research article a local fractional variational iteration...

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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 202650 6 pageshttpdxdoiorg1011552013202650

Research ArticleA Local Fractional Variational Iteration Method for LaplaceEquation within Local Fractional Operators

Yong-Ju Yang1 Dumitru Baleanu234 and Xiao-Jun Yang56

1 School of Mathematics and Statistics Nanyang Normal University Nanyang 4730 61 China2Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University 06530 Ankara Turkey3 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz University PO Box 80204Jeddah 21589 Saudi Arabia

4 Institute of Space Sciences Magurele RO-077125 Bucharest Romania5 Institute of Software Science Zhengzhou Normal University Zhengzhou 450044 China6Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou 221008 China

Correspondence should be addressed to Xiao-Jun Yang dyangxiaojun163com

Received 15 November 2012 Accepted 27 February 2013

Academic Editor Syed Tauseef Mohyud-Din

Copyright copy 2013 Yong-Ju Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The local fractional variational iteration method for local fractional Laplace equation is investigated in this paper The operatorsare described in the sense of local fractional operators The obtained results reveal that the method is very effective

1 Introduction

As it is known the partial differential equations [1 2] andfractional differential equations [3ndash5] appear in many areasof science and engineering As a result various kinds ofanalytical methods and numerical methods were developed[6ndash8] For example the variational iteration method [9ndash15] was applied to solve differential equations [16ndash18] inte-gral equations [19] and numerous applications to differ-ent nonlinear equations in physics and engineering Alsothe fractional variational iteration method [20ndash23] andthe fractional complex transform [24ndash27] were discussedrecently The efficient techniques have successfully addresseda wide class of nonlinear problems for differential equa-tions see [28ndash36] and the references therein We noticethat the developed methods are very convenient efficientand accurate

Recently the local fractional variational iterationmethod[37] is derived from local fractional operators [38ndash48] Themethod which accurately computes the solutions in a localfractional series form or in an exact form presents interest

to applied sciences for problems where the other methodscannot be applied properly

In this paper we investigate the application of localfractional variational iteration method for solving the localfractional Laplace equations [49] with the different fractalconditions

This paper is organized as followsIn Section 2 the basic mathematical tools are reviewed

Section 3 presents briefly the local fractional variationaliteration method based on local fractional variational forfractal Lagrange multipliers Section 4 presents solutions tothe local fractional Laplace equations with differential fractalconditions

2 Mathematical Fundamentals

In this section we present few mathematical fundamentalsof local fractional calculus and introduce the basicnotions of local fractional continuity local fractionalderivative and local fractional integral of nondifferentialfunctions

2 Abstract and Applied Analysis

21 Local Fractional Continuity

Lemma 1 (see [42]) Let 119865 be a subset of the real line and afractal If 119891 (119865 119889) rarr (Ω

1015840 1198891015840) is a bi-Lipschitz mapping then

there is for constants 120588 120591 gt 0 and 119865 sub 119877

120588119904119867119904

(119865) le 119867119904

(119891 (119865)) le 120591119904119867119904

(119865) (1)

such that for all 1199091 1199092

isin 119865

12058812057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572

le1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572

(2)

As a direct result of Lemma 1 one has [42]1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572 (3)

such that1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 lt 120576120572

(4)

where 120572 is fractal dimension of 119865Suppose that there is [38ndash43]

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (5)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then 119891(119909) is called

local fractional continuous at 119909 = 1199090and it is denoted by

lim119909rarr1199090

119891 (119909) = 119891 (1199090) (6)

Suppose that the function 119891(119909) is satisfied the condition (5)for119909 isin (119886 119887) and hence it is called a local fractional continuouson the interval (119886 119887) denoted by

119891 (119909) isin 119862120572

(119886 119887) (7)

22 Local Fractional Derivatives and Integrals Suppose that119891(119909) isin 119862

120572(119886 119887) then the local fractional derivative of 119891(119909)

of order 120572 at 119909 = 1199090is given by [37ndash43]

