Research ArticleExact Solutions of the Time Fractional BBM-Burger Equation byNovel (1198661015840119866)-Expansion Method

Muhammad Shakeel Qazi Mahmood Ul-Hassan Jamshad Ahmad and Tauseef Naqvi

Department of Mathematics Faculty of Sciences HITEC University Taxila Cantt Taxila 47080 Pakistan

Correspondence should be addressed to Muhammad Shakeel muhammadshakeel74yahoocom

Received 15 February 2014 Accepted 27 May 2014 Published 11 September 2014

Academic Editor Hossein Jafari

Copyright copy 2014 Muhammad Shakeel et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The fractional derivatives are used in the sense modified Riemann-Liouville to obtain exact solutions for BBM-Burger equationof fractional order This equation can be converted into an ordinary differential equation by using a persistent fractional complextransform and as a result hyperbolic function solutions trigonometric function solutions and rational solutions are attainedTheperformance of the method is reliable useful and gives newer general exact solutions with more free parameters than the existingmethods Numerical results coupled with the graphical representation completely reveal the trustworthiness of the method

1 Introduction

Differential equations of noninteger order are generalizationsof conventional differential equations of integer order [1]Exploration and applications of integrals and derivatives ofrandomorder are efficiently dealt with the field ofmathemati-cal analysis called as fractional calculus which has engrossedin considerable interest in many disciplines nowadays Thebehavior of many physical systems can be perfectly definedby the fractional theory In recent years we cannot rebuffthe importance of fractional differential equations becauseof their numerous applications in the areas of physics andengineering For example the nonlinear fluctuation of earth-quakes can be modeled with the help of fractional derivativesand the fluid-dynamic traffic model with fractional deriva-tives can eradicate the problems arising from the huge trafficflow [2 3] We apply a generalized fractional complex trans-form [4ndash7] to convert the given fractional order differentialequation to ordinary differential equation On account ofdevelopment of the computer and its exact description ofcountless real-life problems fractional calculus has touchedthe height of fame and success nowadays although it wasinvented three centuries ago by Newton and Leibniz Manyimportant phenomena in electromagnetic acoustics vis-coelasticity electrochemistry and material science are better

described by differential equations of noninteger order [8ndash11] A physical interpretation of the fractional calculus wasgiven in [12ndash14] With the development of symbolic compu-tation software likeMaple many researchers developed andestablishedmany numerical and analytical methods to searchfor exact solutions of nonlinear evolution equations (NLEEs)for example Cole-Hopf transformation [15] Tanh-functionmethod [16ndash19] inverse scattering transform method [20]variational iteration method [21 22] Exp-function method[23ndash26] homogeneous balance method [27 28] and F-expansion method [29 30] which are used for searching theexact solutions

Lately a straight and crisp method called (1198661015840119866)-

expansion method was introduced by Wang et al [31] andconfirmed that it is a powerful method for seeking analyticsolutions of nonlinear evolution equations (NLEEs) Foradditional references see the articles [32ndash37] In order toestablish the efficiency and diligence of (1198661015840119866)-expansionmethod and to extend the range of applicability furtherresearch has been carried out by several researchers Forillustration Zhang et al [38] made an overview of (1198661015840119866)-expansion method for the evolution equations with variablecoefficients Zhang et al [39] also presented an improved(1198661015840119866)-expansion method to look for more broad traveling

wave solutions Zayed [40] offered a new technique of

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 181594 15 pageshttpdxdoiorg1011552014181594

2 Advances in Mathematical Physics

(1198661015840119866)-expansion method where 119866(120585) gratifies the Jacobi

elliptic equation [1198661015840(120585)]2 = 11989021198664(120585) + 119890

11198662(120585) + 119890

0 where

1198902 1198901 and 119890

0are random constants and obtained new exact

solutions Zayed [41] for a second time offered a differentapproach of this method in which 119866(120585) satisfies the Riccatiequation 1198661015840(120585) = 119860119866(120585) + 119861119866

2(120585) where 119860 and 119861 are casual

constantsThe (119866

1015840119866)-expansion method and the transformed

rational function method used by Ma and Lee [42] have acommon idea That is we initially put the given nonlinearevolution equation (NLEE) into the equivalent ordinary dif-ferential equation (ODE) and then ODE can be transformedinto a system of arithmetical polynomials with the influentialconstants By the solutions of the ordinary differential equa-tion we can obtain the exact traveling solutions and rationalsolution of the nonlinear evolution equations

In this article we will apply novel (1198661015840119866)-expansionmethod introduced by Alam et al [43] to solve the timefractional BBM-Burger equation whereas the modifiedRiemann-Liouville derivative given by Jumarie [44] is usedand abundant new families of exact solutions are found TheJumariersquos modified Riemann-Liouville derivative of order 120572 isdefined by the following expression

119863120572

119905119891 (119905)

=

1

Γ (1 minus 120572)

119889

119889119905

int

119905

0

(119905 minus 120585)minus120572

(119891 (120585) minus 119891 (0)) 119889120585

0 lt 120572 lt 1

(119891(119899)(119905))

(120572minus119899)

119899 le 120572 lt 119899 + 1 119899 ge 1

(1)

Some important properties of Jumariersquos derivative are

119863120572

119905119891 (119905) =

Γ (1 + 120574)

Γ (1 + 120574 minus 120572)

119905120574minus120572

(2)

119863120572

119905(119891 (119905) 119892 (119905)) = 119892 (119905)119863

120572

119905119891 (119905) + 119891 (119905)119863

120572

119905119892 (119905) (3)

119863120572

119905119891 [119892 (119905)] = 119891

1015840

119892[119892 (119905)]119863

120572

119905119892 (119905) = 119863

120572

119892119891 [119892 (119905)] (119892

1015840

(119905))

120572

(4)

2 Description of the Method

Consider the fractional partial differential equation in theform

119878 (119906 119906119909 119906119905 119863120572

119905119906 ) = 0 0 lt 120572 le 1 (5)

where 119863120572119905119906 is Jumariersquos modified Riemann-Liouville deriva-

tives of 119906 119906(119909 119905) is an unknown function and 119878 is a poly-nomial in 119906 and its various partial derivatives includingfractional derivatives in which the highest order derivativesand nonlinear terms are involved

The main steps of the method are as follows

Step 1 Li and He [5] projected a fractional complex trans-formation to convert fractional partial differential equationinto ordinary differential equation (ODE) so all analytical

methods devoted to the advanced calculus can be easilyapplied to the fractional calculus The following complextransformation

