Research Article On the Fractional Nagumo Equation with ... · extend the nonlinear Nagumo equation using the concept of fractionalderivative order. It is something very di cult to

$Page 1: Research Article On the Fractional Nagumo Equation with ... · extend the nonlinear Nagumo equation using the concept of fractionalderivative order. It is something very di cult to$
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


(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


(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)


+

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


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



2 minus228554 times 1018












minus5

0 25 25 25 251 minus298096 times 10

12
















5

0 25 25 25 251 minus350932 times 10

12












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


05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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


(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


(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


(119909 0)

minus L [119871 (119908119899

(119909 119905))

+119873 (119908119899

(119909 119905)) + 119896 (119909 119905) ]]]

(11)





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)


= 119906 (119909 0)

+1


119905

0

(119905 minus 120591)1minus120572

[minus120573119906120597119906

120597119909+

120597

120597119909[119886119906

120597119906

120597119909]

+ 120574119906 (1 minus 119906) (119906 minus 1)] 119889120591


119906 (119909 119905) =

infin

sum

119899=0

119901119899

119906119899

(119909 119905) (14)


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)


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)


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)


+

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))


119906 (119909 119905) = 1199060

(119909 119905) + 1199061

(119909 119905) + 1199062




120582 (119904) = minus1

119904120572(21)


(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)


1199060

(119909 119905) = 1199092


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


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)


119906 (119909 119905) = 1199062

(119909 119905) (26)


5 Numerical Results




119909 119905HDM

120572 = 025

NDVIM120572 = 025

HDM120572 = 09

NDVIM120572 = 09

minus10

0 100 100 100 1001 minus114275 times 10

18
















minus5

0 25 25 25 251 minus298096 times 10

12
















5

0 25 25 25 251 minus350932 times 10

12












05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5





05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5



6 Conclusion



05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


05

10

Distance0

5

10

Time

0

05

Distanc

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













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)


= 119906 (119909 0)

+1


119905

0

(119905 minus 120591)1minus120572

[minus120573119906120597119906

120597119909+

120597

120597119909[119886119906

120597119906

120597119909]

+ 120574119906 (1 minus 119906) (119906 minus 1)] 119889120591


119906 (119909 119905) =

infin

sum

119899=0

119901119899

119906119899

(119909 119905) (14)


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)


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)


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)


+

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))


119906 (119909 119905) = 1199060

(119909 119905) + 1199061

(119909 119905) + 1199062




120582 (119904) = minus1

119904120572(21)


(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)


1199060

(119909 119905) = 1199092


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


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)


119906 (119909 119905) = 1199062

(119909 119905) (26)


5 Numerical Results




119909 119905HDM

120572 = 025

NDVIM120572 = 025

HDM120572 = 09

NDVIM120572 = 09

minus10

0 100 100 100 1001 minus114275 times 10

18
















minus5

0 25 25 25 251 minus298096 times 10

12
















5

0 25 25 25 251 minus350932 times 10

12












05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5





05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5



6 Conclusion



05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


05

10

Distance0

5

10

Time

0

05

Distanc

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

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))


119906 (119909 119905) = 1199060

(119909 119905) + 1199061

(119909 119905) + 1199062




120582 (119904) = minus1

119904120572(21)


(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)


1199060

(119909 119905) = 1199092


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


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)


119906 (119909 119905) = 1199062

(119909 119905) (26)


5 Numerical Results




119909 119905HDM

120572 = 025

NDVIM120572 = 025

HDM120572 = 09

NDVIM120572 = 09

minus10

0 100 100 100 1001 minus114275 times 10

18
















minus5

0 25 25 25 251 minus298096 times 10

12
















5

0 25 25 25 251 minus350932 times 10

12












05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5





05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5



6 Conclusion



05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


05

10

Distance0

5

10

Time

0

05

Distanc

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 119905HDM

120572 = 025

NDVIM120572 = 025

HDM120572 = 09

NDVIM120572 = 09

minus10

0 100 100 100 1001 minus114275 times 10

18
















minus5

0 25 25 25 251 minus298096 times 10

12
















5

0 25 25 25 251 minus350932 times 10

12












05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5





05

10

Distance0

5

10

Time

0

0

5

Time

times1018

minus1

minus2

minus3

minus4

u(xt)

minus10

minus5



6 Conclusion



05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


05

10

Distance0

5

10

Time

0

05

Distanc

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05

10

Distance0

5

10

Time

0

05

Distance

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5


05

10

Distance0

5

10

Time

0

05

Distanc

5

Time

times1017

minus1

minus2

minus3

u(xt)

minus10

minus5





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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