Research ArticleA Novel Analytical Technique to Obtain Kink Solutions forHigher Order Nonlinear Fractional Evolution Equations

Qazi Mahmood Ul Hassan Jamshad Ahmad and Muhammad Shakeel

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

Correspondence should be addressed to Qazi Mahmood Ul Hassan qazimahmoodyahoocom

Received 9 February 2014 Revised 15 April 2014 Accepted 16 April 2014 Published 1 July 2014

Academic Editor Xiao-Jun Yang

Copyright copy 2014 Qazi Mahmood Ul Hassan et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We use the fractional derivatives in Caputorsquos sense to construct exact solutions to fractional fifth order nonlinear evolutionequations A generalized fractional complex transform is appropriately used to convert this equation to ordinary differentialequation which subsequently resulted in a number of exact solutions

1 Introduction

The concept of differentiation and integration to nonintegerorder is not new in any caseThe notion of fractional calculusemerged when the ideas of classical calculus were proposedby Leibniz who mentioned it in a letter to LrsquoHospital in 1695The foundation of the earliest more or less systematic studiescan be traced back to the beginning and middle of the 19thcentury by Liouville in 1832 Riemann in 1853 and Holmgrenin 1864 although Euler in 1730 Lagrange in 1772 and othersalso made contributions Recently it has turned out thosedifferential equations involving derivatives of noninteger [12] For example the nonlinear oscillation of earthquakescan be modeled with fractional derivatives [3] There hasbeen some attempt to solve linear problems with multiplefractional derivatives [3 4] Notmuchwork has been done onnonlinear problems and only a few numerical schemes havebeen proposed for solving nonlinear fractional differentialequations [5 6] More recently applications have includedclasses of nonlinear equation with multiorder fractionalderivatives The generalized fractional complex transformwas applied in [7ndash13] to convert fractional order differentialequation to ordinary differential equation Finally by usingExp-function method [14ndash25] we obtain generalized solitarysolutions and periodic solutions Recently the theory of local

fractional integrals and derivatives [26ndash28] is one of usefultools to handle the fractal and continuously nondifferentiablefunctions It is to be tinted that that 119888 = 119889 and 119901 = 119902 are theonly relations that can be obtained by applying Exp-functionmethod [29] to any nonlinear ordinary differential equationMost scientific problems and phenomena in different fieldsof sciences and engineering occur nonlinearly Except in alimited number of these problems are linear this method hasbeen effectively and accurately shown to solve a large class ofnonlinear problems The solution procedure of this methodwith the aid of Maple is of utter simplicity and this methodcan easily extend to other kinds of nonlinear evolutionequations In engineering and science scientific phenomenagive a variety of solutions that are characterized by distinctfeatures Traveling waves appear in many distinct physicalstructures in solitary wave theory [30] such as solitons kinkspeakons cuspons compactons and many others Solitonsare localized traveling waves which are asymptotically zeroat large distances In other words solitons are localizedwave packets with exponential wings or tails Solitons aregenerated from robust balance between nonlinearity anddispersion Solitons exhibit properties typically associatedwith particles Kink waves [30 31] are solitons that rise ordescend from one asymptotic state to another and henceanother type of traveling waves as in the case of the Burgers

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 213482 11 pageshttpdxdoiorg1011552014213482

2 Abstract and Applied Analysis

hierarchy Peakons that are peaked solitary wave solutionsare another type of travelling waves as in the case of Camassa-Holm equation For peakons the traveling wave solutions aresmooth except for a peak at a corner of its crest Peakons arethe points at which spatial derivative changes sign so thatpeakons have a finite jump in 1st derivative of the solutionCuspons are other forms of solitons where solution exhibitscusps at their crests Unlike peakons where the derivatives atthe peak differ only by a sign the derivatives at the jump ofa cuspon diverge The compactons are solitons with compactspatial support such that each compacton is a soliton confinedto a finite core or a soliton without exponential tails orwings Other types of travelling waves arise in science such asnegatons positons and complexitons In this research we usethe Exp-function method along with generalized fractionalcomplex transform to obtain new Kink wavesrsquo solutions for[30ndash32]

2 Preliminaries and Notation [1 2]

In this section we give some basic definitions and propertiesof the fractional calculus theory [1 2] which will be usedfurther in this work For the finite derivative in [119886 119887] wedefine the following fractional integral and derivatives

Definition 1 A real function 119891(119909) 119909 gt 0 is said to be in thespace 119862120583 120583 isin 119877 if there exists a real number (119901 gt 120583) suchthat 119891(119909) = 119909

1199011198911(119909) where 1198911(119909) = 119862(0infin) and it is said to

be in the space 119862119898

120583120583 if 119891119898 isin 119862120583 119898 isin 119873

Definition 2 TheRiemann-Liouville fractional integral oper-ator of order 120572 ge 0 of a function 119891 isin 119862120583 120583 ge minus1 is definedas

