Applied Mathematics and Computation 249 (2014) 81–88

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Travelling wave solutions to some time–space nonlinearevolution equations

http://dx.doi.org/10.1016/j.amc.2014.09.1040096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: younis.pu@gmail.com (M. Younis), hamood84@gmail.com (H.ur Rehman), muzamil.iftikhar@ue.edu.pk (M. Iftikhar).

Muhammad Younis a,⇑, Hamood ur Rehman b, Muzamil Iftikhar b

a Centre for Undergraduate Studies, University of the Punjab, Lahore 54590, Pakistanb Department of Mathematics, University of Education, Okara Campus, Okara 56300, Pakistan

a r t i c l e i n f o a b s t r a c t

Keywords:Travelling wave solutionsFractional complex transformationModified direct algebraic methodFractional order nonlinear equationsFractional calculus

In this article, the modified direct algebraic method has been extended to celebrate thenew complex travelling wave solutions of nonlinear evolution equations of fractionalorder. For finding solutions, the fractional complex transformation has been implementedto convert nonlinear partial fractional differential equations into nonlinear ordinary differ-ential equations. After this, the modified direct algebraic method has been applied success-fully, to construct the complex travelling wave solutions. As far as concerned about theapplications, the nonlinear time–space fractional KdV, foam drainage and coupled burgers’equations of fractional order have been considered.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The nonlinear phenomena has not only shown a variety of applications in electromagnetic, acoustics, electrochemistry,cosmology and fluid mechanics, but has also shown many fundamental physical phenomena in nature that cannot bedescribed without nonlinear evolution equations [1–4].

Knowing the importance of nonlinear evolution equations, a lot of techniques already exist to find solutions in the field ofmathematical physics. In recent years, many new powerful and interesting techniques have been developed to handle thenonlinear equations. For example, adomian decomposition method [5] and generalized differential transform method [6]have been used to find the numerical solutions for fractional order differential equations. ðG0=GÞ-expansion method [7]can also construct the travelling wave solutions of nonlinear evolution equations. This method has been further extended[8,9] to seek the solutions for nonlinear equations of fractional order. Jacobi elliptic function expansion method [10],tanh-function method [11], homotopy perturbation method [12] etc. were also developed for solving the nonlinear evolutionequations. For more references see also [13–16].

In this article, fractional complex transformation [17] has been implemented to convert nonlinear partial differentialequations into nonlinear ordinary differential equations, using Jumarie’s modified Riemann–Liouville derivative [18]. Themodified direct algebraic method [19,20] can be applied to find the exact travelling wave solutions from ODE. For this,the following applications have been considered to find new complex solutions. Firstly, time–space fractional derivative non-linear KdV equation, in the following form [8], has been considered:

82 M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88

@au@taþ au

@bu@xbþ @

3bu@x3b

¼ 0; t > 0; 0 < a; b 6 1: ð1:1Þ

The time–space fractional derivative foam drainage equation [9] can be considered in the following form:

@au@ta¼ 1

2u@2bu@x2b

þ 2u2 @bu@xbþ @bu

@xb

� �2

; t > 0; 0 < a; b 6 1; ð1:2Þ

and in view of [8], the time–space fractional coupled burgers’ equation has the following form:

@au@ta¼ @

2bu@x2b

� 2u@bu@xb� a

@bðuvÞ@xb

@av@ta ¼

@2bv@x2b

� 2v @bv@xb� b

@bðuvÞ@xb

; t > 0; 0 < a; b 6 1: ð1:3Þ

The rest of the article is organized as follows, in Section 2 the basic definitions and properties of the fractional calculushave been considered. In Section 3, the modified direct algebraic method has been proposed to find the new exact solutionsfor NPDEs of fractional order with the help of fractional complex transformation. Some applications, in Section 4, to constructthe new exact solutions of time–space fractional KdV, foam drainage and coupled burgers’ equations of fractional order havebeen considered. The conclusion is drawn in last Section 5.

2. Preliminaries and basic definitions

In this section, the extended method has been applied in the sense of Jumarie’s modified Riemann–Liouville derivative oforder a. For this, some basic definitions and properties of fractional calculus are being considered (for details see also [3]).