119906 (119909 119905) = 119906 (120585) 120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(6)

where 119871 119881 are arbitrary constants with 119871 119881 = 0 permits usto convert (5) into an ordinary differential equation of integerorder in the form

119875 (119906 1199061015840 11990610158401015840 119906101584010158401015840 ) = 0 (7)

where the primes stand for the ordinary derivatives withrespect to 120585

Step 2 Integrate (7) term by term one or more times ifpossible capitulate constant(s) of integration which can becalculated later on

Step 3 Suppose that the solution of (7) can be expressed as

119906 (120585) =

119898

sum

119894=minus119898

120572119894(119896 + Φ (120585))

119894

(8)

where

Φ (120585) =1198661015840(120585)

119866 (120585)

(9)

Herein 120572minus119898

or 120572119898

may be zero but both of them cannotbe zero simultaneously 120572

119894(119894 = 0 plusmn1 plusmn2 plusmn119873) and 119896 are

constants to be determined later and 119866 = 119866(120585) satisfies thesecond order nonlinear ordinary differential equation

11986611986610158401015840= 119860119866119866

1015840+ 1198611198662+ 119862(119866

1015840)

2

(10)

where primes denote the derivative with respect 120585 119860 119861 and119862 are genuine constants

The Cole-Hopf transformation Φ(120585) = ln (119866(120585))120585

=

1198661015840(120585)119866(120585) reduces (10) into the Riccati equation

Φ1015840

(120585) = 119861 + 119860Φ (120585) + (119862 minus 1)Φ2

(120585) (11)

Equation (11) has individual twenty-five solutions (see Zhu[45] for details)

Step 4 The positive integer119898 can be calculated by balancingthe highest order linear termwith the highest order nonlinearterm in (7)

Step 5 Inserting (8) together with (9) and (10) into (7)we obtain polynomials in (119896 + (119866

1015840119866))119894 and (119896 + (119866

1015840119866))minus119894

(119894 = 0 1 2 119873) Collecting each coefficient of the resultedpolynomials to zero acquiesces an overdetermined set ofalgebraic equations for 120572

119894(119894 = 0 plusmn1 plusmn2 plusmn119873) 119896 119871 and

119881

Step 6 After solving the system of algebraic equationsobtained in Step 5 we can obtain the values of the constantsThe solutions of (10) together with the obtained values of theconstants yield abundant exact travelingwave solutions of thenonlinear evolution equation (5)

Advances in Mathematical Physics 3

3 Application of the Method to the TimeFractional BBM-Burger Equation

Consider the following BBM-Burger equation in time frac-tional operator form as

119863120572

119905119906 minus 119906119909119909119905

+ 119906119909+ (

1199062

2

)

119909

= 0 119905 gt 0 0 lt 120572 le 1 (12)

S Kumar and D Kumar [46] used new fractional homotopyanalysis transform method to time fractional BBM-Burgerequation and found the series solution of this equation Songand Zhang [47] and Fakhari et al [48] find different typeof solutions of the fractional BBM-Burger equation by usinghomotopy analysis method

By the use of (4) (12) is converted into an ordinarydifferential equation of integer order and after integratingonce we obtain

(119871 + 119881) 119906 +119871

2

1199062minus 119881119871211990610158401015840+ 1198621= 0 (13)

where 1198621is an integration constant

Considering the homogeneous balance between 11990610158401015840 and

1199062 in (13) we obtain 119898 = 2 Therefore the trial solution (8)

becomes

119906 (120585) = 120572minus2(119896 + Φ (120585))

minus2

+ 120572minus1(119896 + Φ (120585))

minus1

+ 1205720+ 1205721(119896 + Φ (120585)) + 120572

2(119896 + Φ (120585))

2

(14)

Using (14) into (13) left-hand side is converted into polyno-mials in (119896 + (1198661015840119866))119894 and (119896 + (1198661015840119866))minus119894 (119894 = 0 1 2 119873)Equating the coefficients of same power of the resultedpolynomials to zero we attain a set of algebraic equations(which are omitted for the sake of simplicity) for 120572

0 1205721 1205722

120572minus1 120572minus2 119896 119862

1 119871 and 119881 Solving the overdetermined set

of algebraic equations by using the symbolic computationsoftware such as Maple 13 we obtain the following solutionsets

The Set 1 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

1205721= 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

1205722= 12119881119871(119862 minus 1)

2

119881 = 119881 119871 = 119871 119896 = 119896 120572minus1= 0 120572

minus2= 0

1198621= minus

1

2119871

((((minus4119862 + 4) 119861 + 1198602)1198811198712minus 119881 minus 119871)

times (((minus4119862 + 4) 119861 + 1198602)1198811198712+ 119881 + 119871))

(15)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 2 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

120572minus1= minus12 ((1 minus 119862) 119896

2+ 119896119860 minus 119861)119881119871 ((2 minus 2119862) 119896 + 119860)

120572minus2= 12119881119871((1 minus 119862) 119896

2+ 119896119860 minus 119861)

2

119881 = 119881 119871 = 119871 119896 = 119896 1205722= 0 120572

1= 0

1198621= minus

1

2119871

(119881 ((minus4119862 + 4) 119861 + 1198602) 1198712minus 119881 minus 119871)

times (119881 ((minus4119862 + 4) 119861 + 1198602) 1198712+ 119881 + 119871)

(16)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 3 We consider that

1205722= 12119881119871(119862 minus 1)

2

1205720=

minus21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

119896 =119860

2 (119862 minus 1)

120572minus2=

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

1205721= 0 120572

minus1= 0 119881 = 119881 119871 = 119871

1198621= minus

8

119871

(1198811198712((minus4119862 + 4) 119861 + 119860

2) minus

119881

4

minus119871

4

)

times (1198811198712((minus4119862 + 4) 119861 + 119860

2) +

119881

4

+119871

4

)

(17)

where 119881 119871 119860 119861 and 119862 are arbitrary constantsSubstituting (15)ndash(17) into (14) we obtain

1199061(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times (119896 + (1198661015840

119866

)) + 12119881119871(119862 minus 1)2times (119896 + (

1198661015840

119866

))

(18)

4 Advances in Mathematical Physics

1199062(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862 minus 1) 119896 minus 8119861+8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times (119896 + (1198661015840

119866

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times (119896 + (1198661015840

119866

))

minus2

(19)

1199063(120585)

= minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2times (

119860

2(119862 minus 1)

