Liouville fractional derivative. The fractional complex transform [30] is used toanalytically deal with fractional differential equations. This method is extremely simple but effective for solving fractional differentialequations.

a Department of Mathematics - Computer, Art-Science Faculty, Eskisehir Osmangazi University, Eskisehir, Turkeyb 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

The manuscript suggests the�

G0

G

�-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<:

1�.1�˛/

ddt

R t0 .t � �/

�˛.f .�/ � f .0//d� , 0 < ˛ < 1

.f .n/.t//.˛�n/, n � ˛ < nC 1, n � 1.(1)

Modified Riemann–Liouville derivative has many important properties; four of their important formulas are [32]

D˛t x� D�.1C �/

�.1C � � ˛/x��˛ , � > 0, (2)

D˛t .cf .t// D cD˛t f .t/, c D constant (3)

D˛t faf .t/C bg.t/g D aD˛t f .t/C bD˛t g.t/, (4)

where a and b are constant.

D˛t c D 0, c D constant, (5)

which are direct consequences of the equality

d˛x.t/ D �.1C ˛/dx.t/. (6)

The rest of this paper is organized as follows. In Section 2, we describe the algorithm for using the�

G0

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 latticeequations and to the time-fractional relativistic Toda lattice system. In Section 5, some conclusions are given.

2. Description of the�

G0

G

�-expansion method for NLDDEs

In this section, we would like to outline the algorithm for using the�

G0

G

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

consider a system of M fractional NLDDEs in the form

P�

unCp1 .x/ , : : : , unCpk .x/ , : : : , u˛nCp1.x/ , : : : , u˛nCpk

.x/ , : : : , u.r˛/nCp1.x/ , : : : , u.r˛/nCpk

.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 hasQ components ni , the k shift vectors ps 2 ZQ, and u.r˛/ .x/ denotes the collection of modified Riemann–Liouville derivative terms oforder r˛. Using a fractional complex transformation,

unCps .x/ D UnCps .�n/ , �n D

QXiD1

dini C

NXjD1

cj

�.1C ˛/x˛j C �, .s D 1, 2, : : : , k/ , (8)

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

D˛t u D �0

tdud�D˛t �

D˛x u D �0

xdud�D˛x �

, (9)

where �0

t and �0

x are called the sigma indexes (see [33]); without loss of generality we can take �0

t D �0

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

Q�

UnCp1 .�n/ , : : : , UnCpk .�n/ , : : : , U0nCp1.�n/ , : : : , U0nCpk

.�n/ , : : : , U.r/nCp1.�n/ , : : : , U.r/nCpk

.�n/�D 0. (10)

Step 1: We assume the following series expansion as a solution of Equation (10):

Un .�n/ D

mXlD0

˛l

�G0 .�n/

G .�n/

�l

, ˛m ¤ 0, (11)



where m and ˛i are constants to be determined later, and G .�n/ satisfies a second-order linear ordinary differential equation:

d2G .�n/

d�2n

C �dG .�n/

d�nC �G .�n/ D 0, (12)

where � and � are arbitrary constants. Using the general solutions of Equation (12), we obtain the following three cases:

G0 .�n/

G .�n/

8ˆˆ<ˆˆ:

p�2�4�

2

0@ C1 sinh

�p�2�4�

2 �n

�CC2 cosh

�p�2�4�

2 �n

�

C1 cosh

�p�2�4�

2 �n

�CC2 sinh

�p�2�4�

2 �n

�1A � �

2 , �2 � 4� > 0,

p4��2

2

0@�C1 sin

�p4��2

2 �n

�CC2 cos

�p4��2

2 �n

�

C1 cos

�p4��2

2 �n

�CC2 sin

�p4��2

2 �n

�1A � �

2 , �2 � 4� < 0,

C1C1�nCC2

� �2 , �2 � 4� D 0,

(13)

where C1 and C2 are arbitrary constants.Step 2: We derive from (13) and put the formulas