Definition 2.1. Jumarie’s modified Riemann–Liouville derivative, of order a, can be defined as follows:

Das f ðsÞ ¼

1Cð1�aÞ

dds

R s0 ðs� nÞ�aðf ðnÞ � f ð0ÞÞdn; 0 < a < 1;

ðf ðnÞðsÞÞa�n

; n 6 a < nþ 1; n P 1:

(

Moreover, some properties for the modified Riemann–Liouville derivative have also been considered,

Das sr ¼ Cð1þ rÞ

Cð1þ r � aÞ sr�a;

Das ðf ðsÞgðsÞÞ ¼ f ðsÞDa

s gðsÞ þ gðsÞDas tðsÞ;

Das f ½gðsÞ� ¼ f 0g ½gðsÞ�D

as gðsÞ ¼ Da

s f ½gðsÞ�ðg0ðtÞÞa:

The above properties are very useful and help to convert fractional order differential equations into ordinary differen-tial equations. In the following section, the extended modified direct algebraic method has been described to find thesolutions.

3. The modified direct algebraic method

In this section, the modified direct algebraic method [19,20] has been discussed to obtain the solutions of nonlinear par-tial differential equations of fractional order.

For this, we consider the NPDEs of fractional order in the following way:

P u;Dat u;Db

s u;Dcxu; . . . ;Da

t Dat u;Da

t Dbs u;Db

s Dbs u;Db

s Dcxu; . . .

� �¼ 0; for 0 < a;b; c < 1; ð3:1Þ

where u is an unknown function and P is a polynomial of u and its partial fractional derivatives along with the involvement ofhigher order derivatives and nonlinear terms.

To find the exact solutions, the method can be performed using the following steps.

Step 1: First, we convert the NPDEs of fractional order into nonlinear ordinary differential equations using fractional com-plex transformation introduced by Li and He [17]. For this, the travelling wave variable

uðt; x; yÞ ¼ uðnÞ; n ¼ Kta

Cðaþ 1Þ þLxb

Cðbþ 1Þ þMyc

Cðcþ 1Þ ð3:2Þ

M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88 83

where K; L and M are non-zero arbitrary constants, permits to reduce Eq. (3.1) to an ODE of u ¼ uðnÞ in the followingform

Pðu;u0;u00;u000; . . .Þ ¼ 0: ð3:3Þ

Step 2: Suppose that the solution of Eq. (3.1) can be expressed by the following ansatze:

uðnÞ ¼ A0 þXm

j¼0

Ajuj þ Bju�j� �i

; ð3:4Þ

u0 ¼ Bþu2; ð3:5Þ

where B is a parameter to be determined, u ¼ uðnÞ;u0 ¼ dudn .

Step 3: The homogeneous balance can be used, to determine the positive integer m, between the highest order derivativesand the nonlinear terms appearing in (3.4).

Step 4: After the substitution of (3.4) into (3.3), we collect all the terms with the same order of uj together. Equate eachcoefficient of the polynomials of uj to zero, yields the set of algebraic equations for K; L;M;B;A0 andAi;Biði ¼ 1;2; . . . ;mÞ.

Step 5: After solving the system of algebraic equations, the following travelling wave solutions can be celebrated using thegeneral solutions of the Eq. (3.5).

(i) If B < 0

u ¼ �ffiffiffiffiffiffiffi�Bp

tanhffiffiffiffiffiffiffi�Bp

n� �

; or u ¼ �ffiffiffiffiffiffiffi�Bp

cothffiffiffiffiffiffiffi�Bp

n� �

; ð3:6Þ

it depends on initial conditions.

(ii) If B > 0

u ¼ffiffiffiBp

tanffiffiffiBp

n� �

; or u ¼ �ffiffiffiBp

cotffiffiffiBp

n� �

; ð3:7Þ

it depends on initial conditions.

(iii) If B ¼ 0

u ¼ �1n: ð3:8Þ

After substituting the above results into Eq. (3.4), the travelling wave solutions of Eq. (3.1) can be obtained.

4. Applications to modified direct algebraic method

In this section, the modified direct algebraic method has been applied successfully to construct the exact solutions to thefollowing three NPDEs of fractional order.

4.1. Example 1. Nonlinear time–space fractional KdV equation

It may be observed that the fractional complex transform

uðx; tÞ ¼ uðnÞ; n ¼ iKxb

Cðbþ 1Þ þiLta

Cðaþ 1Þ ð4:1Þ

where K and L are arbitrary constants, which permits to reduce the Eq. (1.1) into an ODE of the following form:

C þ iLuþ 12

aiKu2 � iK3u00 ¼ 0: ð4:2Þ

After calculating the homogeneous balance ði:e;m ¼ 2Þ, between the highest order derivatives and nonlinear term pre-sented in the above equation, we have form Eq. (3.4) as follow,

uðnÞ ¼ A0 þ A1uþ A2u2 þ B1u�1 þ B2u�2 ð4:3Þ

where A0;A1;A2;B1;B2;K and L are arbitrary constants. To determine these constants substitute the Eq. (4.3) into (4.2), andcollecting all the terms with the same power of u together, and equating each coefficient equal to zero, yields a set of thefollowing algebraic equations.