+ (1198661015840

119866

))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

+ (1198661015840

119866

))

minus2

(20)

where

120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(21)

Substituting the solutions 119866(120585) of (10) into (18) and simplify-ing we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

(22)

and its graph is shown in Figure 1 Consider

1199063

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) 2

1199064

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585))) 2

Advances in Mathematical Physics 5

minus150000

minus100000

minus50000

0

1

2

3

4

5

1

2

3

4

5

x t

(a) 120572 = 025

minus10

minus8

times105

minus6

minus4

minus2

0

1

2

3

4

5

1

2

3

4

5

x t

(b) 120572 = 050

minus90000

minus80000

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

0

1 1

2 2

3 3

4 4

5 5

x t

(c) 120572 = 075

minus250

minus200

minus150

minus100

minus50

1

2

3

4

5

1

2

3

4

5

x

t

(d) 120572 = 1

Figure 1 (a)ndash(d) show the singular soliton solution for 11990621for different values of parameters

1199065

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

2

1199066

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

(1198661015840119866)-expansion method where 119866(120585) gratifies the Jacobi

elliptic equation [1198661015840(120585)]2 = 11989021198664(120585) + 119890

11198662(120585) + 119890

0 where

1198902 1198901 and 119890

0are random constants and obtained new exact

solutions Zayed [41] for a second time offered a differentapproach of this method in which 119866(120585) satisfies the Riccatiequation 1198661015840(120585) = 119860119866(120585) + 119861119866

2(120585) where 119860 and 119861 are casual

constantsThe (119866

1015840119866)-expansion method and the transformed

rational function method used by Ma and Lee [42] have acommon idea That is we initially put the given nonlinearevolution equation (NLEE) into the equivalent ordinary dif-ferential equation (ODE) and then ODE can be transformedinto a system of arithmetical polynomials with the influentialconstants By the solutions of the ordinary differential equa-tion we can obtain the exact traveling solutions and rationalsolution of the nonlinear evolution equations

In this article we will apply novel (1198661015840119866)-expansionmethod introduced by Alam et al [43] to solve the timefractional BBM-Burger equation whereas the modifiedRiemann-Liouville derivative given by Jumarie [44] is usedand abundant new families of exact solutions are found TheJumariersquos modified Riemann-Liouville derivative of order 120572 isdefined by the following expression

119863120572

119905119891 (119905)

=

1

Γ (1 minus 120572)

119889

119889119905

int

119905

0

(119905 minus 120585)minus120572

(119891 (120585) minus 119891 (0)) 119889120585

0 lt 120572 lt 1

(119891(119899)(119905))

(120572minus119899)

119899 le 120572 lt 119899 + 1 119899 ge 1

(1)

Some important properties of Jumariersquos derivative are

119863120572

119905119891 (119905) =

Γ (1 + 120574)

Γ (1 + 120574 minus 120572)

119905120574minus120572

(2)

119863120572

119905(119891 (119905) 119892 (119905)) = 119892 (119905)119863

120572

119905119891 (119905) + 119891 (119905)119863

120572

119905119892 (119905) (3)

119863120572

119905119891 [119892 (119905)] = 119891

1015840

119892[119892 (119905)]119863

120572

119905119892 (119905) = 119863

120572

119892119891 [119892 (119905)] (119892

1015840

(119905))

120572

(4)

2 Description of the Method

Consider the fractional partial differential equation in theform

119878 (119906 119906119909 119906119905 119863120572

119905119906 ) = 0 0 lt 120572 le 1 (5)

where 119863120572119905119906 is Jumariersquos modified Riemann-Liouville deriva-

tives of 119906 119906(119909 119905) is an unknown function and 119878 is a poly-nomial in 119906 and its various partial derivatives includingfractional derivatives in which the highest order derivativesand nonlinear terms are involved

The main steps of the method are as follows

Step 1 Li and He [5] projected a fractional complex trans-formation to convert fractional partial differential equationinto ordinary differential equation (ODE) so all analytical

methods devoted to the advanced calculus can be easilyapplied to the fractional calculus The following complextransformation

119906 (119909 119905) = 119906 (120585) 120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(6)

where 119871 119881 are arbitrary constants with 119871 119881 = 0 permits usto convert (5) into an ordinary differential equation of integerorder in the form

119875 (119906 1199061015840 11990610158401015840 119906101584010158401015840 ) = 0 (7)

where the primes stand for the ordinary derivatives withrespect to 120585

Step 2 Integrate (7) term by term one or more times ifpossible capitulate constant(s) of integration which can becalculated later on

Step 3 Suppose that the solution of (7) can be expressed as

119906 (120585) =

119898

sum

119894=minus119898

120572119894(119896 + Φ (120585))

119894

(8)

where

Φ (120585) =1198661015840(120585)

119866 (120585)

(9)

Herein 120572minus119898

or 120572119898

may be zero but both of them cannotbe zero simultaneously 120572

119894(119894 = 0 plusmn1 plusmn2 plusmn119873) and 119896 are

constants to be determined later and 119866 = 119866(120585) satisfies thesecond order nonlinear ordinary differential equation

11986611986610158401015840= 119860119866119866

1015840+ 1198611198662+ 119862(119866

1015840)

2

(10)

where primes denote the derivative with respect 120585 119860 119861 and119862 are genuine constants

The Cole-Hopf transformation Φ(120585) = ln (119866(120585))120585

=

1198661015840(120585)119866(120585) reduces (10) into the Riccati equation

Φ1015840

(120585) = 119861 + 119860Φ (120585) + (119862 minus 1)Φ2

(120585) (11)

Equation (11) has individual twenty-five solutions (see Zhu[45] for details)

Step 4 The positive integer119898 can be calculated by balancingthe highest order linear termwith the highest order nonlinearterm in (7)

Step 5 Inserting (8) together with (9) and (10) into (7)we obtain polynomials in (119896 + (119866

1015840119866))119894 and (119896 + (119866

1015840119866))minus119894

(119894 = 0 1 2 119873) Collecting each coefficient of the resultedpolynomials to zero acquiesces an overdetermined set ofalgebraic equations for 120572

119894(119894 = 0 plusmn1 plusmn2 plusmn119873) 119896 119871 and

119881

Step 6 After solving the system of algebraic equationsobtained in Step 5 we can obtain the values of the constantsThe solutions of (10) together with the obtained values of theconstants yield abundant exact travelingwave solutions of thenonlinear evolution equation (5)