G0��n˙y

�G��n˙y

� D�2C

G0.�n/G.�n/

˙�

p�.�2�4�/

2 f

�p�.�2�4�/

2 y

�

1˙ 2p�2�4�

��2C

G0.�n/G.�n/

�f

�p�.�2�4�/

2 y

� � �2

, (14)

where D ˙1 and

f�p

�.�2�4�/2 y

�D

8<:

tanh�p

�2�4�2 y

�, D 1,

tan�p

4��2

2 y�

, D �1.(15)

By a simple computation, we can obtain the identity

�nCps D �n C 's, 's D ps1d1 C ps2d2 C : : :C psQdQ, (16)

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

UnCps .�n/ D

mXlD0

˛l

24 �

2CG0.�n/G.�n/

˙�

p�.�2�4�/

2 f

�p�.�2�4�/

2 's

�

1˙ 2p�2�4�

��2C

G0.�n/G.�n/

�f

�p�.�2�4�/

2 's

� � �2

35

l

. (17)

Step 3: By using the homogeneous balance principle for the highest order nonlinear term(s) and the highest order partial derivativeof Un .�n/ in Equation (10), we can easily determine the degree m of Equation (11). It should be noted that the leading termsof UnCps .�n/, .ps ¤ 0/will not affect the balancing procedure, because UnCps .�n/ can be interpreted as being of degree zero

in�

G0.�n/G.�n/

�and we are interested in balancing the terms of G0.�n/

G.�n/.

Step 4: Then, substituting Equations (11) and (17) together with (12) into Equation (10) and equating the coefficients of�G0.�n/G.�n/

�i.i D 0, 1, 2, 3, .../ to zero, we obtain a system of nonlinear algebraic equations, from which the constants ˛l , di , and

cj can be explicitly determined. Finally, we substitute these values into Equation (11) and obtain various kinds of discreteexact 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

@t˛D un .vn � vn�1/ ,

@˛vn

@t˛D vn .unC1 � un/ ,

(18)

where 0 < ˛ � 1, un D u .n, t/, vn D v .n, t/, and n 2 Z.Using a fractional complex transformation,



un˙1 D Un˙1 .�n/ , �n D d1nC c1t˛

�.1C ˛/C �,

vn˙1 D Vn˙1 .�n/ , �n D d1nC c1t˛

�.1C ˛/C �,

(19)

and substituting (19) with (2) and (9) into (18), Equation (18) turns into

c1U0n � Un .Vn � Vn�1/ D 0

c1V 0n � Vn .UnC1 � Un/ D 0.(20)

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 findthe 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 Vn�1

do not affect the balance:

Un .�n/ D

mXlD0

˛l

�G0 .�n/

G .�n/

�l

, ˛m ¤ 0, (21)

UnCps .�n/ D

mXlD0

˛l

24 �

2CG0.�n/G.�n/

˙�

p�.�2�4�/

2 f

�p�.�2�4�/

2 's

�

1˙ 2p�2�4�

��2C

G0.�n/G.�n/

�f

�p�.�2�4�/

2 's

� � �2

35

l

, (22)

Vn .�n/ D

kXlD0

ˇl

�G0 .�n/

G .�n/

�l

, ˇk ¤ 0, (23)

VnCps .�n/ D

kXlD0

ˇl

24 �

2CG0.�n/G.�n/

˙�

p�.�2�4�/

2 f

�p�.�2�4�/

2 's

�

1˙ 2p�2�4�

��2C

G0.�n/G.�n/

�f

�p�.�2�4�/

2 's

� � �2

35

l

. (24)

In the first equation of Equation system (20), from the highest order nonlinear term UnVn and the highest order partial derivative of Un,we find the balancing term of Vn as follows:

U0n � UnVn ) mC 1 D mC k) k D 1. (25)

In the second equation of Equation system (20), from the highest order nonlinear term UnVn and the highest order partial derivative ofVn, we find the balancing term of Un as follows:

V 0n � UnVn ) kC 1 D mC k) m D 1. (26)

Thus, substituting these balancing terms into (21)–(24), we obtain

Un D ˛0 C ˛1

�G0 .�n/

G .�n/

�, ˛1 ¤ 0, (27)

UnC1 .�n/ D ˛0 C ˛1

264p�.�2�4�/

2

0B@

2p�.�2�4�/

��2C

G0.�n/G.�n/

�C� f

�p�.�2�4�/

2 d1

�

1C

�2p�2�4�

��2C

G0.�n/G.�n/

��f

�p�.�2�4�/

2 d1

�

1CA � �

2

375 , (28)

Vn D ˇ0 C ˇ1

�G0 .�n/

G .�n/

�, ˇ1 ¤ 0 (29)

Vn�1 .�n/ D ˇ0 C ˇ1

264p�.�2�4�/

2

0B@

2p�.�2�4�/

��2C

G0.�n/G.�n/

�� f

�p�.�2�4�/

2 d1

�

1�

�2p�2�4�

��2C

G0.�n/G.�n/

��f

�p�.�2�4�/

2 d1

�

1CA � �

2

375 , (30)

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

and setting the coefficients of�

G0.�n/G.�n/

�ito zero, we derive a system of nonlinear algebraic equations for ˛0, ˛1, ˇ0, ˇ1, d1, and c1. Solving

this algebraic system by the use of Maple, we obtain hyperbolic and trigonometric function solutions for Equation (20) as follows:



In the case when �2 � 4� > 0,

Let tanh�p

�2�4�2 d1

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

�G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: �

�˛1c1

�p�2 � 4� � �

�� 2ˇ1˛0

�D 0,

�G0.�n/G.�n/

�1: ˛1c1�

�p�2 � 4� � �

�� 2˛1c1� � 2ˇ1�˛1 � 2ˇ1˛0� D 0,

�G0.�n/G.�n/

�2: �3˛1c1�C ˛1c1

p.�2 � 4�/ � 2ˇ1˛0 � 2ˇ1˛1� D 0,

�G0.�n/G.�n/

�3: �2˛1 .c1 C ˇ1/ D 0.

(31)

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

G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: ��

�ˇ1c1

�p�2 � 4�C �

�� 2ˇ0˛1

�D 0,

�G0.�n/G.�n/

�1: �ˇ1c1�

�p�2 � 4�C �

�� 2ˇ1c1�C 2ˇ1�˛1 C 2˛1ˇ0� D 0,

�G0.�n/G.�n/

�2: �3ˇ1c1� � ˇ1c1

p.�2 � 4�/C 2ˇ0˛1 C 2ˇ1˛1� D 0,

�G0.�n/G.�n/

�3: �2ˇ1 .c1 � ˛1/ D 0.

(32)

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

˛0 Dc1

��p�2�4� coth

�p�2�4�

2 d1

��

2 , ˛1 D c1,

ˇ0 D�c1

��Cp�2�4� coth

�p�2�4�

2 d1

��

2 , ˇ1 D �c1,

d1 D d1, c1 D c1, D 1.

(33)

Substituting these values into (27) and (29), we have the hyperbolic function solutions as follows:

un,1 .t/ D c1

p�2�4�

2

0@� coth

�p�2�4�

2 d1

�C

C1 sinh

�p�2�4�

2 �n

�CC2 cosh

�p�2�4�

2 �n

�

C1 cosh

�p�2�4�

2 �n

�CC2 sinh

�p�2�4�

2 �n

�1A , (34)

vn,1 .t/ D �c1

p�2�4�

2

0@coth

�p�2�4�

2 d1

�C

C1 sinh

�p�2�4�

2 �n

�CC2 cosh

�p�2�4�

2 �n

�

C1 cosh

�p�2�4�

2 �n

�CC2 sinh

�p�2�4�

2 �n

�1A , (35)

where �n D d1nC c1t˛

�.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

un,2 D c1

�� coth .d1/C coth.d1nC c1

t˛

�.1C˛/ C �/�

,

vn,2 D �c1

�coth .d1/C coth.d1nC c1

t˛

�.1C˛/ C �/�

;

(36)

and if we take �2 � 4� D 4 and C2 D 0 in (27) and (29), the solutions of Equation (18) become

un,3 D c1

�� coth .d1/C tanh.d1nC c1

t˛

�.1C˛/ C �/�

,

vn,3 D �c1

�coth .d1/C tanh.d1nC c1

t˛

�.1C˛/ C �/�

.

