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Research ArticleOn the Fractional Nagumo Equation withNonlinear Diffusion and Convection
Abdon Atangana1 and Suares Clovis Oukouomi Noutchie2
1 Institute for Groundwater Studies Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa
2Ma SIM Focus Area North-West University Mafikeng 2735 South Africa
Correspondence should be addressed to Abdon Atangana abdonatanganayahoofr
Received 2 July 2014 Accepted 20 July 2014 Published 5 August 2014
Academic Editor Dumitru Baleanu
Copyright copy 2014 A Atangana and S C Oukouomi Noutchie This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited
We presented the Nagumo equation using the concept of fractional calculus With the help of two analytical techniques includingthe homotopy decomposition method (HDM) and the new development of variational iteration method (NDVIM) we derived anapproximate solution Both methods use a basic idea of integral transform and are very simple to be used
1 Introduction
TheNagumo equation with linear or nonlinear diffusion andconvection has broadly been useful to population dynamicsecology neurophysiology chemical reactions and flamepropagation [1ndash3] In particular the case where the equationinvolves degenerate nonlinear diffusion is of considerableinterest [4ndash8] In this case a travelling wave front solutionof sharp type is known to exist for exactly one value ofthe wave speed Such wave fronts for instance representcollective motion of populations in particular collective cellspreading invasion in ecology and concentration in chemicalreactions [9ndash15] However it has been showing that the realworld problems describe via fractional order derivative givesbetter prediction [16ndash20] It is therefore important furtherextend the nonlinear Nagumo equation using the concept offractional derivative order
It is something very difficult to obtain the exact solutionof nonlinear equation with fractional order derivative Manyscholars sometimes to avoid this difficulty solve this class ofproblem numerically However even with numerical schemeit is also difficult to provide a numerical solution for nonlinearequations Thus many scholars to access the behaviour ofthe solution of the real problem under study present anapproximate solution of this type of equations
In the literature there exist several analytical techniques[21ndash25] to deal with nonlinear equations including frac-tional type The purpose of this work is to present anapproximate solution for the generalized nonlinear Nagumoequation withnonlinear diffusion and convection via thewell-known variational iteration method (VIM) and thehomotopy decomposition method (HDM) The nonlinearfractional Nagumo equation considered here is given belowas
120597120572
119906
120597119905120572+ 120573119906119899
120597119906
120597119909=
120597
120597119909[120572119906119899
120597119906
120597119909]
+ 120574119906 (1 minus 119906119898
) (119906119898
minus 120575) 0 lt 120572 le 1
(1)
119906 (119909 0) = 119891 (119909) (2)
119906 (0 119905) = 119892 (119905) (3)
where 120572 120573 120574 and 120575 are constants and for the sake ofsimplicity in this paper we consider 119898 = 119899 = 1 = 120575 Theconcept of fractional order is not well by some scholars inorder to accommodate those we present in the next sectionthe basic information regarding this concept
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 963985 7 pageshttpdxdoiorg1011552014963985
2 Abstract and Applied Analysis
2 Basic Information about Fractional Calculus
There exists a vast literature on different definitions of frac-tional derivatives The most popular ones are the Riemann-Liouville and the Caputo derivatives For Caputo we have thefollowing
Definition 1 (see [26ndash30]) A real function119891(119909) 119909 gt 0 is saidto be in the space 119862
120583 120583 isin R if there exists a real number
119901 gt 120583 such that 119891(119909) = 119909119901
ℎ(119909) where ℎ(119909) isin 119862[0 infin) andit is said to be in space 119862
119898
120583if 119891(119898)
isin 119862120583 119898 isin N
Definition 2 (partial derivatives of fractional order [31ndash34])Assume now that 119891(119909) is a function of 119899 variables 119909
119894 119894 =
1 119899 also of class119862 on119863 isin R119899We define partial derivative
of order 120572 for 119891 with respect to 119909119894 the function
119886120597120572
119909119891 =
1
Γ (119898 minus 120572)int
119909119894
119886
(119909119894minus 119905)119898minus120572minus1
120597119898
119909119894
119891 (119909119895)10038161003816100381610038161003816119909119895=119905
119889119905 (4)
where 120597119898
119909119894
is the usual partial derivative of integer order 119898
Definition 3 (see [26ndash30]) The Riemann-Liouville fractionalintegral operator of order120572 ge 0 of a function119891 isin 119862
120583 120583 ge minus1
is defined as
119869120572
119891 (119909) =1
Γ (120572)int
119909
0
(119909 minus 119905)120572minus1
119891 (119905) 119889119905 120572 gt 0 119909 gt 0
1198690
119891 (119909) = 119891 (119909)
(5)
Properties of the operator can be found in [31ndash38] we men-tion only the following
For
119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 120574 gt minus1
119869120572
119869120573
119891 (119909) = 119869120572+120573
119891 (119909)
119869120572
119869120573
119891 (119909) = 119869120573
119869120572
119891 (119909)
119869120572
119909120574
=Γ (120574 + 1)
Γ (120572 + 120574 + 1)119909120572+120574
(6)
3 Basic Information of the HDM and VIM
In this section we shall present the basic information ofthe chosen analytical methods the homotopy decompositionmethod and variational iteration method we shall start withthe homotopy decomposition method
31 Information about the HDM The interested reader canfind the full detail of the methodology of the homo-topy decomposition method in [39ndash42] This relatively newmethod was recently used to solve some nonlinear frac-tional partial differential equations However since the newdevelopment of the variational iteration method using theLaplace transform was recently introduced we shall presentits methodology in the following subsection
32 Some Information about the Variational IterationMethodIn its initial development the essential nature of the methodwas to construct the following correction functional for (2)when 120572 is a natural number
119908119899+1
(119909 119905) = 119908119899
(119909 119905)
+ int
119905
0
120582 (119905 120591) [minus120597119898
119908 (119909 120591)
120597119905119898+ 119871 (119908
119899(119909 120591))
+ 119873 (119908119899
(119909 120591)) + 119896 (119909 120591)] 119889120591
(7)
where 120582(119905 120591) is the so-called Lagrange multiplier [43] and119908119899(119909 119905) is the 119899-approximate solution However this devel-
opment was not suitable for equations with fractional orderderivative [43] Therefore in their work they apply thenew development of the VIM proposed in [44] to findthe Lagrange multiplier In this new VIM the first step ofthe basic character of the method is to apply the Laplacetransform on both sides of (2) to obtain
119904119898
119908 (119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
= L [119871 (119908 (119909 119905)) + 119873 (119908 (119909 119905)) + 119896 (119909 119905)]
(8)
The recursive formula of (8) can now be used to putforward the main recursive method connecting the Lagrangemultiplier as
119908119899+1
(119909 119904)
= 119908119899
(119909 119904)
+ 120582 (119904) [119904119898
119908119899
(119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
minusL [119871 (119908119899
(119909 119905)) + 119873 (119908119899
(119909 119905)) + 119896 (119909 119905)] ]
(9)