Advances in Mathematical Physics 3

3 Application of the Method to the TimeFractional BBM-Burger Equation

Consider the following BBM-Burger equation in time frac-tional operator form as

119863120572

119905119906 minus 119906119909119909119905

+ 119906119909+ (

1199062

2

)

119909

= 0 119905 gt 0 0 lt 120572 le 1 (12)

S Kumar and D Kumar [46] used new fractional homotopyanalysis transform method to time fractional BBM-Burgerequation and found the series solution of this equation Songand Zhang [47] and Fakhari et al [48] find different typeof solutions of the fractional BBM-Burger equation by usinghomotopy analysis method

By the use of (4) (12) is converted into an ordinarydifferential equation of integer order and after integratingonce we obtain

(119871 + 119881) 119906 +119871

2

1199062minus 119881119871211990610158401015840+ 1198621= 0 (13)

where 1198621is an integration constant

Considering the homogeneous balance between 11990610158401015840 and

1199062 in (13) we obtain 119898 = 2 Therefore the trial solution (8)

becomes

119906 (120585) = 120572minus2(119896 + Φ (120585))

minus2

+ 120572minus1(119896 + Φ (120585))

minus1

+ 1205720+ 1205721(119896 + Φ (120585)) + 120572

2(119896 + Φ (120585))

2

(14)

Using (14) into (13) left-hand side is converted into polyno-mials in (119896 + (1198661015840119866))119894 and (119896 + (1198661015840119866))minus119894 (119894 = 0 1 2 119873)Equating the coefficients of same power of the resultedpolynomials to zero we attain a set of algebraic equations(which are omitted for the sake of simplicity) for 120572

0 1205721 1205722

120572minus1 120572minus2 119896 119862

1 119871 and 119881 Solving the overdetermined set

of algebraic equations by using the symbolic computationsoftware such as Maple 13 we obtain the following solutionsets

The Set 1 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

1205721= 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

1205722= 12119881119871(119862 minus 1)

2

119881 = 119881 119871 = 119871 119896 = 119896 120572minus1= 0 120572

minus2= 0

1198621= minus

1

2119871

((((minus4119862 + 4) 119861 + 1198602)1198811198712minus 119881 minus 119871)

times (((minus4119862 + 4) 119861 + 1198602)1198811198712+ 119881 + 119871))

(15)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 2 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

120572minus1= minus12 ((1 minus 119862) 119896

2+ 119896119860 minus 119861)119881119871 ((2 minus 2119862) 119896 + 119860)

120572minus2= 12119881119871((1 minus 119862) 119896

2+ 119896119860 minus 119861)

2

119881 = 119881 119871 = 119871 119896 = 119896 1205722= 0 120572

1= 0

1198621= minus

1

2119871

(119881 ((minus4119862 + 4) 119861 + 1198602) 1198712minus 119881 minus 119871)

times (119881 ((minus4119862 + 4) 119861 + 1198602) 1198712+ 119881 + 119871)

(16)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 3 We consider that

1205722= 12119881119871(119862 minus 1)

2

1205720=

minus21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

119896 =119860

2 (119862 minus 1)

120572minus2=

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

1205721= 0 120572

minus1= 0 119881 = 119881 119871 = 119871

1198621= minus

8

119871

(1198811198712((minus4119862 + 4) 119861 + 119860

2) minus

119881

4

minus119871

4

)

times (1198811198712((minus4119862 + 4) 119861 + 119860

2) +

119881

4

+119871

4

)

(17)

where 119881 119871 119860 119861 and 119862 are arbitrary constantsSubstituting (15)ndash(17) into (14) we obtain

1199061(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times (119896 + (1198661015840

119866

)) + 12119881119871(119862 minus 1)2times (119896 + (

1198661015840

119866

))

(18)

4 Advances in Mathematical Physics

1199062(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862 minus 1) 119896 minus 8119861+8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times (119896 + (1198661015840

119866

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times (119896 + (1198661015840

119866

))

minus2

(19)

1199063(120585)

= minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2times (

119860

2(119862 minus 1)

+ (1198661015840

119866

))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

+ (1198661015840

119866

))

minus2

(20)

where

120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(21)

Substituting the solutions 119866(120585) of (10) into (18) and simplify-ing we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

(22)

and its graph is shown in Figure 1 Consider

1199063

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) 2

1199064

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585))) 2

Advances in Mathematical Physics 5

minus150000

minus100000

minus50000

0

1

2

3

4

5

1

2

3

4

5

x t

(a) 120572 = 025

minus10

minus8

times105

minus6

minus4

minus2

0

1

2

3

4

5

1

2

3

4

5

x t

(b) 120572 = 050

minus90000

minus80000

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

0

1 1

2 2

3 3

4 4

5 5

x t

(c) 120572 = 075

minus250

minus200

minus150

minus100

minus50

1

2

3

4

5

1

2

3

4

5

x

t

(d) 120572 = 1

Figure 1 (a)ndash(d) show the singular soliton solution for 11990621for different values of parameters

1199065

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

2

1199066

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

3 Application of the Method to the TimeFractional BBM-Burger Equation

Consider the following BBM-Burger equation in time frac-tional operator form as

119863120572

119905119906 minus 119906119909119909119905

+ 119906119909+ (

1199062

2

)

119909

= 0 119905 gt 0 0 lt 120572 le 1 (12)

S Kumar and D Kumar [46] used new fractional homotopyanalysis transform method to time fractional BBM-Burgerequation and found the series solution of this equation Songand Zhang [47] and Fakhari et al [48] find different typeof solutions of the fractional BBM-Burger equation by usinghomotopy analysis method

By the use of (4) (12) is converted into an ordinarydifferential equation of integer order and after integratingonce we obtain

(119871 + 119881) 119906 +119871

2

1199062minus 119881119871211990610158401015840+ 1198621= 0 (13)

where 1198621is an integration constant

Considering the homogeneous balance between 11990610158401015840 and

1199062 in (13) we obtain 119898 = 2 Therefore the trial solution (8)

becomes

119906 (120585) = 120572minus2(119896 + Φ (120585))

minus2

+ 120572minus1(119896 + Φ (120585))

minus1

+ 1205720+ 1205721(119896 + Φ (120585)) + 120572

2(119896 + Φ (120585))

2

(14)

Using (14) into (13) left-hand side is converted into polyno-mials in (119896 + (1198661015840119866))119894 and (119896 + (1198661015840119866))minus119894 (119894 = 0 1 2 119873)Equating the coefficients of same power of the resultedpolynomials to zero we attain a set of algebraic equations(which are omitted for the sake of simplicity) for 120572