(37)

In the case when �2 � 4� < 0,



Let tan�p

4��2

2 d1


�G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: �

�˛1c1

�p4� � �2 � �

�� 2ˇ1˛0

�D 0,

�G0.�n/G.�n/

�1: ˛1c1�

�p4� � �2 � �

�� 2˛1c1� � 2ˇ1�˛1 � 2ˇ1˛0� D 0,

�G0.�n/G.�n/

�2: �3˛1c1�C ˛1c1

p4� � �2 � 2ˇ1˛0 � 2ˇ1˛1� D 0,

�G0.�n/G.�n/

�3: �2˛1 .c1 C ˇ1/ D 0.

(38)


G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: ��

�ˇ1c1

�p4� � �2 C �

�� 2ˇ0˛1

�D 0,

�G0.�n/G.�n/

�1: �ˇ1c1�

�p4� � �2 C �

�� 2ˇ1c1�C 2ˇ1�˛1 C 2˛1ˇ0� D 0,

�G0.�n/G.�n/

�2: �3ˇ1c1� � ˇ1c1

p4� � �2 C 2ˇ0˛1 C 2ˇ1˛1� D 0,

�G0.�n/G.�n/

�3: �2ˇ1 .c1 � ˛1/ D 0.

(39)


˛0 Dc1

��p

4��2 cot

�p4��2

2 d1

��

2 , ˛1 D c1,

ˇ0 D�c1

��Cp

4��2 cot

�p4��2

2 d1

��

2 , ˇ1 D �c1,

d1 D d1, c1 D c1, D �1.

(40)

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

un,4 .t/ D c1

p4��2

2

0@� cot

�p4��2

2 d1

�C�C1 sin

�p4��2

2 �n

�CC2 cos

�p4��2

2 �n

�

C1 cos

�p4��2

2 �n

�CC2 sin

�p4��2

2 �n

�1A , (41)

vn,4 .t/ D �c1

p4��2

2

0@cot

�p4��2

2 d1

�C�C1 sin

�p4��2

2 �n

�CC2 cos

�p4��2

2 �n

�

C1 cos

�p4��2

2 �n

�CC2 sin

�p4��2

2 �n

�1A , (42)

where �n D d1nC c1t˛

�.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

un,5 .t/ D c1

�� cot .d1/C cot.d1nC c1

t˛

�.1C˛/ C �/�

,

vn,5 .t/ D �c1

�cot .d1/C cot

�d1nC c1

t˛

�.1C˛/ C ��

;

(43)

and if we take 4� � �2 D 4 and C2 D 0 in (41) and (42), the solutions of Equation (18) become

un,6 .t/ D c1

�� cot .d1/C tan.d1nC c1

t˛

�.1C˛/ C �/�

,

vn,6 .t/ D �c1

�cot .d1/C tan

�d1nC c1

t˛

�.1C˛/ C ��

.