Now considering L[119871(119908119899(119909 119905)) + 119873(119908
119899(119909 119905)) + 119896(119909 119905)] the
restricted term the Lagrange multiplier can be obtained as[43]
120582 (119904) = minus1
119904119898 (10)
Now applying the inverse Laplace transform on both sides of(9) we obtain the following iteration
119908119899+1
(119909 119905)
= 119908119899
(119909 119905)
minus Lminus1
[1
119904119898[119904119898
119908119899
(119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
minus L [119871 (119908119899
(119909 119905))
+119873 (119908119899
(119909 119905)) + 119896 (119909 119905) ]]]
(11)
Abstract and Applied Analysis 3
4 Application to the FractionalNagumo Equation
In this section we present the application of the homotopydecomposition method and the new development of the so-called variational iterationmethod to the nonlinear fractionalNagumo equation We shall start with the HDM
41 Application of the HDM Let us consider the fractionalnonlinear Nagumo Equation with the following initial con-dition
119906 (119909 0) = 1199092
120597120572
119906
120597119905120572+ 120573119906
120597119906
120597119909=
120597
120597119909[119886119906
120597119906
120597119909] + 120574119906 (1 minus 119906) (119906 minus 1)
0 lt 120572 le 1
(12)
Using the steps involved in the HDMwe arrive at the follow-ing119906 (119909 119905)
= 119906 (119909 0)
+1
Γ [1 minus 120572]int
119905
0
(119905 minus 120591)1minus120572
[minus120573119906120597119906
120597119909+
120597
120597119909[119886119906
120597119906
120597119909]
+ 120574119906 (1 minus 119906) (119906 minus 1)] 119889120591
(13)Now assume the solution of the above equation can beexpressed in series form as follows
119906 (119909 119905) =
infin
sum
119899=0
119901119899
119906119899
(119909 119905) (14)
Replacing this in (13) and after comparing the term of thesame power of 119901 we obtain the following recursive formulas
1199060
(119909 119905) = 119906 (119909 119905)
1199061
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731199060
1205971199060
120597119909+
120597
120597119909[1198861199060
1205971199060
120597119909]
+1205741199060
(1 minus 1199060) (1199060
minus 1) ] 119889120591
1199062
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
2+ 1198861198672
2+ 1198861198673
2
+1205741198674
2+ 1205741198675
2minus 1199061] 119889120591
(15)Here
1198671
2= 1199060
(119909 119905) 1205971199091199061
(119909 119905) + 1199061
(119909 119905) 1205971199091199060
(119909 119905)
1198672
2= 21205971199091199060
(119909 119905) 1205971199091199061
(119909 119905)
1198673
2= 1199060
(119909 119905) 120597119909119909
1199061
(119909 119905) + 1199061
(119909 119905) 120597119909119909
1199060
(119909 119905)
1198674
2= 1199062
0(119909 119905) 119906
1(119909 119905) + 119906
3
1+ 1199062
1(119909 119905) 119906
0(119909 119905)
1198675
2= 21199060
(119909 119905) 1199061
(119909 119905)
(16)
The general recursive formula for 119901 ge 3 is given as
119906119899
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
119899+ 1198861198672
119899+ 1198861198673
119899
+1205741198674
119899+ 1205741198675
119899minus 119906119899minus1
] 119889120591
(17)
with
1198671
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198672
119899(119909 119905) =
119899minus1
sum
119895=0
120597119909119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198673
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119909119906119899minus119895
(119909 119905)
1198674
119899(119909 119905) =
119899minus1
sum
119895=0
119895
sum
119896=0
119906119895(119909 119905) 119906
119895minus119896(119909 119905) 119906
119899minus119895minus1(119909 119905)
1198675
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 119906
119899minus119895(119909 119905)
(18)
so that integrating the above set of integral equations weobtain the following
1199060
(119909 119905) = 1199092
1199061
(119909 119905) = minus
119905120572
1199092
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)
Γ (1 + 120572)
1199062
(119909 119905)
= 1199052120572
1199092
((481198862
+ 120574
minus 2119886 (3 + 24119909120573 + 4120574 minus 221199092
120574
+ 119909 (101199091205732
+ 119909120574 (minus2 + 1199092
minus 2120574 + 31199092
120574 minus 1199096
120574)
+2120573 (1 + (2 minus 81199092
+ 31199094
) 120574))))
times1
Γ (1 + 2120572)
4 Abstract and Applied Analysis
+
119905120572
1199094
120574(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)2
Γ (1 + 2120572)
Γ2 (1 + 120572) Γ (1 + 3120572)
minus
1199052120572
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)3
Γ (1 + 3120572)
Γ3 (1 + 120572) Γ (1 + 4120572))
(19)Here we have computed only three terms in the series solu-tion However using the recursive formula we can computethe remaining terms and the approximate solution is givenas follows
119906 (119909 119905) = 1199060
(119909 119905) + 1199061
(119909 119905) + 1199062
(119909 119905) + sdot sdot sdot + sdot sdot sdot (20)
42 New Development of the Variational Iteration Method Inthis subsection we test the efficiency of the new developmentof the variation iteration method by solving the nonlinearfractional equation (1)Therefore following themethodologyof NDVIM we are at the following
The Lagrange multiplier is
120582 (119904) = minus1
119904120572(21)
and the recursive formula is given as119906119899+1
(119909 119905)
= 119906119899
(119909 119905)
minus Lminus1
[1
119904120572[L [minus120573119906
119899
120597119906119899
120597119909
+120597
120597119909[119886119906119899
120597119906119899
120597119909]
+120574119906119899
(1 minus 119906119899) (119906119899
minus 1) ]]]
(22)
with the initial term
1199060
(119909 119905) = 1199092
(23)so that using the iteration formulas we obtain
1199061
(119909 119905) = 1199092
minus119905120572
Γ (1 + 120572)
times (61198861199092
minus 21199093
120573 + 1199092
(1 minus 1199092
) (1199092
minus 1) 120574)
(24)
1199062
(119909 119905)
= (1199092
(1199053120572
(minus1441198863
+ 121198862
times (18119909120573 + 4120574 minus 221199092
120574 + 211199094
120574)
minus 4119886 (12119909120573120574 minus 541199093
120573120574
+ 481199095
120573120574 + 1205742
+ 431199094
1205742
minus 481199096
1205742
+ 181199098
1205742
+ 21199092
(121205732
minus 71205742
))
+ 119909 (21205731205742
+ 521199094
1205731205742
minus 521199096
1205731205742
+ 181199098
1205731205742
+ 261199097
1205743
minus 141199099
1205743
+ 311990911
1205743
+ 41199092
(31205733
minus 51205731205742
)
+ 61199095
(51205732
120574 minus 41205743
) + 2119909 (51205732
120574 minus 1205743
)
+ 1199093
(minus361205732
120574 + 111205743
)))
times Γ (1 + 120572) Γ2
(1 + 2120572) Γ (1 + 4120572)
+ 1199052120572
(1801198862
+ 6119909120573120574 minus 201199093
120573120574 + 141199095
120573120574
+ 1205742
+ 121199094
1205742
minus 101199096
1205742
+ 31199098
1205742
minus 4119886 (23119909120573 + 9120574
minus 251199092
120574 + 161199094
120574)
+21199092
(51205732
minus 31205742
))
times Γ (1 + 3120572) Γ3
(1 + 120572) Γ (1 + 4120572)
+ Γ (1 + 2120572) Γ (1 + 3120572)
times (1199054120572
1199094
120574
times (minus6119886 + 2119909120573 + 120574 minus 21199092
120574 + 1199094
120574)3
times Γ (1 + 3120572)
minus (2119905120572
(16119886 minus 2119909120573 minus 120574 + 21199092
120574 minus 1199094
120574)
minusΓ (1 + 120572) ) Γ (1 + 120572)
timesΓ3
(1 + 120572) Γ (1 + 4120572))))
times (Γ3
(1 + 120572) Γ (1 + 2120572)
times Γ (1 + 3120572) Γ (1 + 4120572) )minus1
(25)
Using the recursive formula the remaining term can beobtained but here due to the length of this termwe computedonly three terms and the approximate solution case given as
119906 (119909 119905) = 1199062
(119909 119905) (26)
In the following section we compare the approximate solu-tion via HDM and NDVIM
5 Numerical Results