0 1205721 1205722

120572minus1 120572minus2 119896 119862

1 119871 and 119881 Solving the overdetermined set

of algebraic equations by using the symbolic computationsoftware such as Maple 13 we obtain the following solutionsets

The Set 1 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

1205721= 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

1205722= 12119881119871(119862 minus 1)

2

119881 = 119881 119871 = 119871 119896 = 119896 120572minus1= 0 120572

minus2= 0

1198621= minus

1

2119871

((((minus4119862 + 4) 119861 + 1198602)1198811198712minus 119881 minus 119871)

times (((minus4119862 + 4) 119861 + 1198602)1198811198712+ 119881 + 119871))

(15)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 2 We consider that

1205720= (119881 (12(119862 minus 1)

21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

120572minus1= minus12 ((1 minus 119862) 119896

2+ 119896119860 minus 119861)119881119871 ((2 minus 2119862) 119896 + 119860)

120572minus2= 12119881119871((1 minus 119862) 119896

2+ 119896119860 minus 119861)

2

119881 = 119881 119871 = 119871 119896 = 119896 1205722= 0 120572

1= 0

1198621= minus

1

2119871

(119881 ((minus4119862 + 4) 119861 + 1198602) 1198712minus 119881 minus 119871)

times (119881 ((minus4119862 + 4) 119861 + 1198602) 1198712+ 119881 + 119871)

(16)

where 119896 119871 119881 119860 119861 and 119862 are arbitrary constants

The Set 3 We consider that

1205722= 12119881119871(119862 minus 1)

2

1205720=

minus21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

119896 =119860

2 (119862 minus 1)

120572minus2=

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

1205721= 0 120572

minus1= 0 119881 = 119881 119871 = 119871

1198621= minus

8

119871

(1198811198712((minus4119862 + 4) 119861 + 119860

2) minus

119881

4

minus119871

4

)

times (1198811198712((minus4119862 + 4) 119861 + 119860

2) +

119881

4

+119871

4

)

(17)

where 119881 119871 119860 119861 and 119862 are arbitrary constantsSubstituting (15)ndash(17) into (14) we obtain

1199061(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times (119896 + (1198661015840

119866

)) + 12119881119871(119862 minus 1)2times (119896 + (

1198661015840

119866

))

(18)

4 Advances in Mathematical Physics

1199062(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862 minus 1) 119896 minus 8119861+8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times (119896 + (1198661015840

119866

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times (119896 + (1198661015840

119866

))

minus2

(19)

1199063(120585)

= minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2times (

119860

2(119862 minus 1)

+ (1198661015840

119866

))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

+ (1198661015840

119866

))

minus2

(20)

where

120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(21)

Substituting the solutions 119866(120585) of (10) into (18) and simplify-ing we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

(22)

and its graph is shown in Figure 1 Consider

1199063

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) 2

1199064

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585))) 2

Advances in Mathematical Physics 5

minus150000

minus100000

minus50000

0

1

2

3

4

5

1

2

3

4

5

x t

(a) 120572 = 025

minus10

minus8

times105

minus6

minus4

minus2

0

1

2

3

4

5

1

2

3

4

5

x t

(b) 120572 = 050

minus90000

minus80000

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

0

1 1

2 2

3 3

4 4

5 5

x t

(c) 120572 = 075

minus250

minus200

minus150

minus100

minus50

1

2

3

4

5

1

2

3

4

5

x

t

(d) 120572 = 1

Figure 1 (a)ndash(d) show the singular soliton solution for 11990621for different values of parameters

1199065

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

2

1199066

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

1199062(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862 minus 1) 119896 minus 8119861+8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times (119896 + (1198661015840

119866

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times (119896 + (1198661015840

119866

))

minus2

(19)

1199063(120585)

= minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2times (

119860

2(119862 minus 1)

+ (1198661015840

119866

))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

+ (1198661015840

119866

))

minus2

(20)

where

120585 = 119871119909 + 119881119905120572

Γ (1 + 120572)

(21)

Substituting the solutions 119866(120585) of (10) into (18) and simplify-ing we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

1199061

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

2

(22)

and its graph is shown in Figure 1 Consider

1199063

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) 2

1199064

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (coth (radicΔ120585)plusmncsch (radicΔ120585))) 2

Advances in Mathematical Physics 5

minus150000

minus100000

minus50000

0

1

2

3

4

5

1

2

3

4

5

x t

(a) 120572 = 025

minus10

minus8

times105

minus6

minus4

minus2

0

1

2

3

4

5

1

2

3

4

5

x t

(b) 120572 = 050

minus90000

minus80000

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

0

1 1

2 2

3 3

4 4

5 5

x t

(c) 120572 = 075

minus250

minus200

minus150

minus100

minus50

1

2

3

4

5

1

2

3

4

5

x

t

(d) 120572 = 1

Figure 1 (a)ndash(d) show the singular soliton solution for 11990621for different values of parameters

1199065

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

2

1199066

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

minus150000

minus100000

minus50000

0

1

2

3

4

5

1

2

3

4

5

x t

(a) 120572 = 025

minus10

minus8

times105

minus6

minus4

minus2

0

1

2

3

4

5

1

2

3

4

5

x t

(b) 120572 = 050

minus90000

minus80000

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

0

1 1

2 2

3 3

4 4

5 5

x t

(c) 120572 = 075

minus250

minus200

minus150

minus100

minus50

1

2

3

4

5

1

2

3

4

5

x

t

(d) 120572 = 1

Figure 1 (a)ndash(d) show the singular soliton solution for 11990621for different values of parameters

1199065

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896

minus 8119861 + 8119861119862 + 1198602) 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

4 (119862 minus 1)

times (2119860 + radicΔ

times (tanh(radicΔ120585

4

) + coth(radicΔ120585

4

)))

2

1199066

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

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Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicΔ (1198652+1198672)minus119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585)+119861

]]

]

2

1199067

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+ 1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicΔ (1198652+1198672)+119865radicΔ cosh (radicΔ120585)

119865 sinh (radicΔ120585) + 119861

]]

]

2

(23)

where 119865 and119867 are real constants Consider the following

1199068

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 cosh (radicΔ1205852)radicΔ sinh (radicΔ1205852) minus 119860 cosh (radicΔ1205852)

2

1199069

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sinh (radicΔ1205852)radicΔ cosh (radicΔ1205852)minus119860 sinh (radicΔ1205852)

2

11990610

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 cosh (radicΔ120585)radicΔ sinh (radicΔ120585)minus119860 cosh (radicΔ120585) plusmn 119894radicΔ