119869120572(119909) =

1

Γ (120572)

int

119909

0

(119909 minus 119905)120572minus1

119891 (119905) 119889119905

120572 gt 0 119909 gt 0 1198690(119909) = 119891 (119909)

(1)

Properties of the operator 119869119886 can be found in [1] we mentiononly the following

For 119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 and 120574 ge minus1

119869120572119869120573119891 (119909) = 119869

120572+119861119891 (119909)

119869120572119869119861119891 (119909) = 119869

119861119869120572119891 (119909)

119869120572119909120574=

Γ (120574 + 1)

Γ (120572 + 120574 + 1)

119909120572+120574

(2)

The Riemann-Liouville derivative has certain disadvantageswhen trying to model real-world phenomena with fractionaldifferential equationsTherefore wewill introduce amodifiedfractional differential operator proposed by M Caputo in hiswork on the theory of viscoelasticity [2]

Definition 3 For 119898 to be the smallest integer that exceeds 120572the Caputo time fractional derivative operator of order 120572 gt 0

defined as

119862

119886119863

120572

119905119891 (119905) = 119869

119898minus120572119863119898119891 (119905)

=

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1

119891119898

(119905) 119889119905

(3)

for 119898 minus 1 lt 120572 le 1119898 119898 isin 119873 119905 gt 0 119891 isin 119862119898

minus1

3 Chain Rule for Fractional Calculus andFractional Complex Transform

In [7] the authors used the following chain rule 120597120572119906120597119905120572

=

(120597119906120597119904)(120597120572119904120597119905120572) to convert a fractional differential equation

with Jumariersquos modification of Riemann-Liouville derivativeinto its classical differential partner In [10] the authorsshowed that this chain rule is invalid and showed followingrelation [8]

119863119886

119905119906 = 120590

1015840

119905

119889119906

119889120578

119863119886

119905120578 119863

119886

119909119906 = 120590

1015840

119909

119889119906

119889120578

119863119886

119909120578 (4)

To determine 120590119904 consider a special case as follows

119904 = 119905120572 119906 = 119904

119898 (5)

and we have

120597120572119906

120597119905120572

=

Γ (1 + 119898120572) 119905119898120572minus120572

Γ (1 + 119898120572 minus 120572)

= 120590 sdot

120597119906

120597119904

= 120590119898119905119898120572minus120572

(6)

Thus one can calculate 120590119904 as

120590119904 =Γ (1 + 119898120572)

Γ (1 + 119898120572 minus 120572)

(7)

Other fractional indexes (1205901015840

119909 1205901015840

119910 1205901015840

119911) can determine in sim-

ilar way Li and He [3 7ndash9] proposed the following frac-tional complex transform for converting fractional differen-tial equations into ordinary differential equations so thatall analytical methods for advanced calculus can be easilyapplied to fractional calculus

119906 (119909 119905) = 119906 (120578) 120578 =

119896119909120573

Γ (1 + 120573)

+

120596119905120572

Γ (1 + 120572)

+

119872119909120574

Γ (1 + 120574)

(8)

where 119896 120596 and 119872 are constants

4 Exp-Function Method [33ndash36]Consider the general nonlinear partial differential equationof fractional order

119875 (119906 119906119905 119906119909 119906119909119909 119906119909119909119909 119863120572

119905119906119863120572

119909119906119863120572

119909119909119906 ) = 0

0 lt 120572 le 1

(9)

Abstract and Applied Analysis 3

where 119863120572

119905119906 119863120572119909119906 and 119863

120572

119909119909119906 are the fractional derivative of 119906

with respect to 119905 119909 119909119909 respectivelyUse

119906 (119909 119905) = 119906 (120578) 120578 =

119896119909120573

Γ (1 + 120573)

+

120596119905120572

Γ (1 + 120572)

+

119872119909120574

Γ (1 + 120574)

(10)

where 119896 120596 and 119872 are constantsThen (9) becomes

119876(119906 1199061015840 11990610158401015840 119906101584010158401015840

119906119894V) = 0 (11)

where the prime denotes derivative with respect to 120578 Inaccordance with Exp-function method we assume that thewave solution can be expressed in the following form

119906 (120578) =

sum119889

119899minus119888119886119899 exp [119899120578]

sum119902

119898minus119901119887119898 exp [119898120578]

(12)

where 119901 119902 119888 and 119889 are positive integers which are known tobe further determined and 119886119899 and 119887119898 are unknown constantsEquation (8) can be rewritten as

119906 (120578) =

119886119888 exp (119888120578) + sdot sdot sdot + 119886minus119889 exp (minus119889120578)

119887119901 exp (119901120578) + sdot sdot sdot + 119887minus119902 exp (minus119902120578)

(13)