We devote this section to the comparison of the numericalsolutions obtained via theHDMand theNDVIM for differentvalues of the fractional order derivative In this case wechose 120572 = 15 120574 = 1 and 119886 = 4 The following figuresshow the numerical solution of the time fractional nonlinear
Abstract and Applied Analysis 5
Table 1 Comparison of numerical values for the approximate solutions via HDM and NDVIM
119909 119905HDM
120572 = 025
NDVIM120572 = 025
HDM120572 = 09
NDVIM120572 = 09
minus10
0 100 100 100 1001 minus114275 times 10
18
minus114319 times 1018
minus324381 times 1017
minus324545 times 1017
2 minus228554 times 1018
minus228628 times 1018
minus393359 times 1018
minus393465 times 1018
6 minus685676 times 1018
minus685845 times 1018
minus205326 times 1020
minus205347 times 1020
10 minus11428 times 1019
minus114305 times 1019
minus129153 times 1021
minus129161 times 1021
minus5
0 25 25 25 251 minus298096 times 10
12
minus299989 times 1012
minus845746 times 1011
minus852756 times 1011
2 minus59636 times 1012
minus599543 times 1012
minus102663 times 1013
minus103118 times 1013
6 minus178972 times 1013
minus179697 times 1013
minus536272 times 1014
minus537156 times 1014
10 minus298327 times 1013
minus299391 times 1013
minus337376 times 1015
minus337727 times 1015
5
0 25 25 25 251 minus350932 times 10
12
minus353141 times 1012
minus995678 times 1011
minus100386 times 1012
2 minus702052 times 1012
minus705766 times 1012
minus120856 times 1013
minus121387 times 1013
6 minus210687 times 1013
minus211533 times 1013
minus631281 times 1014
minus632312 times 1014
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 1 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 09
Nagumo equation Figure 1 show the approximate solutionobtained via the HDM for 120572 = 09 Figure 2 shows theapproximate solution via the NDVIM Figure 3 Show theapproximate solution obtained via the HDM for 120572 = 025
and Figure 4 shows the approximate solution via theNDVIMTable 1 shows the comparison of the numerical values of thesolution obtained via theHDMand theNDVIM respectivelyfor different values of alpha
Both methods used the idea of iteration the initial com-ponents are obtained as the Taylor series of the exact solutionOn one hand the new development of variational iterationmethod makes use of the Laplace transform the Lagrangemultiplier and finally the inverse Laplace transform On the
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 2 Approximate solution of the time-fractional nonlinearNagumo equation via the NDVIM for the value of alpha equal to09
other hand the HDM uses just a simple integral and theperturbation technique Both techniques are simple to imple-ment and are very accurate
6 Conclusion
The Nagumo equation is a very complex equation for whichthe exact solution does not exist The Nagumo equation wasextended to the concept of fractional order derivative Theresulting equation was further analyzed within the frame-work of the homotopy decomposition method and the newdevelopment of variational iteration method Both methodsuse a simple idea of integral transformThe numerical results
6 Abstract and Applied Analysis
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
Submit your manuscripts athttpwwwhindawicom
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International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
2 Basic Information about Fractional Calculus
There exists a vast literature on different definitions of frac-tional derivatives The most popular ones are the Riemann-Liouville and the Caputo derivatives For Caputo we have thefollowing
Definition 1 (see [26ndash30]) A real function119891(119909) 119909 gt 0 is saidto be in the space 119862
120583 120583 isin R if there exists a real number
119901 gt 120583 such that 119891(119909) = 119909119901
ℎ(119909) where ℎ(119909) isin 119862[0 infin) andit is said to be in space 119862
119898
120583if 119891(119898)
isin 119862120583 119898 isin N
Definition 2 (partial derivatives of fractional order [31ndash34])Assume now that 119891(119909) is a function of 119899 variables 119909
119894 119894 =
1 119899 also of class119862 on119863 isin R119899We define partial derivative
of order 120572 for 119891 with respect to 119909119894 the function
119886120597120572
119909119891 =
1
Γ (119898 minus 120572)int
119909119894
119886
(119909119894minus 119905)119898minus120572minus1
120597119898
119909119894
119891 (119909119895)10038161003816100381610038161003816119909119895=119905
119889119905 (4)
where 120597119898
119909119894
is the usual partial derivative of integer order 119898
Definition 3 (see [26ndash30]) The Riemann-Liouville fractionalintegral operator of order120572 ge 0 of a function119891 isin 119862
120583 120583 ge minus1
is defined as
119869120572
119891 (119909) =1
Γ (120572)int
119909
0
(119909 minus 119905)120572minus1
119891 (119905) 119889119905 120572 gt 0 119909 gt 0
1198690
119891 (119909) = 119891 (119909)
(5)
Properties of the operator can be found in [31ndash38] we men-tion only the following
For
119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 120574 gt minus1
119869120572
119869120573
119891 (119909) = 119869120572+120573
119891 (119909)
119869120572
119869120573
119891 (119909) = 119869120573
119869120572
119891 (119909)
119869120572
119909120574
=Γ (120574 + 1)
Γ (120572 + 120574 + 1)119909120572+120574
(6)
3 Basic Information of the HDM and VIM
In this section we shall present the basic information ofthe chosen analytical methods the homotopy decompositionmethod and variational iteration method we shall start withthe homotopy decomposition method
31 Information about the HDM The interested reader canfind the full detail of the methodology of the homo-topy decomposition method in [39ndash42] This relatively newmethod was recently used to solve some nonlinear frac-tional partial differential equations However since the newdevelopment of the variational iteration method using theLaplace transform was recently introduced we shall presentits methodology in the following subsection
32 Some Information about the Variational IterationMethodIn its initial development the essential nature of the methodwas to construct the following correction functional for (2)when 120572 is a natural number
119908119899+1
(119909 119905) = 119908119899
(119909 119905)
+ int
119905
0
120582 (119905 120591) [minus120597119898
119908 (119909 120591)
120597119905119898+ 119871 (119908
119899(119909 120591))
+ 119873 (119908119899
(119909 120591)) + 119896 (119909 120591)] 119889120591
(7)
where 120582(119905 120591) is the so-called Lagrange multiplier [43] and119908119899(119909 119905) is the 119899-approximate solution However this devel-
opment was not suitable for equations with fractional orderderivative [43] Therefore in their work they apply thenew development of the VIM proposed in [44] to findthe Lagrange multiplier In this new VIM the first step ofthe basic character of the method is to apply the Laplacetransform on both sides of (2) to obtain
119904119898
119908 (119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
= L [119871 (119908 (119909 119905)) + 119873 (119908 (119909 119905)) + 119896 (119909 119905)]
(8)
The recursive formula of (8) can now be used to putforward the main recursive method connecting the Lagrangemultiplier as
119908119899+1
(119909 119904)
= 119908119899
(119909 119904)
+ 120582 (119904) [119904119898
119908119899
(119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
minusL [119871 (119908119899
(119909 119905)) + 119873 (119908119899
(119909 119905)) + 119896 (119909 119905)] ]
(9)
Now considering L[119871(119908119899(119909 119905)) + 119873(119908
119899(119909 119905)) + 119896(119909 119905)] the
restricted term the Lagrange multiplier can be obtained as[43]
120582 (119904) = minus1