2

11990611

1(120585) = (119881 (12(119862 minus 1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

times 119896+

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585)minus119860 sinh (radicΔ120585) plusmn radicΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sinh (radicΔ120585)radicΔ cosh (radicΔ120585) minus 119860 sinh (radicΔ120585) plusmn radicΔ

2

(24)

WhenΔ = 1198602minus4119861119862+4119861 lt 0 and119860(119862minus1) = 0 (or 119861(119862minus1) =

0)

11990612

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896+1

2 (119862 minus 1)

(minus119860+radicminusΔ tan(radicminusΔ120585

2

))

2

11990613

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

+ 12119881119871(119862 minus 1)2

times 119896minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

2

11990614

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) 2

11990615

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585)))

+ 12119881119871(119862 minus 1)2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicminusΔ (cot (radicminusΔ120585) plusmn csch (radicminusΔ120585))) 2

11990616

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus119860+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861]]

]

2

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

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Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

11990617

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus 1198672) minus 119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585) + 119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

times minus119860+

plusmnradicminusΔ(1198652minus 1198672)minus119865radicminusΔ cos (radicminusΔ120585)

119865 sin(radicminusΔ120585) + 119861]]

]

2

11990618

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times[[

[

119896 +1

2 (119862 minus 1)

times

minus 119860

+

plusmnradicminusΔ (1198652minus1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861

]]

]

+ 12119881119871(119862 minus 1)2

times[[

[

119896 +1

2 (119862 minus 1)

timesminus119860+

plusmnradicminusΔ(1198652minus 1198672)+119865radicminusΔ cos (radicminusΔ120585)

119865 sin (radicminusΔ120585)+119861]]

]

2

(25)

where 119865 and 119867 are real constants such that 1198652 minus 1198672gt 0

Consider the following

11990619

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896 minus

2119861 cos (radicminusΔ1205852)radicminusΔ sin (radicminusΔ1205852) + 119860 cos (radicminusΔ1205852)

2

11990620

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

+ 12119881119871(119862 minus 1)2

times 119896+

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)

2

11990621

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

times 119896 minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896minus

2119861 cos (radicminusΔ120585)radicminusΔ sin (radicminusΔ120585)+119860 cos (radicminusΔ120585)plusmnradicminusΔ

2

11990622

1(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

+ 12119881119871(119862 minus 1)2

times 119896 +

2119861 sin (radicminusΔ1205852)radicminusΔ cos (radicminusΔ1205852)minus119860 sin (radicminusΔ1205852)plusmnradicminusΔ

2

(26)

When 119861 = 0 and 119860(119862 minus 1) = 0

11990623

1(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus1198601198881

(119862 minus 1) 1198881+ cosh (119860120585) minus sinh (119860120585)

2

(27)

and its graph is shown in Figure 2 Consider

11990624

1(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh (119860120585) + sinh (119860120585)

+ 12119881119871(119862 minus 1)2

times 119896 minus119860 (cosh (119860120585) + sinh (119860120585))

(119862 minus 1) 1198881+ cosh(119860120585) + sinh(119860120585)

2

(28)

where 1198881is an arbitrary constant

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

1(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

+ 12119881119871 (119862 minus 1) (minus2119896119862 + 2119896 + 119860)

times 119896 minus1

(119862 minus 1) 120585 + 1198882

+ 12119881119871(119862 minus 1)2

times 119896 minus1

(119862 minus 1)120585 + 1198882

2

(29)

where 1198882is an arbitrary constant

Substituting the solutions 119866(120585) of (10) into (19) andsimplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ tanh(radicΔ120585

2

))

minus2

1199062

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicΔ coth(radicΔ120585

2

))

minus2

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

2

15

1

05

0

minus05

1

23

45

6

78

9

10

tx

10

8

6

4

2

0

(a) 120572 = 025

2

15

1

05

0

minus05

1

234

56

78910

t x

10

8

6

4

2

0

(b) 120572 = 050

2

15

1

05

0

minus05

1

23

4

56

78910

t x

10

8

6

4

2

0

(c) 120572 = 075

2

1

0

1234

56

78

910

t x

10

8

6

4

2

0

(d) 120572 = 1

Figure 2 (a)ndash(d) show the bell-shape secℎ2 solitary traveling wave solution for 119906231for different values of parameters

1199063

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

times (119860 + radicΔ (tanh (radicΔ120585) plusmn 119894sech (radicΔ120585))) minus2

(30)the other families of exact solutions of (12) are omitted forconvenience

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)11990612

2(120585)

= (119881 (12(119862 minus 1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

(minus119860 + radicminusΔ tan(radicminusΔ120585

2

))

minus2

(31)

and its graph is given in Figure 3 Consider

11990613

2(120585)

= (119881 (12(119862 minus 1)21198962minus 12119860 (119862 minus 1) 119896 minus 8119861 + 8119861119862 + 119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

2 (119862 minus 1)

(119860 + radicminusΔ cot(radicminusΔ120585

2

))

minus2

11990614

2(120585)

= (119881 (12(119862minus1)21198962minus12119860 (119862minus1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 +1

2 (119862 minus 1)

times (minus119860+radicminusΔ (tan (radicminusΔ120585)plusmnsec (radicminusΔ120585))) minus2

(32)

When 119860 = 119861 = 0 and (119862 minus 1) = 0 the solution of (12) is

11990625

2(120585) = (119881 (12(119862minus1)

21198962minus12119860 (119862 minus 1) 119896minus8119861+8119861119862+119860

2)

times 1198712minus 119881 minus 119871) (119871)

minus1

minus 12119881119871 ((1 minus 119862) 1198962+ 119896119860 minus 119861) ((2 minus 2119862) 119896 + 119860)

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus1

+ 12119881119871((1 minus 119862) 1198962+ 119896119860 minus 119861)

2

times 119896 minus1

(119862 minus 1)120585 + 1198882

minus2

(33)

where 1198882is an arbitrary constant

We can write down the other families of exact solutionsof (12) which are omitted for practicality

Finally substituting the solutions 119866(120585) of (10) into (20)and simplifying we obtain the following solutions

When Δ = 1198602minus 4119861119862 + 4119861 gt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