This equivalent formulation plays an important and funda-mental for finding the analytic solution of problems 119888 and 119901

can be determined by [29]

5 Solution Procedure

Consider the following new fifth order nonlinear (2 + 1)-dimensional evolution equations of fractional order

1198633120572

119905119906 minus (119863

120572

119905119906)119909119909119909119909

minus (119863120572

119905119906)119909119909

minus 4(119906119909(119863120572

119905119906)119909)119909= 0 (14)

Using (8) in (14) then it can be converted to an ordinarydifferential equation Consider

minus1205963 + 120596119896

4119906(V)

+ 1198962120596 + 4120596119896

3 = 0 (15)

where the prime denotes the derivative with respect to 120578 Thesolution of (15) can be expressed in form (13) To determinethe value of 119888 and 119901 by using [26]

119901 = 119888 119902 = 119889 (16)

Case 1 We can freely choose the values of 119888 and 119889 but wewill illustrate that the final solution does not strongly dependupon the choice of values of 119888 and 119889 For simplicity we set119901 = 119888 = 1 and 119902 = 119889 = 1 (15) reduces to

119906 (120578) =

1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198871 exp [120578] + 1198860 + 119887minus1 exp [minus120578]

(17)

Substituting (17) into (15) we have

1

119860

[11988841198861 exp [4120578] = 1198883 exp [3120578] + 1198882 exp [2120578] + 1198881 exp [120578]

+ 1198880 + 119888minus1 exp [minus120578] + 119888minus2 exp [minus2120578]

+119888minus3 exp [minus3120578] + 119888minus4 exp [minus4120578]] = 0

(18)

where 119860 = (1198871 exp(120578) + 1198870 + 119887minus1 exp(minus120578))4 and 119888119894 are con-

stants obtained by Maple software 16 Equating the coeffi-cients of exp(119899120578) to be zero we obtain

119888minus4 = 0 119888minus3 = 0 119888minus2 = 0 119888minus1 = 0

1198880 = 0 1198881 = 0 1198882 = 0 1198883 = 0 1198884 = 0

(19)

For solution of (19) we have five solution sets satisfying thegiven (15)

1st Solution Set Consider

120596 = minusradic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(20)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 1)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(21)

2nd Solution Set Consider

120596 = radic41198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 1198860 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(22)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 2)

119906 (119909 119905) = (119887minus1 (61198961198871 + 1198861) 119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

1198871

+1198861119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

+1198871119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

minus1

(23)

4 Abstract and Applied Analysis

55

5

45

35

4

3

minus10 minus5 0 5 10

x

120572 = 025 t = 1

t

0

04

5

45

4

35

3

minus10 minus5 0 5 10

x

120572 = 050 t = 1

minus10 minus5 0 5 10

x

120572 = 075 t = 1

minus10 minus5 0 5 10

x

120572 = 1 t = 1 120572 = 1

55

5

4

3

55

5

4

3

55

5

4

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

45

35

45

35

45

35

120572 = 025

120572 = 050

120572 = 075

Figure 1 Kink wavesrsquo solutions of (14) for 1st solution set

Abstract and Applied Analysis 5

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

minus5

minus5

minus5

minus5

Figure 2 Kink wavesrsquo solutions of (14) for 2nd solution set

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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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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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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 Abstract and Applied Analysis

hierarchy Peakons that are peaked solitary wave solutionsare another type of travelling waves as in the case of Camassa-Holm equation For peakons the traveling wave solutions aresmooth except for a peak at a corner of its crest Peakons arethe points at which spatial derivative changes sign so thatpeakons have a finite jump in 1st derivative of the solutionCuspons are other forms of solitons where solution exhibitscusps at their crests Unlike peakons where the derivatives atthe peak differ only by a sign the derivatives at the jump ofa cuspon diverge The compactons are solitons with compactspatial support such that each compacton is a soliton confinedto a finite core or a soliton without exponential tails orwings Other types of travelling waves arise in science such asnegatons positons and complexitons In this research we usethe Exp-function method along with generalized fractionalcomplex transform to obtain new Kink wavesrsquo solutions for[30ndash32]

2 Preliminaries and Notation [1 2]

In this section we give some basic definitions and propertiesof the fractional calculus theory [1 2] which will be usedfurther in this work For the finite derivative in [119886 119887] wedefine the following fractional integral and derivatives

Definition 1 A real function 119891(119909) 119909 gt 0 is said to be in thespace 119862120583 120583 isin 119877 if there exists a real number (119901 gt 120583) suchthat 119891(119909) = 119909

1199011198911(119909) where 1198911(119909) = 119862(0infin) and it is said to

be in the space 119862119898

120583120583 if 119891119898 isin 119862120583 119898 isin 119873

Definition 2 TheRiemann-Liouville fractional integral oper-ator of order 120572 ge 0 of a function 119891 isin 119862120583 120583 ge minus1 is definedas

