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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
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Differential EquationsInternational Journal of
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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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
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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
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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