1199061

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ tanh(radicΔ120585

2

)))

minus2

1199062

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicΔ coth(radicΔ120585

2

)))

minus2

1199063

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Advances in Mathematical Physics

minus16

minus14

minus12

minus1

minus8

minus6

minus4

minus2

times107

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

x t

(a) 120572 = 025

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

minus3

times106

minus2

minus1

(b) 120572 = 050

minus180000

minus160000

minus140000

minus120000

minus100000

minus80000

minus60000

minus40000

minus20000

minus30

minus20

minus10

0

10

20

30 10

9

8

7

6

xt

(c) 120572 = 075

minus30

minus20

minus100

10

20

30 10

9

8

7

6

x t

minus50000

minus40000

minus30000

minus20000

minus10000

(d) 120572 = 1

Figure 3 (a)ndash(d) show singular periodic traveling wave solution of 119906122for different values of parameters

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicΔ tanh (radicΔ120585) plusmn 119894sech (radicΔ120585) )minus2

(34)

Others families of exact solutions are omitted for the sake ofsimplicity

When Δ = 1198602minus 4119861119862 + 4119861 lt 0 and 119860(119862 minus 1) = 0 (or

119861(119862 minus 1) = 0)

11990612

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ tan(radicminusΔ120585

2

)))

minus2

(35)

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 13

minus10

minus8

minus6

minus4

minus2

times107

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(a) 120572 = 025

minus4

times107

minus3

minus2

minus1

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(b) 120572 = 050

minus25

times107

minus2

minus15

minus1

minus05

minus10

minus5

0

5

10 1009

0807

0605

0403

0201

x t

(c) 120572 = 075

minus70000

minus60000

minus50000

minus40000

minus30000

minus20000

minus10000

minus10

minus5

0

5

10 1009

0807

060504

0302

01

x t

(d) 120572 = 1

Figure 4 (a)ndash(d) show singular periodic traveling wave solution of 119906123for different values of parameters

and its graph is given in Figure 4 Consider

11990613

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

(radicminusΔ cot(radicminusΔ120585

2

)))

minus2

11990614

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (1

2 (119862 minus 1)

times radicminusΔ tan (radicminusΔ120585) plusmn sec (radicminusΔ120585) )minus2

(36)

When (119862 minus 1) = 0 and 119860 = 119861 = 0 the solution of (12) is

11990625

3(120585) = minus

21198811198712((4 minus 4119862) 119861 + 119860

2) minus 119881 minus 119871

119871

+ 12119881119871(119862 minus 1)2

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Advances in Mathematical Physics

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

2

+

3119881119871((4 minus 4119862) 119861 + 1198602)

2

4(1 minus 119862)2

times (119860

2(119862 minus 1)

minus1

(119862 minus 1)120585 + 1198882

)

minus2

(37)where 119888

2is an arbitrary constant

Other exact solutions of (12) are omitted here for conve-nience

4 Discussion

If we replace119860 by minus119860 and 119861 by minus119861 and put119862 = 0 in (10) thenthe novel (1198661015840119866)-expansion method coincides with Akbaret alrsquos generalized and improved (119866

1015840119866)-expansion method

[36] On the other hand if we put 119896 = 0 in (8) and 119862 = 0

in (10) then the method is identical to the improved (1198661015840119866)-expansionmethod presented by Zhang et al [39] Again if weset 119896 = 0 119862 = 0 and the negative exponents of (1198661015840119866) arezero in (8) then the method turn out into the basic (1198661015840119866)-expansion method introduced by Wang et al [31] At theend if we put 119862 = 0 in (10) and 120572

119894(119894 = 1 2 3 119873) are

functions of 119909 and 119905 instead of constants then the method istransformed into the generalized (1198661015840119866)-expansion methoddeveloped by Zhang et al [38] Thus the methods presentedin [31 36 38 39] are only special cases of the novel (1198661015840119866)-expansion method So our method is more general havingmore free parameters than the existing methods

5 Conclusions

A novel (1198661015840119866)-expansion method is applied to fractionalpartial differential equation successfully As applicationsabundant new exact solutions for time fractional BBM-Burger equation have been successfully obtainedThenonlin-ear fractional complex transformation for 120585 is very importantwhich ensures that a certain fractional partial differentialequation can be converted into another ordinary differentialequation of integer order The obtained solutions are moregeneral with more free parameters Thus novel (1198661015840119866)-expansion method would be a powerful mathematical toolfor solving nonlinear evolution equations To the best ofour knowledge the solutions obtained in this article are notreported in literature previously

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Nikitin and T A Barannyk ldquoSolitary wave and other solu-tions for nonlinear heat equationsrdquo Central European Journal ofMathematics vol 2 no 5 pp 840ndash858 2004

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] J H He ldquoSome applications of nonlinear fractional differentialequations and their applicationsrdquo Bulletin of Science Technologyamp Society vol 15 no 2 pp 86ndash90 1999

[4] Z B Li and J H He ldquoApplication of the fractional complextransform to fractional differential equationsrdquo Nonlinear Sci-ence Letter A vol 2 no 3 pp 121ndash126 2011

[5] Z Li and J He ldquoFractional complex transform for fractionaldifferential equationsrdquoMathematical ampComputational Applica-tions vol 15 no 2 pp 970ndash973 2010

[6] J He S K Elagan and Z B Li ldquoGeometrical explanation ofthe fractional complex transform and derivative chain rule forfractional calculusrdquo Physics Letters A vol 376 no 4 pp 257ndash259 2012

[7] R W Ibrahim ldquoFractional complex transforms for fractionaldifferential equationsrdquo Advances in Difference Equations vol2012 article 192 2012

[8] H Jafari A KademD Baleanu and T Yilmaz ldquoSolutions of thefractional davey-stewartson equations with variational iterationmethodrdquo Romanian Reports in Physics vol 64 no 2 pp 337ndash346 2012

[9] H Jafari and S Seifi ldquoSolving a system of nonlinear fractionalpartial differential equations using homotopy analysis methodrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 5 pp 1962ndash1969 2009

[10] S Das R Kumar P K Gupta and H Jafari ldquoApproximateanalytical solutions for fractional space- and time-partial differ-ential equations using homotopy analysis methodrdquoApplicationsand Applied Mathematics vol 5 no 10 pp 1641ndash1659 2010

[11] H Jafari and C M Khalique ldquoAnalytical solutions of nonlin-ear fractional differential equations using variational iterationmethodrdquo Journal of Nonlinear Systems and Applications vol 2no 3-4 pp 148ndash151 2011

[12] Y Hao H M Srivastava and H Jafari ldquoHelmholtz anddiffusion equations associated with local fractional derivativeoperators involving the CANtorian and CANtor-type cylindri-cal coordinatesrdquo Advances in Mathematical Physics vol 2013Article ID 754248 5 pages 2013