119869120572(119909) =

1

Γ (120572)

int

119909

0

(119909 minus 119905)120572minus1

119891 (119905) 119889119905

120572 gt 0 119909 gt 0 1198690(119909) = 119891 (119909)

(1)

Properties of the operator 119869119886 can be found in [1] we mentiononly the following

For 119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 and 120574 ge minus1

119869120572119869120573119891 (119909) = 119869

120572+119861119891 (119909)

119869120572119869119861119891 (119909) = 119869

119861119869120572119891 (119909)

119869120572119909120574=

Γ (120574 + 1)

Γ (120572 + 120574 + 1)

119909120572+120574

(2)

The Riemann-Liouville derivative has certain disadvantageswhen trying to model real-world phenomena with fractionaldifferential equationsTherefore wewill introduce amodifiedfractional differential operator proposed by M Caputo in hiswork on the theory of viscoelasticity [2]

Definition 3 For 119898 to be the smallest integer that exceeds 120572the Caputo time fractional derivative operator of order 120572 gt 0

defined as

119862

119886119863

120572

119905119891 (119905) = 119869

119898minus120572119863119898119891 (119905)

=

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1

119891119898

(119905) 119889119905

(3)

for 119898 minus 1 lt 120572 le 1119898 119898 isin 119873 119905 gt 0 119891 isin 119862119898

minus1

3 Chain Rule for Fractional Calculus andFractional Complex Transform

In [7] the authors used the following chain rule 120597120572119906120597119905120572

=

(120597119906120597119904)(120597120572119904120597119905120572) to convert a fractional differential equation

with Jumariersquos modification of Riemann-Liouville derivativeinto its classical differential partner In [10] the authorsshowed that this chain rule is invalid and showed followingrelation [8]

119863119886

119905119906 = 120590

1015840

119905

119889119906

119889120578

119863119886

119905120578 119863

119886

119909119906 = 120590

1015840

119909

119889119906

119889120578

119863119886

119909120578 (4)

To determine 120590119904 consider a special case as follows

119904 = 119905120572 119906 = 119904

119898 (5)

and we have

120597120572119906

120597119905120572

=

Γ (1 + 119898120572) 119905119898120572minus120572

Γ (1 + 119898120572 minus 120572)

= 120590 sdot

120597119906

120597119904

= 120590119898119905119898120572minus120572

(6)

Thus one can calculate 120590119904 as

120590119904 =Γ (1 + 119898120572)

Γ (1 + 119898120572 minus 120572)

(7)

Other fractional indexes (1205901015840

119909 1205901015840

119910 1205901015840

119911) can determine in sim-

ilar way Li and He [3 7ndash9] proposed the following frac-tional complex transform for converting fractional differen-tial equations into ordinary differential equations so thatall analytical methods for advanced calculus can be easilyapplied to fractional calculus

119906 (119909 119905) = 119906 (120578) 120578 =

119896119909120573

Γ (1 + 120573)

+

120596119905120572

Γ (1 + 120572)

+

119872119909120574

Γ (1 + 120574)

(8)

where 119896 120596 and 119872 are constants

4 Exp-Function Method [33ndash36]Consider the general nonlinear partial differential equationof fractional order

119875 (119906 119906119905 119906119909 119906119909119909 119906119909119909119909 119863120572

119905119906119863120572

119909119906119863120572

119909119909119906 ) = 0

0 lt 120572 le 1

(9)

Abstract and Applied Analysis 3

where 119863120572

119905119906 119863120572119909119906 and 119863

120572

119909119909119906 are the fractional derivative of 119906

with respect to 119905 119909 119909119909 respectivelyUse

119906 (119909 119905) = 119906 (120578) 120578 =

119896119909120573

Γ (1 + 120573)

+

120596119905120572

Γ (1 + 120572)

+

119872119909120574

Γ (1 + 120574)

(10)

where 119896 120596 and 119872 are constantsThen (9) becomes

119876(119906 1199061015840 11990610158401015840 119906101584010158401015840

119906119894V) = 0 (11)

where the prime denotes derivative with respect to 120578 Inaccordance with Exp-function method we assume that thewave solution can be expressed in the following form

119906 (120578) =

sum119889

119899minus119888119886119899 exp [119899120578]

sum119902

119898minus119901119887119898 exp [119898120578]

(12)

where 119901 119902 119888 and 119889 are positive integers which are known tobe further determined and 119886119899 and 119887119898 are unknown constantsEquation (8) can be rewritten as

119906 (120578) =

119886119888 exp (119888120578) + sdot sdot sdot + 119886minus119889 exp (minus119889120578)

119887119901 exp (119901120578) + sdot sdot sdot + 119887minus119902 exp (minus119902120578)

(13)

This equivalent formulation plays an important and funda-mental for finding the analytic solution of problems 119888 and 119901

can be determined by [29]