119904119898 (10)
Now applying the inverse Laplace transform on both sides of(9) we obtain the following iteration
119908119899+1
(119909 119905)
= 119908119899
(119909 119905)
minus Lminus1
[1
119904119898[119904119898
119908119899
(119909 119904) minus 119904119898minus1
119908 (119909 0) minus sdot sdot sdot 119908119898minus1
(119909 0)
minus L [119871 (119908119899
(119909 119905))
+119873 (119908119899
(119909 119905)) + 119896 (119909 119905) ]]]
(11)
Abstract and Applied Analysis 3
4 Application to the FractionalNagumo Equation
In this section we present the application of the homotopydecomposition method and the new development of the so-called variational iterationmethod to the nonlinear fractionalNagumo equation We shall start with the HDM
41 Application of the HDM Let us consider the fractionalnonlinear Nagumo Equation with the following initial con-dition
119906 (119909 0) = 1199092
120597120572
119906
120597119905120572+ 120573119906
120597119906
120597119909=
120597
120597119909[119886119906
120597119906
120597119909] + 120574119906 (1 minus 119906) (119906 minus 1)
0 lt 120572 le 1
(12)
Using the steps involved in the HDMwe arrive at the follow-ing119906 (119909 119905)
= 119906 (119909 0)
+1
Γ [1 minus 120572]int
119905
0
(119905 minus 120591)1minus120572
[minus120573119906120597119906
120597119909+
120597
120597119909[119886119906
120597119906
120597119909]
+ 120574119906 (1 minus 119906) (119906 minus 1)] 119889120591
(13)Now assume the solution of the above equation can beexpressed in series form as follows
119906 (119909 119905) =
infin
sum
119899=0
119901119899
119906119899
(119909 119905) (14)
Replacing this in (13) and after comparing the term of thesame power of 119901 we obtain the following recursive formulas
1199060
(119909 119905) = 119906 (119909 119905)
1199061
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731199060
1205971199060
120597119909+
120597
120597119909[1198861199060
1205971199060
120597119909]
+1205741199060
(1 minus 1199060) (1199060
minus 1) ] 119889120591
1199062
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
2+ 1198861198672
2+ 1198861198673
2
+1205741198674
2+ 1205741198675
2minus 1199061] 119889120591
(15)Here
1198671
2= 1199060
(119909 119905) 1205971199091199061
(119909 119905) + 1199061
(119909 119905) 1205971199091199060
(119909 119905)
1198672
2= 21205971199091199060
(119909 119905) 1205971199091199061
(119909 119905)
1198673
2= 1199060
(119909 119905) 120597119909119909
1199061
(119909 119905) + 1199061
(119909 119905) 120597119909119909
1199060
(119909 119905)
1198674
2= 1199062
0(119909 119905) 119906
1(119909 119905) + 119906
3
1+ 1199062
1(119909 119905) 119906
0(119909 119905)
1198675
2= 21199060
(119909 119905) 1199061
(119909 119905)
(16)
The general recursive formula for 119901 ge 3 is given as
119906119899
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
119899+ 1198861198672
119899+ 1198861198673
119899
+1205741198674
119899+ 1205741198675
119899minus 119906119899minus1
] 119889120591
(17)
with
1198671
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198672
119899(119909 119905) =
119899minus1
sum
119895=0
120597119909119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198673
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119909119906119899minus119895
(119909 119905)
1198674
119899(119909 119905) =
119899minus1
sum
119895=0
119895
sum
119896=0
119906119895(119909 119905) 119906
119895minus119896(119909 119905) 119906
119899minus119895minus1(119909 119905)
1198675
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 119906
119899minus119895(119909 119905)
(18)
so that integrating the above set of integral equations weobtain the following
1199060
(119909 119905) = 1199092
1199061
(119909 119905) = minus
119905120572
1199092
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)
Γ (1 + 120572)
1199062
(119909 119905)
= 1199052120572
1199092
((481198862
+ 120574
minus 2119886 (3 + 24119909120573 + 4120574 minus 221199092
120574
+ 119909 (101199091205732
+ 119909120574 (minus2 + 1199092
minus 2120574 + 31199092
120574 minus 1199096
120574)
+2120573 (1 + (2 minus 81199092
+ 31199094
) 120574))))
times1
Γ (1 + 2120572)
4 Abstract and Applied Analysis
+
119905120572
1199094
120574(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)2
Γ (1 + 2120572)
Γ2 (1 + 120572) Γ (1 + 3120572)
minus
1199052120572
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)3
Γ (1 + 3120572)
Γ3 (1 + 120572) Γ (1 + 4120572))
(19)Here we have computed only three terms in the series solu-tion However using the recursive formula we can computethe remaining terms and the approximate solution is givenas follows
119906 (119909 119905) = 1199060
(119909 119905) + 1199061
(119909 119905) + 1199062
(119909 119905) + sdot sdot sdot + sdot sdot sdot (20)
42 New Development of the Variational Iteration Method Inthis subsection we test the efficiency of the new developmentof the variation iteration method by solving the nonlinearfractional equation (1)Therefore following themethodologyof NDVIM we are at the following
The Lagrange multiplier is
120582 (119904) = minus1
119904120572(21)
and the recursive formula is given as119906119899+1
(119909 119905)
= 119906119899
(119909 119905)
minus Lminus1
[1
119904120572[L [minus120573119906
119899
120597119906119899
120597119909
+120597
120597119909[119886119906119899
120597119906119899
120597119909]
+120574119906119899
(1 minus 119906119899) (119906119899
minus 1) ]]]
(22)
with the initial term
1199060
(119909 119905) = 1199092
(23)so that using the iteration formulas we obtain
1199061
(119909 119905) = 1199092
minus119905120572
Γ (1 + 120572)
times (61198861199092
minus 21199093
120573 + 1199092
(1 minus 1199092
) (1199092
minus 1) 120574)
(24)
1199062
(119909 119905)
= (1199092
(1199053120572
(minus1441198863
+ 121198862
times (18119909120573 + 4120574 minus 221199092
120574 + 211199094
120574)
minus 4119886 (12119909120573120574 minus 541199093
120573120574
+ 481199095
120573120574 + 1205742
+ 431199094
1205742
minus 481199096
1205742
+ 181199098
1205742
+ 21199092
(121205732
minus 71205742
))
+ 119909 (21205731205742
+ 521199094
1205731205742
minus 521199096
1205731205742
+ 181199098
1205731205742
+ 261199097
1205743
minus 141199099
1205743
+ 311990911
1205743
+ 41199092
(31205733
minus 51205731205742
)
+ 61199095
(51205732
120574 minus 41205743
) + 2119909 (51205732
120574 minus 1205743
)
+ 1199093
(minus361205732
120574 + 111205743
)))
times Γ (1 + 120572) Γ2
(1 + 2120572) Γ (1 + 4120572)
+ 1199052120572
(1801198862
+ 6119909120573120574 minus 201199093
120573120574 + 141199095
120573120574
+ 1205742
+ 121199094
1205742
minus 101199096
1205742
+ 31199098
1205742
minus 4119886 (23119909120573 + 9120574
minus 251199092
120574 + 161199094
120574)
+21199092
(51205732
minus 31205742
))
times Γ (1 + 3120572) Γ3
(1 + 120572) Γ (1 + 4120572)
+ Γ (1 + 2120572) Γ (1 + 3120572)
times (1199054120572
1199094
120574
times (minus6119886 + 2119909120573 + 120574 minus 21199092
120574 + 1199094
120574)3
times Γ (1 + 3120572)
minus (2119905120572
(16119886 minus 2119909120573 minus 120574 + 21199092
120574 minus 1199094
120574)
minusΓ (1 + 120572) ) Γ (1 + 120572)
timesΓ3
(1 + 120572) Γ (1 + 4120572))))
times (Γ3
(1 + 120572) Γ (1 + 2120572)
times Γ (1 + 3120572) Γ (1 + 4120572) )minus1
(25)
Using the recursive formula the remaining term can beobtained but here due to the length of this termwe computedonly three terms and the approximate solution case given as
119906 (119909 119905) = 1199062
(119909 119905) (26)
In the following section we compare the approximate solu-tion via HDM and NDVIM
5 Numerical Results