84 M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88

C þ iLA0 þ12

aiKðA20 þ 2A1B1 þ 2A2B2Þ � 2iK3ðB2 þ 2A2BÞ ¼ 0;

LA1 þ aKðA0A1 þ A2B1Þ � 2K3A1B ¼ 0;

LB1 þ aKðA0B1 þ A1B2Þ � 2K3B1B ¼ 0;

LA2 þ12

aKðA21 þ 2A0A2Þ � 8K3A2B ¼ 0;

LB2 þ12

aKðB21 þ 2A0B2Þ � 8K3B2B ¼ 0;

aKA2 � 2K3 ¼ 0;

aKB2 � 2K3B2 ¼ 0;

aKA2 � 12K3 ¼ 0;

aKB2 � 12K3B2 ¼ 0:

For solving the above equations, the following different cases to Eq. (1.1) can be obtained.Case 1:

C ¼ aK

6B2 6B2A20 � 4BA0B2 � 3iB2A2

0 þ iB22

� �; B2 ¼

12K2B2

a;

L ¼ iK aA0 � 8K2B� �

; A1 ¼ A2 ¼ B1 ¼ 0:

Where A0;B2;B; a and K are arbitrary constants and B2 – 0.Case 2:

C ¼ i2aK

2iL2 � 16LK3B� L2 þ 16iLK3Bþ 16K6B2� �

; A2 ¼12K2

a;

A0 ¼23

B� iL

12K3 ; A1 ¼ B1 ¼ B2 ¼ 0:

Where a;A0;A2;B;K and L are arbitrary constants and A2 – 0.Case 3:

C ¼ i2aK

2iL2 � 16LK3Bþ L2 þ 16iLK3Bþ 256K6B2� �

; A2 ¼12K2

a; B2 ¼

12K2B2

a;

A0 ¼23

B� iL

12K3 ; A1 ¼ B1 ¼ 0:

Where A0 and A2 are arbitrary constants and A2 – 0.Consequent to case 1, the travelling wave solutions of the Eq. (1.1) have the following results.

For B < 0, we have

u1ðx; tÞ ¼ u1ðnÞ ¼8K2B

a� iL

aKþ 12K2B3

atanh2 in

2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaKA0 þ iL

2K

r !;

u2ðx; tÞ ¼ u2ðnÞ ¼8K2B

a� iL

aKþ 12K2B3

acoth2 in

2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaKA0 þ iL

2K

r !;

where n ¼ iKxb

Cðbþ 1Þ �KðaA0 � 8K2BÞta

Cðaþ 1Þ :

For B > 0, we have

u3ðx; tÞ ¼ u3ðnÞ ¼8K2B

a� iL

aKþ 12K2B3

atan2 n

2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaKA0 þ iL

2K

r !;

u4ðx; tÞ ¼ u4ðnÞ ¼8K2B

a� iL

aKþ 12K2B3

acot2 n

2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaKA0 þ iL

2K

r !;

where n ¼ iKxb

Cðbþ 1Þ �KðaA0 � 8K2BÞta

Cðaþ 1Þ :

M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88 85

For B ¼ 0, we have

u5ðx; tÞ ¼ u5ðnÞ ¼8K2B

a� iL

aK; where n ¼ iKxb

Cðbþ 1Þ :

These are the new exact complex solutions to Eq. (1.1). Similarly, the variety of exact solutions can be constructed from cases2 and 3.

4.2. Example 2. Nonlinear time–space fractional derivative foam drainage equation

Apply the fractional complex transformation (4.1) on the Eq. (1.2), the following form can be obtained:

iLu0 þ 12

K2uu00 � 2iKu2u0 þ K2ðu0Þ2 ¼ 0: ð4:4Þ

After calculating the homogeneous balance, which is m ¼ 1. We have

uðnÞ ¼ A0 þ A1uþ B1u�1 ð4:5Þ

where A0;A1;K and L are arbitrary constants. To determine these constants, equate the coefficients of w�1;w�2;w�3 and w�4

equal to zero, yields the set of algebraic equations.

iLðA1B� B1Þ � 2iKðA0A1B� A20B1 þ A2

1BB1 � A1B21Þ þ K2ðA1BB1 þ A1B� B2

1Þ ¼ 0;

iLA1 þ K2A1ðA1Bþ B1Þ � 2iKðA20A1 þ A3

1Bþ A21B1Þ þ 2K2A1ðA1B� B1Þ ¼ 0;

iLBB1 þ K2BB1ðA1B� 2B1 � 1Þ � 2iKB1ðA20B� A1BB1 þ B2

1 þ 2A1BÞ ¼ 0;