(44)



4. Time-fractional relativistic Toda lattice system

In this section, we will consider the time-fractional relativistic Toda lattice system [35]:

D�t un D .1C ˛un/ .vn � vn�1/ ,

D�t un D vn .unC1 � un C ˛vnC1 � ˛vn�1/ ,(45)

where 0 < � � 1, un D u .n, t/, vn D v .n, t/, and n 2 Z.Using a fractional complex transformation,

un˙1 D Un˙1 .�n/ , �n D d1nC c1t�

�.1C�/ C �,

vn˙1 D Vn˙1 .�n/ , �n D d1nC c1t�

�.1C�/ C �,(46)

and substituting (46) with (2) and (9) into (45), Equation (45) turns into

c1U0n � .1C ˛Un/ .Vn � Vn�1/ D 0,

c1V 0n � Vn .UnC1 � Un C ˛VnC1 � ˛Vn�1/ D 0.(47)

In this case, ps D 1, 's D d1, and prime denotes derivative with respect to �n. By using the homogeneous balance principle for thehighest order nonlinear term and the highest order partial derivative of Un and Vn in Equation (47), we find the balancing terms. Letm be the balancing term of Un and k be the balancing term of Vn. Here the terms UnC1 and Un�1 do not affect the balance. In the firstequation of Equation system (47), from the highest order nonlinear term UnVn and the highest order partial derivative of Un, we findthe balancing term of Vn as follows:

U0n � UnVn ) mC 1 D mC k) k D 1. (48)

In the second equation of Equation system (47), from the highest order nonlinear term UnVn and the highest order partial derivative ofVn, we find the balancing term of Un as follows:

V 0n � UnVn ) kC 1 D mC k) m D 1. (49)

Thus, substituting these balancing terms into (48)–(49), we obtain

Un D ˛0 C ˛1

�G0 .�n/

G .�n/

�, ˛1 ¤ 0, (50)

UnC1 .�n/ D ˛0 C ˛1

264p�.�2�4�/

2

0B@

2p�.�2�4�/

��2C

G0.�n/G.�n/

�C� f

�p�.�2�4�/

2 d1

�

1C

�2p�2�4�

��2C

G0.�n/G.�n/

��f

�p�.�2�4�/

2 d1

�

1CA � �

2

375 , (51)

Vn D ˇ0 C ˇ1

�G0 .�n/

G .�n/

�, ˇ1 ¤ 0, (52)

Vn�1 .�n/ D ˇ0 C ˇ1

264p�.�2�4�/

2

0B@

2p�.�2�4�/

��2C

G0.�n/G.�n/

�� f

�p�.�2�4�/

2 d1

�

1�

�2p�2�4�

��2C

G0.�n/G.�n/

��f

�p�.�2�4�/

2 d1

�

1CA � �

2

375 , (53)

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

inator, and setting the coefficients of�

G0.�n/G.�n/

�ito zero, we derive a system of nonlinear algebraic equations for ˛0, ˛1, ˇ0, ˇ1, d1, and c1.

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,



Let tanh�p

�2�4�2 d1


�G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: �

�˛1c1

�p�2 � 4� � �

�� 2ˇ1.1C ˛0˛/

�D 0,

�G0.�n/G.�n/

�1: �˛1c1.�

2 C 2�C �/ � 2ˇ1.˛˛0�C ˛�˛1 C �/ D 0,

�G0.�n/G.�n/

�2: ˛1c1

�p.�2 � 4�/ � 3�

�� 2ˇ1˛˛1� � 2ˇ1.1C ˛˛0/ D 0,

�G0.�n/G.�n/

�3: �2˛1 .c1 C ˇ1˛/ D 0.

(54)

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

G0.�n/G.�n/

�i.0 � i � 4/:

�G0.�n/G.�n/

�0: ��

�ˇ1c1.4�C 2�2 � �2/C 2ˇ0

�˛1

p�2 � 4� � ˛1�

C2˛ˇ1

p�2 � 4�

��D 0,

�G0.�n/G.�n/

�1: ˇ1c1�..1 � 2/�2 � 4�.1C 2//C 2˛1

2.ˇ0�2 C 2ˇ0�C ˇ1��/

�2p�2 � 4�.ˇ1�C ˇ0�/.˛1 C 2˛ˇ1/ D 0,

�G0.�n/G.�n/

�2: 2

��4ˇ1c1� � 5ˇ1c1�

2 C 2ˇ1˛1�2 C 6ˇ0˛1�C 4ˇ1˛1�

�C ˇ1c1

��2 � 4�

��2p�2 � 4�

�ˇ0˛1 C ˇ1˛1�C 2˛ˇ2

1�C 2ˇ0˛ˇ1

�D 0,

�G0.�n/G.�n/

�3: �22 .4ˇ1c1� � 2ˇ0˛1 � 3˛1ˇ1�/ � 2

p�2 � 4�.˛1ˇ1 C 2˛ˇ2

1/ D 0,

�G0.�n/G.�n/

�4: �4ˇ1

2.c1 � ˛1/ D 0.

(55)


˛0 Dˇ1 ˛

�p�2�4� coth

�p�2�4�

2 d1

��

�

2 � 1˛

, ˛1 D �ˇ1˛,

ˇ0 Dˇ1

��p�2�4� coth

�p�2�4�

2 d1

��

2 , ˇ1 D ˇ1,

d1 D d1, c1 D �ˇ1˛, D 1.

(56)

Substituting these values into Equations (50) and (52), we have the hyperbolic solutions of Equation (45):

un,1 .t/ D ˛ˇ1

p�2�4�

2

0@coth

�p�2�4�

2 d1

��

C1 sinh

�p�2�4�

2 �n

�CC2 cosh

�p�2�4�

2 �n

�

C1 cosh

�p�2�4�

2 �n

�CC2 sinh

�p�2�4�

2 �n

�1A (57)

vn,1 .t/ D �ˇ1

p�2�4�

2

0@coth

�p�2�4�

2 d1

�C

C1 sinh

�p�2�4�

2 �n

�CC2 cosh

�p�2�4�

2 �n

�

C1 cosh

�p�2�4�

2 �n

�CC2 sinh

�p�2�4�

2 �n

�1A (58)

where �n D d1nC c1t�

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

un,2 D ˛ˇ1

�coth .d1/ � coth.d1nC c1

t�

�.1C�/ C �/�

,

vn,2 D �ˇ1

�coth .d1/C coth.d1nC c1

t�

�.1C�/ C �/�

;

(59)



and if we take �2 � 4� D 4 and C2 D 0 in (57) and (58), the solutions of Equation (45) become

un,3 D ˛ˇ1

�coth .d1/ � tanh.d1nC c1

t�

�.1C�/ C �/�

,

vn,3 D �ˇ1

�coth .d1/C tanh.d1nC c1

t�

�.1C�/ C �/�

.

(60)

In the case when �2 � 4� < 0,

Let tan�p

4��2

2 d1


�G0.�n/G.�n/

�i.0 � i � 3/:

�G0.�n/G.�n/

�0: �.˛1c1

�p4� � �2 � �

�� 2ˇ1.1C ˛0˛// D 0,

�G0.�n/G.�n/

�1: �˛1c1.�

2 C 2�C �/ � 2ˇ1.˛˛0�C ˛�˛1 � �/ D 0,

�G0.�n/G.�n/

�2: ˛1c1

�p4� � �2 � 3�

�� 2ˇ1˛˛1� � 2ˇ1.1C ˛˛0/ D 0,

�G0.�n/G.�n/

�3: �2˛1 .c1 C ˇ1˛/ D 0.

(61)


G0.�n/G.�n/

�i.0 � i � 4/:

�G0.�n/G.�n/

�0: ��

�ˇ1c1.4�C 2�2 � �2/C 2ˇ0

�˛1

p4� � �2 � ˛1�

C2˛ˇ1

p4� � �2

��D 0,

�G0.�n/G.�n/

�1: ˇ1c1�

�.1 � 2/�2 � 4�.1C 2/

�C 2˛1

2�ˇ0�

2 C 2ˇ0�C ˇ1��

�2p

4� � �2 .ˇ1�C ˇ0�/ .˛1 C 2˛ˇ1/ D 0,

�G0.�n/G.�n/

�2: 2

��4ˇ1c1� � 5ˇ1c1�

2 C 2ˇ1˛1�2 C 6ˇ0˛1�C 4ˇ1˛1�

�C ˇ1c1

��2 � 4�

��2p

4� � �2�ˇ0˛1 C ˇ1˛1�C 2˛ˇ2

1�C 2ˇ0˛ˇ1

�D 0,

�G0.�n/G.�n/

�3: �22 .4ˇ1c1� � 2ˇ0˛1 � 3˛1ˇ1�/ � 2

p4� � �2.˛1ˇ1 C 2˛ˇ2

1/ D 0,

�G0.�n/G.�n/

�4: �4ˇ1

2.c1 � ˛1/ D 0.

(62)


˛0 Dˇ1 ˛

�p

4��2 cot

�p4��2

2 d1

��

�

2 � 1˛

, ˛1 D �ˇ1˛,

ˇ0 Dˇ1

��p

4��2 cot

�p4��2

2 d1

��

2 , ˇ1 D ˇ1,

d1 D d1, c1 D �ˇ1˛, D 1.

(63)

Substituting these values into Equations (50) and (52), we have the trigonometric solutions of Equation (45):

un,4 .t/ D ˛ˇ1

p4��2

2

0@cot

�p4��2

2 d1

��C1 sin

�p4��2

2 �n

�CC2 cos

�p4��2

2 �n

�

C1 cos

�p4��2

2 �n

�CC2 sin

�p4��2

2 �n

�1A (64)

vn,4 .t/ D �ˇ1

p4��2

2

0@cot

�p4��2

2 d1

�C�C1 sin

�p4��2

2 �n

�CC2 cos

�p4��2

2 �n

�

C1 cos

�p4��2

2 �n

�CC2 sin

�p4��2

2 �n

�1A (65)

where �n D d1nC c1t�

�.1C�/ C �, C1, and C2 are arbitrary constants.



Similarly, if we take 4� � �2 D 4 and C1 D 0 in (64) and (65), the solutions of Equation (45) become

un,5 .t/ D ˛ˇ1

�cot .d1/ � cot.d1nC c1

t�

�.1C�/ C �/�

,

vn,5 .t/ D �ˇ1

�cot .d1/C cot.d1nC c1

t�

�.1C�/ C �/�

;

(66)

and if we take 4� � �2 D 4 and C2 D 0 in (64) and (65), the solutions of Equation (45) become

un,6 .t/ D ˛ˇ1

�cot .d1/ � tan.d1nC c1

t�

�.1C�/ C �/�

,

vn,6 .t/ D �ˇ1

�cot .d1/C tan.d1nC c1

t�

�.1C�/ C �/�

.

(67)

5. Conclusion

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

References1. Fermi E, Pasta J, Ulam S. Collected Papers of Enrico Fermi II. University of Chicago Press: IL, 1965.2. Marquii P, Bilbault JM, Rernoissnet M. Observation of nonlinear localized modes in an electrical lattice. Physical Review E 1995; 51(6):6127–6133.3. Kadalbajoo MK, Sharma KK. Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift. Nonlinear

Analysis 2005; 63:1909–1924.4. Ablowitz MJ, Ladik J. Nonlinear differential-difference equations. Journal of Mathematical Physics 1975; 16:598–603.5. Eisenberg HS, Silberberg Y, Morandotti R, Boyd AR, Aitchison JS. Discrete spatial optical solitons in waveguide arrays. Physical Review Letters 1998;

81(16):3383–3386.6. Ma WX, Maruno K. Complexiton solutions of the Toda lattice equation. Physica A 2004; 343:219–237.7. Wu L, Xie LD, Zhang JF. Adomian decomposition method for nonlinear differential-difference equations. Communications in Nonlinear Science and

Numerical Simulation 2009; 14(1):12–18.8. Wang Z. Discrete tanh method for nonlinear difference-differential equations. Computer Physics Communications 2009; 180:1104–1108.9. Zhu SD. Exp-function method for the discrete mKdV lattice. International Journal of Nonlinear Sciences and Numerical Simulation 2007; 8:465–469.