We devote this section to the comparison of the numericalsolutions obtained via theHDMand theNDVIM for differentvalues of the fractional order derivative In this case wechose 120572 = 15 120574 = 1 and 119886 = 4 The following figuresshow the numerical solution of the time fractional nonlinear
Abstract and Applied Analysis 5
Table 1 Comparison of numerical values for the approximate solutions via HDM and NDVIM
119909 119905HDM
120572 = 025
NDVIM120572 = 025
HDM120572 = 09
NDVIM120572 = 09
minus10
0 100 100 100 1001 minus114275 times 10
18
minus114319 times 1018
minus324381 times 1017
minus324545 times 1017
2 minus228554 times 1018
minus228628 times 1018
minus393359 times 1018
minus393465 times 1018
6 minus685676 times 1018
minus685845 times 1018
minus205326 times 1020
minus205347 times 1020
10 minus11428 times 1019
minus114305 times 1019
minus129153 times 1021
minus129161 times 1021
minus5
0 25 25 25 251 minus298096 times 10
12
minus299989 times 1012
minus845746 times 1011
minus852756 times 1011
2 minus59636 times 1012
minus599543 times 1012
minus102663 times 1013
minus103118 times 1013
6 minus178972 times 1013
minus179697 times 1013
minus536272 times 1014
minus537156 times 1014
10 minus298327 times 1013
minus299391 times 1013
minus337376 times 1015
minus337727 times 1015
5
0 25 25 25 251 minus350932 times 10
12
minus353141 times 1012
minus995678 times 1011
minus100386 times 1012
2 minus702052 times 1012
minus705766 times 1012
minus120856 times 1013
minus121387 times 1013
6 minus210687 times 1013
minus211533 times 1013
minus631281 times 1014
minus632312 times 1014
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 1 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 09
Nagumo equation Figure 1 show the approximate solutionobtained via the HDM for 120572 = 09 Figure 2 shows theapproximate solution via the NDVIM Figure 3 Show theapproximate solution obtained via the HDM for 120572 = 025
and Figure 4 shows the approximate solution via theNDVIMTable 1 shows the comparison of the numerical values of thesolution obtained via theHDMand theNDVIM respectivelyfor different values of alpha
Both methods used the idea of iteration the initial com-ponents are obtained as the Taylor series of the exact solutionOn one hand the new development of variational iterationmethod makes use of the Laplace transform the Lagrangemultiplier and finally the inverse Laplace transform On the
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 2 Approximate solution of the time-fractional nonlinearNagumo equation via the NDVIM for the value of alpha equal to09
other hand the HDM uses just a simple integral and theperturbation technique Both techniques are simple to imple-ment and are very accurate
6 Conclusion
The Nagumo equation is a very complex equation for whichthe exact solution does not exist The Nagumo equation wasextended to the concept of fractional order derivative Theresulting equation was further analyzed within the frame-work of the homotopy decomposition method and the newdevelopment of variational iteration method Both methodsuse a simple idea of integral transformThe numerical results
6 Abstract and Applied Analysis
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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 3
4 Application to the FractionalNagumo Equation
In this section we present the application of the homotopydecomposition method and the new development of the so-called variational iterationmethod to the nonlinear fractionalNagumo equation We shall start with the HDM
41 Application of the HDM Let us consider the fractionalnonlinear Nagumo Equation with the following initial con-dition
119906 (119909 0) = 1199092
120597120572
119906
120597119905120572+ 120573119906
120597119906
120597119909=
120597
120597119909[119886119906
120597119906
120597119909] + 120574119906 (1 minus 119906) (119906 minus 1)
0 lt 120572 le 1
(12)
Using the steps involved in the HDMwe arrive at the follow-ing119906 (119909 119905)
= 119906 (119909 0)
+1
Γ [1 minus 120572]int
119905
0
(119905 minus 120591)1minus120572
[minus120573119906120597119906
120597119909+
120597
120597119909[119886119906
120597119906
120597119909]
+ 120574119906 (1 minus 119906) (119906 minus 1)] 119889120591
(13)Now assume the solution of the above equation can beexpressed in series form as follows
119906 (119909 119905) =
infin
sum
119899=0
119901119899
119906119899
(119909 119905) (14)
Replacing this in (13) and after comparing the term of thesame power of 119901 we obtain the following recursive formulas
1199060
(119909 119905) = 119906 (119909 119905)
1199061
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731199060
1205971199060
120597119909+
120597
120597119909[1198861199060
1205971199060
120597119909]
+1205741199060
(1 minus 1199060) (1199060
minus 1) ] 119889120591
1199062
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
2+ 1198861198672
2+ 1198861198673
2
+1205741198674
2+ 1205741198675
2minus 1199061] 119889120591
(15)Here
1198671
2= 1199060
(119909 119905) 1205971199091199061
(119909 119905) + 1199061
(119909 119905) 1205971199091199060
(119909 119905)
1198672
2= 21205971199091199060
(119909 119905) 1205971199091199061
(119909 119905)
1198673
2= 1199060
(119909 119905) 120597119909119909
1199061
(119909 119905) + 1199061
(119909 119905) 120597119909119909
1199060
(119909 119905)
1198674
2= 1199062
0(119909 119905) 119906
1(119909 119905) + 119906
3
1+ 1199062
1(119909 119905) 119906
0(119909 119905)
1198675
2= 21199060
(119909 119905) 1199061
(119909 119905)
(16)
The general recursive formula for 119901 ge 3 is given as
119906119899
(119909 119905)
=1
Γ [120572]int
119905
0
(119905 minus 120591)120572minus1
[minus1205731198671
119899+ 1198861198672
119899+ 1198861198673
119899
+1205741198674
119899+ 1205741198675
119899minus 119906119899minus1
] 119889120591
(17)
with
1198671
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198672
119899(119909 119905) =
119899minus1
sum
119895=0
120597119909119906119895(119909 119905) 120597
119909119906119899minus119895
(119909 119905)
1198673
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 120597
119909119909119906119899minus119895
(119909 119905)
1198674
119899(119909 119905) =
119899minus1
sum
119895=0
119895
sum
119896=0
119906119895(119909 119905) 119906
119895minus119896(119909 119905) 119906
119899minus119895minus1(119909 119905)
1198675
119899(119909 119905) =
119899minus1
sum
119895=0
119906119895(119909 119905) 119906
119899minus119895(119909 119905)
(18)
so that integrating the above set of integral equations weobtain the following
1199060
(119909 119905) = 1199092
1199061
(119909 119905) = minus
119905120572
1199092
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)
Γ (1 + 120572)
1199062
(119909 119905)
= 1199052120572
1199092
((481198862
+ 120574
minus 2119886 (3 + 24119909120573 + 4120574 minus 221199092
120574
+ 119909 (101199091205732
+ 119909120574 (minus2 + 1199092
minus 2120574 + 31199092
120574 minus 1199096
120574)
+2120573 (1 + (2 minus 81199092
+ 31199094
) 120574))))
times1
Γ (1 + 2120572)
4 Abstract and Applied Analysis
+
119905120572
1199094
120574(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)2
Γ (1 + 2120572)
Γ2 (1 + 120572) Γ (1 + 3120572)
minus
1199052120572
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)3
Γ (1 + 3120572)
Γ3 (1 + 120572) Γ (1 + 4120572))
(19)Here we have computed only three terms in the series solu-tion However using the recursive formula we can computethe remaining terms and the approximate solution is givenas follows
119906 (119909 119905) = 1199060
(119909 119905) + 1199061
(119909 119905) + 1199062
(119909 119905) + sdot sdot sdot + sdot sdot sdot (20)
42 New Development of the Variational Iteration Method Inthis subsection we test the efficiency of the new developmentof the variation iteration method by solving the nonlinearfractional equation (1)Therefore following themethodologyof NDVIM we are at the following