5 Solution Procedure

Consider the following new fifth order nonlinear (2 + 1)-dimensional evolution equations of fractional order

1198633120572

119905119906 minus (119863

120572

119905119906)119909119909119909119909

minus (119863120572

119905119906)119909119909

minus 4(119906119909(119863120572

119905119906)119909)119909= 0 (14)

Using (8) in (14) then it can be converted to an ordinarydifferential equation Consider

minus1205963 + 120596119896

4119906(V)

+ 1198962120596 + 4120596119896

3 = 0 (15)

where the prime denotes the derivative with respect to 120578 Thesolution of (15) can be expressed in form (13) To determinethe value of 119888 and 119901 by using [26]

119901 = 119888 119902 = 119889 (16)

Case 1 We can freely choose the values of 119888 and 119889 but wewill illustrate that the final solution does not strongly dependupon the choice of values of 119888 and 119889 For simplicity we set119901 = 119888 = 1 and 119902 = 119889 = 1 (15) reduces to

119906 (120578) =

1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198871 exp [120578] + 1198860 + 119887minus1 exp [minus120578]

(17)

Substituting (17) into (15) we have

1

119860

[11988841198861 exp [4120578] = 1198883 exp [3120578] + 1198882 exp [2120578] + 1198881 exp [120578]

+ 1198880 + 119888minus1 exp [minus120578] + 119888minus2 exp [minus2120578]

+119888minus3 exp [minus3120578] + 119888minus4 exp [minus4120578]] = 0

(18)

where 119860 = (1198871 exp(120578) + 1198870 + 119887minus1 exp(minus120578))4 and 119888119894 are con-

stants obtained by Maple software 16 Equating the coeffi-cients of exp(119899120578) to be zero we obtain

119888minus4 = 0 119888minus3 = 0 119888minus2 = 0 119888minus1 = 0

1198880 = 0 1198881 = 0 1198882 = 0 1198883 = 0 1198884 = 0

(19)

For solution of (19) we have five solution sets satisfying thegiven (15)

1st Solution Set Consider

120596 = minusradic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(20)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 1)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(21)

2nd Solution Set Consider

120596 = radic41198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 1198860 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(22)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 2)

119906 (119909 119905) = (119887minus1 (61198961198871 + 1198861) 119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

1198871

+1198861119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

+1198871119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

minus1

(23)

4 Abstract and Applied Analysis

55

5

45

35

4

3

minus10 minus5 0 5 10

x

120572 = 025 t = 1

t

0

04

5

45

4

35

3

minus10 minus5 0 5 10

x

120572 = 050 t = 1

minus10 minus5 0 5 10

x

120572 = 075 t = 1

minus10 minus5 0 5 10

x

120572 = 1 t = 1 120572 = 1

55

5

4

3

55

5

4

3

55

5

4

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

45

35

45

35

45

35

120572 = 025

120572 = 050

120572 = 075

Figure 1 Kink wavesrsquo solutions of (14) for 1st solution set

Abstract and Applied Analysis 5

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

minus5

minus5

minus5

minus5

Figure 2 Kink wavesrsquo solutions of (14) for 2nd solution set

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

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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

Abstract and Applied Analysis 3

where 119863120572

119905119906 119863120572119909119906 and 119863

120572

119909119909119906 are the fractional derivative of 119906

with respect to 119905 119909 119909119909 respectivelyUse

119906 (119909 119905) = 119906 (120578) 120578 =

119896119909120573

Γ (1 + 120573)

+

120596119905120572

Γ (1 + 120572)

+

119872119909120574

Γ (1 + 120574)

(10)

where 119896 120596 and 119872 are constantsThen (9) becomes

119876(119906 1199061015840 11990610158401015840 119906101584010158401015840

119906119894V) = 0 (11)

where the prime denotes derivative with respect to 120578 Inaccordance with Exp-function method we assume that thewave solution can be expressed in the following form

119906 (120578) =

sum119889

119899minus119888119886119899 exp [119899120578]

sum119902

119898minus119901119887119898 exp [119898120578]

(12)

where 119901 119902 119888 and 119889 are positive integers which are known tobe further determined and 119886119899 and 119887119898 are unknown constantsEquation (8) can be rewritten as

119906 (120578) =

119886119888 exp (119888120578) + sdot sdot sdot + 119886minus119889 exp (minus119889120578)

119887119901 exp (119901120578) + sdot sdot sdot + 119887minus119902 exp (minus119902120578)

(13)

This equivalent formulation plays an important and funda-mental for finding the analytic solution of problems 119888 and 119901

can be determined by [29]

5 Solution Procedure

Consider the following new fifth order nonlinear (2 + 1)-dimensional evolution equations of fractional order