K2A0B1B2 þ 4iKA0B21B ¼ 0;

KB� 4iðA1B� B1 þ BÞ ¼ 0;

K2B2B21 þ iKB3

1B ¼ 0;

K2A21 � iKA3

1 ¼ 0;KB� 4iB1 ¼ 0;KB� 4iA1 ¼ 0:

For solving the above equations, the following different cases to Eq. (1.2) can be obtained.Case 1:

B ¼ iK; L ¼ �iK; K ¼ K; A0 ¼ 0; A1 ¼ 0; B1 ¼ �1

Where K is any arbitrary constant.Case 2:

A1 ¼1B; A1 ¼ �iK; L ¼ �iA2

1; A0 ¼ 0; A1 ¼ A1; B1 ¼ 0

Where A1 is any arbitrary constant.Case 3:

B ¼ B; A1 ¼ �iK; L ¼ 0; A0 ¼ 0; A1 ¼ A1; B1 ¼ 0

Where A1 is any arbitrary constant.Consequent to case 1, the travelling wave solutions of the Eq. (1.2) have the following results.For B < 0, we have

u1ðx; tÞ ¼ u1ðnÞ ¼ iffiffiffiBp

tanh iffiffiffiBp

n� �� ��1

;

u2ðx; tÞ ¼ u2ðnÞ ¼ iffiffiffiBp

coth iffiffiffiBp

n� �� ��1

;

where n ¼ iKxb

Cðbþ 1Þ þKta

Cðaþ 1Þ :

For B > 0, we have

u3ðx; tÞ ¼ u3ðnÞ ¼ �ffiffiffiBp

tanffiffiffiBp

n� �� ��1

;

u4ðx; tÞ ¼ u4ðnÞ ¼ffiffiffiBp

cotffiffiffiBp

n� �� ��1

;

where n ¼ iKxb

Cðbþ 1Þ þKta

Cðaþ 1Þ :

86 M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88

For B ¼ 0, we have

u5ðx; tÞ ¼ u5ðnÞ ¼iKxb

Cðbþ 1Þ þKta

Cðaþ 1Þ :

These are the new exact complex solutions to Eq. (1.2). Similarly, the variety of exact solutions can be constructed for cases 2and 3.

4.3. Example 3. Nonlinear time–space fractional coupled burgers’ equation

Applying the fractional complex transformation on the Eq. (1.3), which reduces into the following form:

c0 þ iLuþ K2u0 þ iKu2 þ aiKðuvÞ ¼ 0

c1 þ iLv þ K2v 0 þ iKv2 þ biKðuvÞ ¼ 0: ð4:6Þ

Where c0 and c1 are constants of integration. Now by calculating homogeneous balance, which is m1 ¼ m2 ¼ 1. We have thefollowing forms of the Eq. (3.4),

uðnÞ ¼ A0 þ A1uþ B1u�1;

uðnÞ ¼ C0 þ C1uþ D1u�1;

where A0;A1; B1; C0;C1 and D1 are arbitrary constants. After equating the coefficients of uj equal to zero, the following set ofalgebraic equations can be obtained.

c0 þ iLA0 þ K2ðA1B� B1Þ þ iKA20 þ 2iKA1B1 þ iaKðA0C0 þ A1D1 þ B1C1Þ ¼ 0;

iLA1 þ 2iKA0A1 þ iaKðA0C1 þ A1C0Þ ¼ 0;iLB1 þ 2iKA0B1 þ iaKðA0D1 þ B1C0Þ ¼ 0;

� K2BB1 þ iKðB21 þ aB1D1Þ ¼ 0;

K2A1 þ iKA21 þ iaKA1C1 ¼ 0;

K2C1 þ iKC21 þ iaKA1C1 ¼ 0;

� K2DD1 þ iKðD21 þ bB1D1Þ ¼ 0;

iLD1 þ 2iKC0D1 þ ibKðC0B1 þ D1A0Þ ¼ 0;iLB1 þ 2iKC0C1 þ ibKðC0A1 þ C1A0Þ ¼ 0;

c1 þ iLC0 þ K2ðC1D� D1Þ þ iKC20 þ 2iKC1D1 þ ibKðC0A0 þ C1B1 þ D1A1Þ ¼ 0;

After solving the above equations, the following different cases to Eq. (1.3) can be obtained.Case 1:

c0 ¼18

KL2ðLK2B� 2iÞ; A1 ¼ �iK; D ¼ �iKD1; A0 ¼

12

LK; L ¼ �iA21;