The Lagrange multiplier is
120582 (119904) = minus1
119904120572(21)
and the recursive formula is given as119906119899+1
(119909 119905)
= 119906119899
(119909 119905)
minus Lminus1
[1
119904120572[L [minus120573119906
119899
120597119906119899
120597119909
+120597
120597119909[119886119906119899
120597119906119899
120597119909]
+120574119906119899
(1 minus 119906119899) (119906119899
minus 1) ]]]
(22)
with the initial term
1199060
(119909 119905) = 1199092
(23)so that using the iteration formulas we obtain
1199061
(119909 119905) = 1199092
minus119905120572
Γ (1 + 120572)
times (61198861199092
minus 21199093
120573 + 1199092
(1 minus 1199092
) (1199092
minus 1) 120574)
(24)
1199062
(119909 119905)
= (1199092
(1199053120572
(minus1441198863
+ 121198862
times (18119909120573 + 4120574 minus 221199092
120574 + 211199094
120574)
minus 4119886 (12119909120573120574 minus 541199093
120573120574
+ 481199095
120573120574 + 1205742
+ 431199094
1205742
minus 481199096
1205742
+ 181199098
1205742
+ 21199092
(121205732
minus 71205742
))
+ 119909 (21205731205742
+ 521199094
1205731205742
minus 521199096
1205731205742
+ 181199098
1205731205742
+ 261199097
1205743
minus 141199099
1205743
+ 311990911
1205743
+ 41199092
(31205733
minus 51205731205742
)
+ 61199095
(51205732
120574 minus 41205743
) + 2119909 (51205732
120574 minus 1205743
)
+ 1199093
(minus361205732
120574 + 111205743
)))
times Γ (1 + 120572) Γ2
(1 + 2120572) Γ (1 + 4120572)
+ 1199052120572
(1801198862
+ 6119909120573120574 minus 201199093
120573120574 + 141199095
120573120574
+ 1205742
+ 121199094
1205742
minus 101199096
1205742
+ 31199098
1205742
minus 4119886 (23119909120573 + 9120574
minus 251199092
120574 + 161199094
120574)
+21199092
(51205732
minus 31205742
))
times Γ (1 + 3120572) Γ3
(1 + 120572) Γ (1 + 4120572)
+ Γ (1 + 2120572) Γ (1 + 3120572)
times (1199054120572
1199094
120574
times (minus6119886 + 2119909120573 + 120574 minus 21199092
120574 + 1199094
120574)3
times Γ (1 + 3120572)
minus (2119905120572
(16119886 minus 2119909120573 minus 120574 + 21199092
120574 minus 1199094
120574)
minusΓ (1 + 120572) ) Γ (1 + 120572)
timesΓ3
(1 + 120572) Γ (1 + 4120572))))
times (Γ3
(1 + 120572) Γ (1 + 2120572)
times Γ (1 + 3120572) Γ (1 + 4120572) )minus1
(25)
Using the recursive formula the remaining term can beobtained but here due to the length of this termwe computedonly three terms and the approximate solution case given as
119906 (119909 119905) = 1199062
(119909 119905) (26)
In the following section we compare the approximate solu-tion via HDM and NDVIM
5 Numerical Results
We devote this section to the comparison of the numericalsolutions obtained via theHDMand theNDVIM for differentvalues of the fractional order derivative In this case wechose 120572 = 15 120574 = 1 and 119886 = 4 The following figuresshow the numerical solution of the time fractional nonlinear
Abstract and Applied Analysis 5
Table 1 Comparison of numerical values for the approximate solutions via HDM and NDVIM
119909 119905HDM
120572 = 025
NDVIM120572 = 025
HDM120572 = 09
NDVIM120572 = 09
minus10
0 100 100 100 1001 minus114275 times 10
18
minus114319 times 1018
minus324381 times 1017
minus324545 times 1017
2 minus228554 times 1018
minus228628 times 1018
minus393359 times 1018
minus393465 times 1018
6 minus685676 times 1018
minus685845 times 1018
minus205326 times 1020
minus205347 times 1020
10 minus11428 times 1019
minus114305 times 1019
minus129153 times 1021
minus129161 times 1021
minus5
0 25 25 25 251 minus298096 times 10
12
minus299989 times 1012
minus845746 times 1011
minus852756 times 1011
2 minus59636 times 1012
minus599543 times 1012
minus102663 times 1013
minus103118 times 1013
6 minus178972 times 1013
minus179697 times 1013
minus536272 times 1014
minus537156 times 1014
10 minus298327 times 1013
minus299391 times 1013
minus337376 times 1015
minus337727 times 1015
5
0 25 25 25 251 minus350932 times 10
12
minus353141 times 1012
minus995678 times 1011
minus100386 times 1012
2 minus702052 times 1012
minus705766 times 1012
minus120856 times 1013
minus121387 times 1013
6 minus210687 times 1013
minus211533 times 1013
minus631281 times 1014
minus632312 times 1014
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 1 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 09
Nagumo equation Figure 1 show the approximate solutionobtained via the HDM for 120572 = 09 Figure 2 shows theapproximate solution via the NDVIM Figure 3 Show theapproximate solution obtained via the HDM for 120572 = 025
and Figure 4 shows the approximate solution via theNDVIMTable 1 shows the comparison of the numerical values of thesolution obtained via theHDMand theNDVIM respectivelyfor different values of alpha
Both methods used the idea of iteration the initial com-ponents are obtained as the Taylor series of the exact solutionOn one hand the new development of variational iterationmethod makes use of the Laplace transform the Lagrangemultiplier and finally the inverse Laplace transform On the
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 2 Approximate solution of the time-fractional nonlinearNagumo equation via the NDVIM for the value of alpha equal to09
other hand the HDM uses just a simple integral and theperturbation technique Both techniques are simple to imple-ment and are very accurate
6 Conclusion
The Nagumo equation is a very complex equation for whichthe exact solution does not exist The Nagumo equation wasextended to the concept of fractional order derivative Theresulting equation was further analyzed within the frame-work of the homotopy decomposition method and the newdevelopment of variational iteration method Both methodsuse a simple idea of integral transformThe numerical results
6 Abstract and Applied Analysis
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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
+
119905120572
1199094
120574(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)2
Γ (1 + 2120572)
Γ2 (1 + 120572) Γ (1 + 3120572)
minus
1199052120572
(minus6119886 + 2119909120573 + (minus1 + 1199092
)2
120574)3
Γ (1 + 3120572)
Γ3 (1 + 120572) Γ (1 + 4120572))
(19)Here we have computed only three terms in the series solu-tion However using the recursive formula we can computethe remaining terms and the approximate solution is givenas follows
119906 (119909 119905) = 1199060
(119909 119905) + 1199061
(119909 119905) + 1199062
(119909 119905) + sdot sdot sdot + sdot sdot sdot (20)
42 New Development of the Variational Iteration Method Inthis subsection we test the efficiency of the new developmentof the variation iteration method by solving the nonlinearfractional equation (1)Therefore following themethodologyof NDVIM we are at the following
The Lagrange multiplier is
120582 (119904) = minus1
119904120572(21)
and the recursive formula is given as119906119899+1
(119909 119905)
= 119906119899
(119909 119905)
minus Lminus1
[1
119904120572[L [minus120573119906
119899
120597119906119899
120597119909
+120597
120597119909[119886119906119899
120597119906119899
120597119909]
+120574119906119899
(1 minus 119906119899) (119906119899
minus 1) ]]]
(22)
with the initial term
1199060
(119909 119905) = 1199092
(23)so that using the iteration formulas we obtain
1199061
(119909 119905) = 1199092
minus119905120572
Γ (1 + 120572)
times (61198861199092
minus 21199093
120573 + 1199092
(1 minus 1199092
) (1199092
minus 1) 120574)
(24)
1199062
(119909 119905)
= (1199092
(1199053120572
(minus1441198863
+ 121198862
times (18119909120573 + 4120574 minus 221199092
120574 + 211199094
120574)
minus 4119886 (12119909120573120574 minus 541199093
120573120574
+ 481199095
120573120574 + 1205742
+ 431199094
1205742
minus 481199096
1205742
+ 181199098
1205742
+ 21199092
(121205732
minus 71205742
))
+ 119909 (21205731205742
+ 521199094