1198633120572

119905119906 minus (119863

120572

119905119906)119909119909119909119909

minus (119863120572

119905119906)119909119909

minus 4(119906119909(119863120572

119905119906)119909)119909= 0 (14)

Using (8) in (14) then it can be converted to an ordinarydifferential equation Consider

minus1205963 + 120596119896

4119906(V)

+ 1198962120596 + 4120596119896

3 = 0 (15)

where the prime denotes the derivative with respect to 120578 Thesolution of (15) can be expressed in form (13) To determinethe value of 119888 and 119901 by using [26]

119901 = 119888 119902 = 119889 (16)

Case 1 We can freely choose the values of 119888 and 119889 but wewill illustrate that the final solution does not strongly dependupon the choice of values of 119888 and 119889 For simplicity we set119901 = 119888 = 1 and 119902 = 119889 = 1 (15) reduces to

119906 (120578) =

1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198871 exp [120578] + 1198860 + 119887minus1 exp [minus120578]

(17)

Substituting (17) into (15) we have

1

119860

[11988841198861 exp [4120578] = 1198883 exp [3120578] + 1198882 exp [2120578] + 1198881 exp [120578]

+ 1198880 + 119888minus1 exp [minus120578] + 119888minus2 exp [minus2120578]

+119888minus3 exp [minus3120578] + 119888minus4 exp [minus4120578]] = 0

(18)

where 119860 = (1198871 exp(120578) + 1198870 + 119887minus1 exp(minus120578))4 and 119888119894 are con-

stants obtained by Maple software 16 Equating the coeffi-cients of exp(119899120578) to be zero we obtain

119888minus4 = 0 119888minus3 = 0 119888minus2 = 0 119888minus1 = 0

1198880 = 0 1198881 = 0 1198882 = 0 1198883 = 0 1198884 = 0

(19)

For solution of (19) we have five solution sets satisfying thegiven (15)

1st Solution Set Consider

120596 = minusradic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(20)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 1)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(21)

2nd Solution Set Consider

120596 = radic41198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 1198860 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(22)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 2)

119906 (119909 119905) = (119887minus1 (61198961198871 + 1198861) 119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

1198871

+1198861119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590radic41198962+1119896119905

120572Γ(1+120572))

+1198871119890119896119909minus(120590radic41198962+1119896119905

120572Γ(1+120572))

)

minus1

(23)

4 Abstract and Applied Analysis

55

5

45

35

4

3

minus10 minus5 0 5 10

x

120572 = 025 t = 1

t

0

04

5

45

4

35

3

minus10 minus5 0 5 10

x

120572 = 050 t = 1

minus10 minus5 0 5 10

x

120572 = 075 t = 1

minus10 minus5 0 5 10

x

120572 = 1 t = 1 120572 = 1

55

5

4

3

55

5

4

3

55

5

4

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

45

35

45

35

45

35

120572 = 025

120572 = 050

120572 = 075

Figure 1 Kink wavesrsquo solutions of (14) for 1st solution set

Abstract and Applied Analysis 5

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

minus5

minus5

minus5

minus5

Figure 2 Kink wavesrsquo solutions of (14) for 2nd solution set

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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 Abstract and Applied Analysis

55

5

45

35

4

3

minus10 minus5 0 5 10

x

120572 = 025 t = 1

t

0

04

5

45

4

35

3

minus10 minus5 0 5 10

x

120572 = 050 t = 1

minus10 minus5 0 5 10

x

120572 = 075 t = 1

minus10 minus5 0 5 10

x

120572 = 1 t = 1 120572 = 1

55

5

4

3

55

5

4

3

55

5

4

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

t

0

04

5

45

4

35

3

4

x20

minus2minus4

45

35

45

35

45

35

120572 = 025

120572 = 050

120572 = 075

Figure 1 Kink wavesrsquo solutions of (14) for 1st solution set

Abstract and Applied Analysis 5

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

minus5

minus5

minus5

minus5

Figure 2 Kink wavesrsquo solutions of (14) for 2nd solution set

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

Abstract and Applied Analysis 5

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

7

6

5

4

3

2

minus10 0 5 10

x

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

4

x

t

0

04

20

minus2minus4

7

6

5

4

3

2

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

minus5

minus5

minus5

minus5

Figure 2 Kink wavesrsquo solutions of (14) for 2nd solution set

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

6 Abstract and Applied Analysis

3rd Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (31198961198870 + 1198860)

1198870

1198860 = 1198860 1198861 = 0 119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 0

(24)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 3)

119906 (119909 119905) =

119887minus1 (31198961198870 + 1198860) 119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

1198870 + 1198860

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198870

(25)

4th Solution Set Consider

120596 = radic1198962+ 1119896

119886minus1=1

9

1

119896211988741

(91198962119886011988701198873

1minus9119896211988611198872

01198872

1minus31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus61198961198862

11198872

01198871 minus 119886

2

011988611198872

1+ 21198860119886

2

111988701198871 minus 119886

3

11198872

0)