C1 ¼iK; b ¼ 2; c1 ¼ C0 ¼ B1 ¼ a ¼ 0:

Where D1;A1;K are arbitrary constants.Case 2:

B1 ¼ �BC1; K ¼ �iC1; L ¼ 0; c0 ¼ �C21B1ð1þ aÞ; c1 ¼ C2

1ðC1D� bB1Þ;A0 ¼ A1 ¼ C0 ¼ D1 ¼ 0:

Where B1;C1;D; a; b are any arbitrary constants.Case 3:

c0 ¼ iKA20 þ 4K2B1; B1 ¼ �iKB; A1 ¼ iK; L ¼ �2KA0; C0 ¼ C1 ¼ D1 ¼ 0:

Where A0;K; a; b are arbitrary constants.Case 4:

B1 ¼ �BA1; K ¼ �iA1; L ¼ 2iA0A1; K ¼ �iA1; b ¼ 2; c0 ¼ A20A1 � 4A2

1B1;

c1 ¼ A21ð2B1 � A1DÞ; C0 ¼ D1 ¼ 0; C1 ¼ �A1:

Where A0;A1;B1;C1;D; a; b are any arbitrary constants.Consequent to case 1, the travelling wave solutions to the Eq. (1.3) have the following results.

M. Younis et al. / Applied Mathematics and Computation 249 (2014) 81–88 87

For B < 0, we have

u1ðx; tÞ ¼ u1ðnÞ ¼12

LK þ 1K

ffiffiffiBp

tanh iffiffiffiBp

n� �

;

v1ðx; tÞ ¼ v1ðnÞ ¼�1K

ffiffiffiBp

tanh iffiffiffiBp

n� �

þ KD

ffiffiffiBp

tanh iffiffiffiBp

n� �� ��1

;

u2ðx; tÞ ¼ u2ðnÞ ¼12

LK þ iK

ffiffiffiBp

coth iffiffiffiBp

n� �

;

v2ðx; tÞ ¼ v2ðnÞ ¼�1K

ffiffiffiBp

coth iffiffiffiBp

n� �

þ KD

ffiffiffiBp

coth iffiffiffiBp

n� �� ��1

;

where n ¼ iKxb

Cðbþ 1Þ þA2

1ta

Cðaþ 1Þ :

For B > 0, we have

u3ðx; tÞ ¼ u1ðnÞ ¼12

LK � iK

ffiffiffiBp

tanffiffiffiBp

n� �

;

v3ðx; tÞ ¼ v1ðnÞ ¼iK

ffiffiffiBp

tanffiffiffiBp

n� �

� KD

ffiffiffiBp

tanffiffiffiBp

n� �� ��1

;

u4ðx; tÞ ¼ u2ðnÞ ¼12

LK þ iK

ffiffiffiBp

cotffiffiffiBp

n� �

;

v4ðx; tÞ ¼ v2ðnÞ ¼�iK

ffiffiffiBp

cotffiffiffiBp

n� �

þ KD

ffiffiffiBp

cotffiffiffiBp

n� �� ��1

;

where n ¼ iKxb

Cðbþ 1Þ þA2

1ta

Cðaþ 1Þ :

For B ¼ 0, we have

u5ðx; tÞ ¼ u5ðnÞ ¼12

LK þ iKn

; v5ðx; tÞ ¼ v5ðnÞ ¼ �i

Kn� iKn

D;

where n ¼ iKxb

Cðbþ 1Þ þA2

1ta

Cðaþ 1Þ :

These are the new exact complex solutions to Eq. (1.3). Similarly, the variety of exact solutions can be constructed for cases 2,3 and 4.

5. Conclusion

The modified direct algebraic method has been extended successfully to construct the travelling wave solutions of non-linear partial differential equation of fractional order, in the sense of modified Riemann–Liouville derivative. First, the frac-tional complex transformation has been used to convert the fractional order partial differential equations into ordinarydifferential equations. Then, the modified direct algebraic method has been used successfully to find the exact travellingwave solutions. The three applications have been considered to construct the new exact travelling wave solutions for thenonlinear time–space fractional KdV equation, time–space fractional derivative foam drainage equation and time–spacefractional burgers’ equation. It can also be concluded that the proposed method is very simple, reliable and propose a varietyof exact solutions to NPDEs of fractional order. Since, the homogeneous balancing principle has been used, so it can beclaimed that this method can be applied to other fractional order partial differential equations where the homogeneous bal-ancing principle is satisfied.

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