1205731205742
minus 521199096
1205731205742
+ 181199098
1205731205742
+ 261199097
1205743
minus 141199099
1205743
+ 311990911
1205743
+ 41199092
(31205733
minus 51205731205742
)
+ 61199095
(51205732
120574 minus 41205743
) + 2119909 (51205732
120574 minus 1205743
)
+ 1199093
(minus361205732
120574 + 111205743
)))
times Γ (1 + 120572) Γ2
(1 + 2120572) Γ (1 + 4120572)
+ 1199052120572
(1801198862
+ 6119909120573120574 minus 201199093
120573120574 + 141199095
120573120574
+ 1205742
+ 121199094
1205742
minus 101199096
1205742
+ 31199098
1205742
minus 4119886 (23119909120573 + 9120574
minus 251199092
120574 + 161199094
120574)
+21199092
(51205732
minus 31205742
))
times Γ (1 + 3120572) Γ3
(1 + 120572) Γ (1 + 4120572)
+ Γ (1 + 2120572) Γ (1 + 3120572)
times (1199054120572
1199094
120574
times (minus6119886 + 2119909120573 + 120574 minus 21199092
120574 + 1199094
120574)3
times Γ (1 + 3120572)
minus (2119905120572
(16119886 minus 2119909120573 minus 120574 + 21199092
120574 minus 1199094
120574)
minusΓ (1 + 120572) ) Γ (1 + 120572)
timesΓ3
(1 + 120572) Γ (1 + 4120572))))
times (Γ3
(1 + 120572) Γ (1 + 2120572)
times Γ (1 + 3120572) Γ (1 + 4120572) )minus1
(25)
Using the recursive formula the remaining term can beobtained but here due to the length of this termwe computedonly three terms and the approximate solution case given as
119906 (119909 119905) = 1199062
(119909 119905) (26)
In the following section we compare the approximate solu-tion via HDM and NDVIM
5 Numerical Results
We devote this section to the comparison of the numericalsolutions obtained via theHDMand theNDVIM for differentvalues of the fractional order derivative In this case wechose 120572 = 15 120574 = 1 and 119886 = 4 The following figuresshow the numerical solution of the time fractional nonlinear
Abstract and Applied Analysis 5
Table 1 Comparison of numerical values for the approximate solutions via HDM and NDVIM
119909 119905HDM
120572 = 025
NDVIM120572 = 025
HDM120572 = 09
NDVIM120572 = 09
minus10
0 100 100 100 1001 minus114275 times 10
18
minus114319 times 1018
minus324381 times 1017
minus324545 times 1017
2 minus228554 times 1018
minus228628 times 1018
minus393359 times 1018
minus393465 times 1018
6 minus685676 times 1018
minus685845 times 1018
minus205326 times 1020
minus205347 times 1020
10 minus11428 times 1019
minus114305 times 1019
minus129153 times 1021
minus129161 times 1021
minus5
0 25 25 25 251 minus298096 times 10
12
minus299989 times 1012
minus845746 times 1011
minus852756 times 1011
2 minus59636 times 1012
minus599543 times 1012
minus102663 times 1013
minus103118 times 1013
6 minus178972 times 1013
minus179697 times 1013
minus536272 times 1014
minus537156 times 1014
10 minus298327 times 1013
minus299391 times 1013
minus337376 times 1015
minus337727 times 1015
5
0 25 25 25 251 minus350932 times 10
12
minus353141 times 1012
minus995678 times 1011
minus100386 times 1012
2 minus702052 times 1012
minus705766 times 1012
minus120856 times 1013
minus121387 times 1013
6 minus210687 times 1013
minus211533 times 1013
minus631281 times 1014
minus632312 times 1014
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 1 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 09
Nagumo equation Figure 1 show the approximate solutionobtained via the HDM for 120572 = 09 Figure 2 shows theapproximate solution via the NDVIM Figure 3 Show theapproximate solution obtained via the HDM for 120572 = 025
and Figure 4 shows the approximate solution via theNDVIMTable 1 shows the comparison of the numerical values of thesolution obtained via theHDMand theNDVIM respectivelyfor different values of alpha
Both methods used the idea of iteration the initial com-ponents are obtained as the Taylor series of the exact solutionOn one hand the new development of variational iterationmethod makes use of the Laplace transform the Lagrangemultiplier and finally the inverse Laplace transform On the
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 2 Approximate solution of the time-fractional nonlinearNagumo equation via the NDVIM for the value of alpha equal to09
other hand the HDM uses just a simple integral and theperturbation technique Both techniques are simple to imple-ment and are very accurate
6 Conclusion
The Nagumo equation is a very complex equation for whichthe exact solution does not exist The Nagumo equation wasextended to the concept of fractional order derivative Theresulting equation was further analyzed within the frame-work of the homotopy decomposition method and the newdevelopment of variational iteration method Both methodsuse a simple idea of integral transformThe numerical results
6 Abstract and Applied Analysis
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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
Table 1 Comparison of numerical values for the approximate solutions via HDM and NDVIM
119909 119905HDM
120572 = 025
NDVIM120572 = 025
HDM120572 = 09
NDVIM120572 = 09
minus10
0 100 100 100 1001 minus114275 times 10
18
minus114319 times 1018
minus324381 times 1017
minus324545 times 1017
2 minus228554 times 1018
minus228628 times 1018
minus393359 times 1018
minus393465 times 1018
6 minus685676 times 1018
minus685845 times 1018
minus205326 times 1020
minus205347 times 1020
10 minus11428 times 1019
minus114305 times 1019
minus129153 times 1021
minus129161 times 1021
minus5
0 25 25 25 251 minus298096 times 10
12
minus299989 times 1012
minus845746 times 1011
minus852756 times 1011
2 minus59636 times 1012
minus599543 times 1012
minus102663 times 1013
minus103118 times 1013
6 minus178972 times 1013
minus179697 times 1013
minus536272 times 1014
minus537156 times 1014
10 minus298327 times 1013
minus299391 times 1013
minus337376 times 1015
minus337727 times 1015
5
0 25 25 25 251 minus350932 times 10
12
minus353141 times 1012
minus995678 times 1011
minus100386 times 1012
2 minus702052 times 1012
minus705766 times 1012
minus120856 times 1013
minus121387 times 1013
6 minus210687 times 1013
minus211533 times 1013
minus631281 times 1014
minus632312 times 1014
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 1 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 09
Nagumo equation Figure 1 show the approximate solutionobtained via the HDM for 120572 = 09 Figure 2 shows theapproximate solution via the NDVIM Figure 3 Show theapproximate solution obtained via the HDM for 120572 = 025
and Figure 4 shows the approximate solution via theNDVIMTable 1 shows the comparison of the numerical values of thesolution obtained via theHDMand theNDVIM respectivelyfor different values of alpha
Both methods used the idea of iteration the initial com-ponents are obtained as the Taylor series of the exact solutionOn one hand the new development of variational iterationmethod makes use of the Laplace transform the Lagrangemultiplier and finally the inverse Laplace transform On the
05
10
Distance0
5
10
Time
0
0
5
Time
times1018
minus1
minus2
minus3
minus4
u(xt)
minus10
minus5
Figure 2 Approximate solution of the time-fractional nonlinearNagumo equation via the NDVIM for the value of alpha equal to09
other hand the HDM uses just a simple integral and theperturbation technique Both techniques are simple to imple-ment and are very accurate
6 Conclusion
The Nagumo equation is a very complex equation for whichthe exact solution does not exist The Nagumo equation wasextended to the concept of fractional order derivative Theresulting equation was further analyzed within the frame-work of the homotopy decomposition method and the newdevelopment of variational iteration method Both methodsuse a simple idea of integral transformThe numerical results
6 Abstract and Applied Analysis
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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
05
10
Distance0
5
10
Time
0
05
Distance