1198871 = 1198871 1198870 = 1198870

1198860 = 1198860 1198861 = 1198861

119887minus1 =1

9

3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1+ 21198860119886111988701198871 minus 119886

2

11198872

0

119896211988731

(26)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 4)

119906 (119909 119905) = (

1

9

1

119896211988741

( (91198962119886011988701198873

1minus 9119896211988611198872

01198872

1

minus 31198961198862

01198873

1+ 9119896119886011988611198870119887

2

1

minus 61198961198862

11198872

01198871 minus 119886

2

011988611198872

1

+211988601198862

111988701198871 minus 119886

3

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

)

+1198860 + 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

times (

1

9

( (3119896119886011988701198872

1minus 31198961198861119887

2

01198871 minus 119886

2

01198872

1

+21198860119886111988701198871 minus 1198862

11198872

0)

times119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

) (11989621198873

1)

minus1

+1198870 + 1198871119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

)

minus1

(27)

5th Solution Set Consider

120596 = radic1198962+ 1119896 119886minus1 =

119887minus1 (61198961198871 + 1198861)

1198871

1198860 = 0 1198861 = 1198861 119887minus1 = 119887minus1 1198870 = 0 1198871 = 1198871

(28)

We therefore obtained the following generalized solitarysolution 119906(119909 119905) of (14) (Figure 5)

119906 (119909 119905)=

119887minus1119890minus119896119909+(120590radic1198962+1119896119905

120572Γ(1+120572))

+ 1198861119890119896119909minus(120590radic1198962+1119896119905

120572Γ(1+120572))

119887minus1119890minus119896119909+(120590radic1198962+1119896119905120572Γ(1+120572))

+ 1198871119890119896119909minus(120590radic1198962+1119896119905120572Γ(1+120572))

(29)

Case 2 If 119901 = 119888 = 2 and 119902 = 119889 = 1 then trial solution (14)reduces to

119906 (120578) =

1198862 exp [2120578] + 1198861 exp [120578] + 1198860 + 119886minus1 exp [minus120578]

1198872 exp [2120578] + 1198871 exp [120578] + 1198870 + 119887minus1 exp [minus120578]

(30)

Proceeding as before we obtain the following

1st Solution Consider

119886minus1 = 119886minus1 1198860 =

119886minus11198870

119887minus1

1198861 =

119886minus11198871

119887minus1

1198862 =

119886minus11198872

119887minus1

119887minus1 = 119887minus1 1198870 = 1198870 1198871 = 1198871 1198872 = 1198872

(31)

Hence we get the generalized solitary wave solution 119906(119909 119905) of(14) (Figure 6)

119906 (119909 119905) = (119886minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198870

119887minus1

+

119886minus11198870

119887minus1

119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+

119886minus11198872

119887minus1

1198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

times (119887minus1119890minus119896119909+(120590120596119896119905

120572Γ(1+120572))

+ 1198870

+ 1198871119890119896119909minus(120590120596119896119905

120572Γ(1+120572))

+11988721198902119896119909minus2(120590120596119896119905

120572Γ(1+120572))

)

minus1

(32)

In both cases for different choices of 119888 119901 119889 and 119902 we getthe same soliton solutions which clearly illustrate that finalsolution does not strongly depend on these parameters

6 Conclusions

Exp-function method is applied to construct solitary solu-tions of the nonlinear new fifth order evolution equationsof fractional orders The reliability of proposed algorithm isfully supported by the computational work the subsequentresults and graphical representations It is observed that Exp-functionmethod is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems offractional orders

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

Abstract and Applied Analysis 7

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

55

5

45

35

4

3

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

t

0

04

4

35

45

3

25

2

4

x20

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 3 Kink wavesrsquo solutions of (14) for 3rd solution set

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

8 Abstract and Applied Analysis

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

45

4

35

25

3

2

minus10 minus5 0 5 10

x

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

t

0

04

4

35

45

3

25

2

4

x2

0

minus2minus4

120572 = 025 t = 1 120572 = 025

120572 = 050 t = 1 120572 = 050

120572 = 075 t = 1 120572 = 075

120572 = 1 t = 1 120572 = 1

Figure 4 Kink wavesrsquo solutions of (14) for 4th solution set

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

Abstract and Applied Analysis 9

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

6

5

4

3

2

1

minus10 minus5 0 5 10

x

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

4

x

t

0

04

20

minus2minus4

6

5

4

3

2

1

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 5 Kink wavesrsquo solutions of (14) for 5th solution set

10 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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 Abstract and Applied Analysis

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus5 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