[13] X J Yang ldquoLocal fractional integral transformsrdquo Progress inNonlinear Science vol 4 pp 1ndash225 2011

[14] X J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[15] A H Salas and C A Gomez S ldquoApplication of the Cole-Hopftransformation for finding exact solutions to several forms ofthe seventh-order KdV equationrdquo Mathematical Problems inEngineering vol 2010 Article ID 194329 14 pages 2010

[16] W Malfliet ldquoThe tanh method a tool for solving certainclasses of nonlinear evolution and wave equationsrdquo Journal ofComputational and Applied Mathematics vol 164-165 pp 529ndash541 2004

[17] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[18] A M Wazwaz ldquoThe extended tanh method for new compactand noncompact solutions for the KP-BBM and the ZK-BBMequationsrdquo Chaos Solitons and Fractals vol 38 no 5 pp 1505ndash1516 2008

[19] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 15

[20] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering vol 149 of London Mathemat-ical Society Lecture Note Series Cambridge University PressCambridge UK 1991

[21] H Jafari andH Tajadodi ldquoHes variational iterationmethod forsolving fractional Riccati differential equationrdquo InternationalJournal ofDifferential Equations vol 2010Article ID ID7647388 pages 2010

[22] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University-Sciencevol 23 no 4 pp 413ndash417 2011

[23] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp700ndash708 2006

[24] H Naher F A Abdullah and M Ali Akbar ldquoNew travelingwave solutions of the higher dimensional nonlinear partialdifferential equation by the exp-function methodrdquo Journal ofAppliedMathematics vol 2012Article ID 575387 14 pages 2012

[25] S T Mohyud-Din ldquoSolutions of nonlinear differential equa-tions by Exp-function methodrdquoWorld Applied Sciences Journalvol 7 pp 116ndash147 2009

[26] S T Mohyud-Din M A Noor and A Waheed ldquoExp-functionmethod for generalized travelling solutions of calogero-degasperis-fokas equationrdquo Zeitschrift fur Naturforschung AJournal of Physical Sciences vol 65 no 1 pp 78ndash84 2010

[27] X Zhao LWang andW Sun ldquoThe repeated homogeneous bal-ance method and its application to nonlinear partial differentialequationsrdquo Chaos Solitons and Fractals vol 28 no 2 pp 448ndash453 2006

[28] F Zhaosheng ldquoComment on lsquoOn the extended applicationsof homogeneous balance methodrsquordquo Applied Mathematics andComputation vol 158 no 2 pp 593ndash596 2004

[29] M Wang and X Li ldquoApplications of 119865-expansion to periodicwave solutions for a new Hamiltonian amplitude equationrdquoChaos Solitons and Fractals vol 24 no 5 pp 1257ndash1268 2005

[30] M A Abdou ldquoThe extended 119865-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons amp Fractals vol 31 no 1 pp 95ndash104 2007

[31] MWang X Li and J Zhang ldquoThe (119866Ecircź119866)-expansion method

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[32] M A Akbar N HM Ali and S T Mohyud-Din ldquoThe alterna-tive1198661015840G-expansionmethod with generalized Riccati equationapplication to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equationrdquo International Journal of Physical Sciences vol7 no 5 pp 743ndash752 2012

[33] H Jafari N Kadkhoda and E Salehpour ldquoApplication of(G1015840G)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[34] E J Parkes ldquoObservations on the basic (1198661015840119866)-expansionmethod for finding solutions to nonlinear evolution equationsrdquoApplied Mathematics and Computation vol 217 no 4 pp 1759ndash1763 2010

[35] M Ali Akbar and N H M Ali ldquoThe modified alternative(G1015840G)-expansion method for finding the exact solutions ofnonlinear PDEs in mathematical physicsrdquo International Journalof Physical Sciences vol 6 no 35 pp 7910ndash7920 2011

[36] M A Akbar N H M Ali and E M E Zayed ldquoA gener-alized and improved (GG)-expansion method for nonlinearevolution equationsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 459879 22 pages 2012

[37] E Salehpour H Jafari and N Kadkhoda ldquoApplication of(GrsquoG)-expansion method to nonlinear Lienard equationrdquoIndian Journal of Science and Technology vol 5 no 4 pp 2554ndash2556 2012

[38] J Zhang X Wei and Y Lu ldquoA generalized 119866

1015840

119866-expansionmethod and its applicationsrdquo Physics Letters A vol 372 no 20pp 3653ndash3658 2008

[39] J Zhang F Jiang andX Zhao ldquoAn improved (1198661015840119866)-expansionmethod for solving nonlinear evolution equationsrdquo Interna-tional Journal of Computer Mathematics vol 87 no 8 pp 1716ndash1725 2010

[40] E M E Zayed ldquoNew traveling wave solutions for higherdimensional nonlinear evolution equations using a generalized(GG)-expansion methodrdquo Journal of Physics A Mathematicaland Theoretical vol 42 no 19 Article ID 195202 2009

[41] E M E Zayed ldquoThe G1015840G-expansion method combined withthe Riccati equation for finding exact solutions of nonlinearPDEsrdquo Journal of Applied Mathematics Informatics vol 29 no1-2 pp 351ndash367 2011

[42] WMa and J Lee ldquoA transformed rational functionmethod andexact solutions to the 3+1 dimensional Jimbo-Miwa equationrdquoChaos Solitons amp Fractals vol 42 no 3 pp 1356ndash1363 2009

[43] M N Alam M A Akbar and S T Mohyud-Din ldquoA novel(1198661015840G)-expansionmethod and its application to the Boussinesqequationrdquo Chinese Physics B vol 23 no 2 Article ID 0202032013

[44] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

[45] S Zhu ldquoThe generalizing Riccati equation mapping methodin non-linear evolution equation application to (2 + 1)-dimensional Boiti-Leon-Pempinelle equationrdquo Chaos Solitonsand Fractals vol 37 no 5 pp 1335ndash1342 2008

[46] S Kumar andD Kumar ldquoFractionalmodelling for BBM-Burgerequation by using new homotopy analysis transform methodrdquoJournal of the Association of Arab Universities for Basic andApplied Sciences 2013

[47] L Song and H Zhang ldquoSolving the fractional BBM-Burgersequation using the homotopy analysis methodrdquo Chaos Solitonsand Fractals vol 40 no 4 pp 1616ndash1622 2009

[48] A Fakhari G Domairry and E Brahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of