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 3 Approximate solution of the time-fractional nonlinearNagumo equation via the HDM for the value of alpha equal to 025
05
10
Distance0
5
10
Time
0
05
Distanc
5
Time
times1017
minus1
minus2
minus3
u(xt)
minus10
minus5
Figure 4 We present in Table 1 the numerical values of theapproximate solutions obtained via both methods for differentvalues of alpha
are presented to test the efficiency and the accuracy of bothmethods From their iteration formulas one can concludethat these twomethods are simple to be used and are powerfulweapons to handle fractional nonlinear equation type
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Abdon Atangana would like to thank Claude Leon founda-tion for their financial support
References
[1] D G Aronson ldquoThe role of the diffusion in mathematical pop-ulation biology skellam revisitedrdquo in Mathematics in Biologyand Medicine vol 57 of Lecture Notes in BiomathematicsSpringer Berlin Germany 1985
[2] J D Murray Mathematical Biology vol 19 Springer BerlinGermany 1989
[3] M A Lewis and P Kareiva ldquoAllee dynamics and the spread ofinvading organismsrdquoTheoretical Population Biology vol 43 no2 pp 141ndash158 1993
[4] Y Hosono ldquoTraveling wave solutions for some density depen-dent diffusion equationsrdquo Japan Journal of AppliedMathematicsvol 3 no 1 pp 163ndash196 1986
[5] P Grindrod and B D Sleeman ldquoWeak travelling fronts forpopulation models with density dependent dispersionrdquoMathe-matical Methods in the Applied Sciences vol 9 no 4 pp 576ndash586 1987
[6] F Sanchez-Garduno and P K Maini ldquoTravelling wave phe-nomena in non-linear diffusion degenerateNagumo equationsrdquoJournal ofMathematical Biology vol 35 no 6 pp 713ndash728 1997
[7] R Laister A T Peplow andR E Beardmore ldquoFinite time extin-ction in nonlinear diffusion equationsrdquo Applied MathematicsLetters vol 17 no 5 pp 561ndash567 2004
[8] M B AMansour ldquoAccurate computation of traveling wave sol-utions of some nonlinear diffusion equationsrdquo Wave Motionvol 44 no 3 pp 222ndash230 2006
[9] J Naoumo S Arimoto and S Yoshizawa ldquoAn active pulsetransmission line simulating nerve axonrdquo Proceedings of theInstitute of Radio Engineers vol 50 pp 2061ndash2070 1962
[10] A C Scott ldquoNeunstor propagation on a tunnel diode loadedtransmission linerdquo Proceedings of the IEEE vol 51 pp 240ndash2491963
[11] J Nagumo S Yoshizawa and S Arimoto ldquoB is table transmis-sion linesrdquo IEEETransactions onCircuitTheory vol 12 pp 400ndash412 1965
[12] J McKean ldquoNagumorsquos equationrdquo Advances in Mathematics vol4 pp 209ndash223 1970
[13] S Hastings ldquoThe existence of periodic solutions to Nagumorsquosequationrdquo The Quarterly Journal of Mathematics vol 25 pp369ndash378 1974
[14] C Zhi-Xiong andG Ben-Yu ldquoAnalytic solutions of theNagumoequationrdquo IMA Journal of Applied Mathematics vol 48 no 2pp 107ndash115 1992
[15] M B A Mansour ldquoOn the sharp front-type solution of theNagumo equation with nonlinear diffusion and convectionrdquoPRAMANA Journal of Physics vol 80 no 3 pp 533ndash538 2013
[16] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID853127 9 pages 2013
[17] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004
[18] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independentmdashpart IIrdquo Geophysical Journal Interna-tional vol 13 no 5 pp 529ndash539 1967
[19] A Cloot and J F Botha ldquoA generalised groundwater flow equa-tion using the concept of non-integer order derivativesrdquoWaterSA vol 32 no 1 pp 1ndash7 2006
Abstract and Applied Analysis 7
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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
[20] D A Benson S W Wheatcraft and M M Meerschaert ldquoApp-lication of a fractional advection-dispersion equationrdquo WaterResources Research vol 36 no 6 pp 1403ndash1412 2000
[21] G Wu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013
[22] MMatinfar andM Ghanbari ldquoThe application of themodifiedvariational iteration method on the generalized Fisherrsquos equa-tionrdquo Journal of Applied Mathematics and Computing vol 31no 1-2 pp 165ndash175 2009
[23] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008
[24] J S Duan R Rach D Baleanu and A M Wazwaz ldquoA reviewof the Adomian decomposition method and its applications tofractional differential equationsrdquoCommunications in FractionalCalculus vol 3 no 2 pp 73ndash99 2012
[25] M Javidi and M A Raji ldquoCombinaison of Laplace transformand homotopy perturbation method to solve the parabolicpartial differential equationsrdquo Communications in FractionalCalculus vol 3 pp 10ndash19 2012
[26] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-IntersciencePublication John Wiley amp Sons New York NY USA 1993
[27] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002
[28] A Anatoly J Juan and M S Hari Theory and Applicationof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006
[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Complexity Nonlin-earity and Chaos World Scientific 2012
[30] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherland 2006
[31] M Madani M Fathizadeh Y Khan and A Yildirim ldquoOn thecoupling of the homotopy perturbation method and LaplacetransformationrdquoMathematical andComputerModelling vol 53no 9-10 pp 1937ndash1945 2011
[32] Y Khan and H Latifizadeh ldquoApplication of new optimalhomotopy perturbation and Adomian decomposition methodsto the MHD non-Newtonian fluid flow over a stretching sheetrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 24 no 1 pp 124ndash136 2014
[33] D Baleanu O G Mustafa and R P Agarwal ldquoOn the solutionset for a class of sequential fractional differential equationsrdquoJournal of Physics A Mathematical and Theoretical vol 43 no38 Article ID 385209 7 pages 2010
[34] D Baleanu H Mohammadi and S Rezapour ldquoSome existenceresults on nonlinear fractional differential equationsrdquo Philo-sophical Transactions of the Royal Society of London A vol 371no 1990 Article ID 20120144 2013
[35] J-F Cheng and Y-M Chu ldquoFractional difference equationswith real variablerdquo Abstract and Applied Analysis vol 2012Article ID 918529 24 pages 2012
[36] F Jarad T Abdeljawad D Baleanu and K Bicen ldquoOn thestability of some discrete fractional nonautonomous systemsrdquo
Abstract and Applied Analysis vol 2012 Article ID 476581 9pages 2012
[37] TAbdeljawad andD Baleanu ldquoCaputo q-fractional initial valueproblems and a q-analogueMittag-Leffler functionrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 16no 12 pp 4682ndash4688 2011
[38] K Diethelm E Scalas and D Baleanu Fractional CalculusSeries on Complexity of Series on Complexity Nonlinearity andChaos vol 3 World Scientific Publishing 2012
[39] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science amp Climatic Change vol 3 no 115 p 2157 2012
[40] A Atangana and A Secer ldquoThe time-fractional coupled-Kort-eweg-de-Vries equationsrdquo Abstract and Applied Analysis vol2013 Article ID 947986 8 pages 2013
[41] A Atangana and E Alabaraoye ldquoSolving system of fractionalpartial differential equations arisen in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationrdquo Advances in Difference Equations vol 2013article 94 14 pages 2013
[42] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters ampMathematics with Applications vol 61 no 8 pp 1963ndash1967 2011
[43] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 no 2 pp 59ndash752011
[44] G Wu and D Baleanu ldquoVariational iteration method for frac-tional calculusmdasha universal approach by Laplace transformrdquoAdvances in Difference Equations vol 2013 article 18 2013
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
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of