042857142850

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5

minus10 0 5 10

x

042857142857

042857142856

042857142855

042857142854

042857142853

042857142852

042857142851

minus5 4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

4

x

t

0

04

20

minus2minus4

02857

02856

02855

02854

02853

02852

02851

120572 = 025 t = 1

120572 = 050 t = 1

120572 = 075 t = 1

120572 = 1 t = 1 120572 = 1

120572 = 025

120572 = 050

120572 = 075

Figure 6 Kink wavesrsquo solutions of (14) for 1st solution set of Case 2

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

Abstract and Applied Analysis 11

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 vol 198 Aca-demic Press San 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] K Diethelm and Y Luchko ldquoNumerical solution of linearmulti-term initial value problems of fractional orderrdquo Journal ofComputational Analysis and Applications vol 6 no 3 pp 243ndash263 2004

[5] A Atangana and N Bildik ldquoExistence and numerical solutionof the Volterra fractional integral equations of the second kindrdquoMathematical Problems in Engineering vol 2013 Article ID981526 11 pages 2013

[6] A Atangana and D Baleanu ldquoNumerical solution of a kindof fractional parabolic equations via two difference schemesrdquoAbstract and Applied Analysis vol 2013 Article ID 828764 8pages 2013

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

[8] Z-B Li and J-H He ldquoFractional complex transform for frac-tional differential equationsrdquo Mathematical amp ComputationalApplications vol 15 no 5 pp 970ndash973 2010

[9] J-H 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

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

[11] J-H He ldquoAn elementary introduction to recently developedasymptotic methods and nanomechanics in textile engineer-ingrdquo International Journal of Modern Physics B vol 22 no 21pp 3487ndash3578 2008

[12] Z B Li and W H Zhu ldquoexact solutions of time-fractionalheat conduction equation by the fractional complex transformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[13] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[14] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[15] S T Mohyud-Din M A Noor and K I Noor ldquoSome relativelynew techniques for nonlinear problemsrdquo Mathematical Prob-lems in Engineering vol 2009Article ID 234849 25 pages 2009

[16] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for generalized travelling solutions of master partialdifferential equationsrdquo Acta Applicandae Mathematicae vol104 no 2 pp 131ndash137 2008

[17] T Ozis and C Koroglu ldquoA novel approach for solving the Fisherequation using Exp-function methodrdquo Physics Letters A vol372 no 21 pp 3836ndash3840 2008

[18] X-H Wu and J-H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[19] X-H Wu and J-H He ldquoSolitary solutions periodic solutionsand compacton-like solutions using the Exp-function methodrdquoComputers ampMathematics with Applications vol 54 no 7-8 pp966ndash986 2007

[20] E Yusufoglu ldquoNew solitonary solutions for the MBBN equa-tions using exp-function methodrdquo Physics Letters A vol 372pp 442ndash446 2008

[21] S Zhang ldquoApplication of exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons ampFractals vol 365 pp 448ndash455 2007

[22] S D Zhu ldquoExp-functionmethod for theHybrid-Lattice systemInterrdquo International Journal of Nonlinear Sciences andNumericalSimulation vol 8 pp 461ndash464 2007

[23] S D Zhu ldquoExp-function method for the discrete m KdV lat-ticerdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 pp 465ndash468 2007

[24] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 465ndash470 1988

[25] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

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

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong 2011

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

[29] A Ebaid ldquoAn improvement on the Exp-function method whenbalancing the highest order linear and nonlinear termsrdquo Journalof Mathematical Analysis and Applications vol 392 no 1 pp 1ndash5 2012

[30] A M Wazwaz ldquoNew higher-dimensional fifth-order nonlinearequations with multiple soliton solutionsrdquo Physica Scripta vol84 Article ID 025007 2011

[31] W-X Ma A Abdeljabbar and M G Asaad ldquoWronskian andGrammian solutions to a (3 + 1)-dimensional generalized KPequationrdquo Applied Mathematics and Computation vol 217 no24 pp 10016ndash10023 2011

[32] A-M Wazwaz Partial Differential Equations and SolitaryWaves Theory SpringerHEP Berlin Germany 2009

[33] H Jafari N Kadkhoda and C M Khalique ldquoTravelling wavesolutions of nonlinear evolution equations using the simplestequationmethodrdquoComputersampMathematics with Applicationsvol 64 no 6 pp 2084ndash2088 2012

[34] M A Abdou A A Soliman and S T Basyony ldquoNew applica-tion of Exp-function method for improved Boussinesq equa-tionrdquo Physics Letters A vol 369 pp 469ndash475 2007

[35] J-H He and M A Abdou ldquoNew periodic solutions fornonlinear evolution equations using Exp-function methodrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1421ndash1429 2007

[36] M A Noor S T Mohyud-Din and A Waheed ldquoExp-functionmethod for solving Kuramoto-Sivashinsky and Boussinesqequationsrdquo Journal of Applied Mathematics and Computing vol29 no 1-2 pp 1ndash13 2009

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

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