10. Baldwin D, Goktas U, Hereman W. Symbolic computation of hyperbolic tangent solutions for nonlinear differential–difference equations. ComputerPhysics Communications 2004; 162(3):203–217.

11. Zhang S, Dong L, Ba J, Sun Y. The .G0=G/-expansion method for nonlinear differential–difference equations. Physics Letters A 2009; 373:905–910.12. Aslan I. Discrete exact solutions to some nonlinear differential-difference equations via the .G0=G/-expansion method,. Applied Mathematics and

Computation 2009; 215(8):3140–3147.13. Zhang S. Discrete Jacobi elliptic function expansion method for nonlinear difference equation. Physica Scripta 2009; 80:045002–045010.14. Gepreel KA. Rational Jacobi elliptic solutions for nonlinear difference differential equations. Nonlinear Science Letters A 2011; 2:151–158.15. Zhang J, Wei X, Lu Y. A generalized .G0=G/-expansion method and its applications. Physics Letters A 2008; 372:3653–3658.16. Wang ML, Li XZ, Zhang JL. The .G0=G/-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics.

Physics Letters A 2008; 372:417–423.17. Tang B, He Y, Wei L, Wang S. Variable-coefficient discrete .G0=G/-expansion method for nonlinear differential–difference equations. Physics Letters A

2011; 375:3355–3361.18. Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley: New York, 1993.19. Podlubny I. Fractional Differential Equations. Academic Press: California, 1999.20. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Elsevier: Amsterdam, 2006.21. Gepreel KA, Omran S. Exact solutions for nonlinear partial fractional differential equations. Chinese Physics B 2012; 21:110204.22. Bekir A, Güner Ö. Exact solutions of nonlinear fractional differential equations by .G0=G/-expansion method,. Chinese Physics B 2013; 22(11):110202.23. Bekir A, Güner Ö, Cevikel AC. Fractional complex transform and exp-function methods for fractional differential equations. Abstract and Applied

Analysis 2013; 2013:426462.24. Zheng B. Exp-function method for solving fractional partial differential equations. The Scientific World Journal 2013; 2013:465723.25. Güner Ö, Cevikel AC. A procedure to construct exact solutions of nonlinear fractional differential equations. The Scientific World Journal 2014;

2014:489495.26. Lu B. The first integral method for some time fractional differential equations. Journal of Mathematical Analysis and Applications 2012; 395:684–693.27. Zhang S, Zhang H-Q. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A 2011; 375:1069–1073.



28. Liu W, Chen K. The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana-Journalof Physics 2013; 81(3):377–384.

29. Bulut H, Baskonus HM, Pandir Y. The modified trial equation method for fractional wave equation and time fractional generalized burgers equation.Abstract and Applied Analysis 2013; 2013:636802.

30. He JH, Elegan SK, Li ZB. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics LettersA 2012; 376:257–259.

31. Jumarie G. Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathemat-ics with Applications 2006; 51:1367–1376.

32. Jumarie G. Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution. Journal of AppliedMathematics and Computing 2007; 24(1–2):31–48.

33. Saad M, Elagan SK, Hamed YS, Sayed M. Using a complex transformation to get an exact solutions for fractional generalized coupled MKDV and KDVequations. International Journal of Basic & Applied Sciences 2013; 13(01):23–25.

34. Bakkyaraj T, Sahadevan R. An approximate solution to some classes of fractional nonlinear partial differential difference equation using adomiandecomposition method. Journal of Fractional Calculus and Applications 2014; 5(1):37–52.

35. Feng QH. Exact solutions for fractional differential-difference equations by an extended Riccati sub-ode method. Communications in TheoreticalPhysics 2013; 59(5):521–527.


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Exact solutions of some systems of fractional differential-difference equations