119863119909

(120572)119891 (1199090) = 119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(8)

where Δ120572

(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

There is [38ndash40]

119891 (119909) isin 119863119909

(120572)

(119886 119887) (9)

if

119891(120572)

(119909) = 119863119909

(120572)119891 (119909) (10)

for any 119909 isin (119886 119887)Local fractional derivative of high order is written in the

form [38ndash40]

119891(119896120572)

(119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

119863119909

(120572)sdot sdot sdot 119863119909

(120572)119891 (119909)

(11)

and local fractional partial derivative of high order is [38ndash40]

120597119896120572

120597119909119896120572119891 (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597120572

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909)

(12)

Let a function 119891(119909) satisfy the condition (7) Localfractional integral of 119891(119909) of order 120572 in the interval [119886 119887] isgiven by [37ndash43]

119886119868119887

(120572)119891 (119909) =

1

Γ (1 + 120572)int

119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) (Δ119905119895)120572

(13)

where Δ119905119895

= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1

] 119895 = 0 119873 minus 1 1199050

= 119886 119905119873

= 119887 is a partitionof the interval [119886 119887] For other definition of local fractionalderivative see [44ndash48]

There exists [38ndash40]

119891 (119909) isin 119868119909

(120572)

(119886 119887) (14)

if

119891(120572)

(119909) =119886119868119909

(120572)119891 (119909) (15)

for any 119909 isin (119886 119887)Local fractionalmultiple integrals of119891(119909) is written in the

form [40]

1199090119868119909

(119896120572)119891 (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

1199090119868119909

(120572)sdot sdot sdot1199090

119868119909

(120572)119891 (119909)

(16)

if (7) is valid for 119909 isin (119886 119887)

3 Local Fractional VariationalIteration Method

In this section we introduce the local fractional variationaliterationmethod derived from the local fractional variationalapproach for fractal Lagrange multipliers [40]

Let us consider the local fractional variational approachin the one-dimensional case through the following localfractional functional which reads [40]

119868 (119910) =119886119868119887

(120572)119891 (119909 119910 (119909) 119910

(120572)

(119909)) (17)

where 119910(120572)

(119909) is taken in local fractional differential operatorand 119886 le 119909 le 119887

The local fractional variational derivative is given by [40]

120575120572

119868 =119886119868119887

(120572)(

120597119891

120597119910minus

119889120572

119889119909120572(

120597119891

120597119910(120572))) 120578 (119909) (18)

where 120575120572 is local fractional variation signal and 120578(119886) = 120578(119887) =

0

Abstract and Applied Analysis 3

The nonlinear local fractional equation reads as

119871120572

119906 + 119873120572

119906 = 0 (19)

where 119871120572and 119873

120572are linear and nonlinear local fractional

operators respectivelyLocal fractional variational iteration algorithm can be

written as [37]

119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119906119899

(119904)] (20)

Here we can construct a correction functional as follows [37]

119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119899

(119904)] (21)

where 119899is considered as a restricted local fractional varia-

tion and 120585120572 is a fractal Lagrange multiplier that is 120575

120572119899

= 0

[37 40]Having determined the fractal Lagrangian multipliers

the successive approximations 119906119899+1

119899 ge 0 of the solution119906 will be readily obtained upon using any selective fractalfunction 119906

0 Consequently we have the solution

119906 = lim119899rarrinfin

119906119899 (22)

Here this technology is called the local fractional variationalmethod [37] We notice that the classical variation is recov-ered in case of local fractional variation when the fractaldimension is equal to 1 Besides the convergence of localfractional variational process and its algorithms were takeninto account [37]

4 Solutions to Local Fractional LaplaceEquation in Fractal Timespace

The local fractional Laplace equation (see [38ndash40] and thereferences therein) is one of the important differential equa-tions with local fractional derivatives In the following weconsider solutions to local fractional Laplace equations infractal timespace

Case 1 Let us start with local fractional Laplace equationgiven by

1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (23)

and subject to the fractal value conditions

120597120572

120597119905120572119879 (119909 0) = 0 119879 (119909 0) = minus119864

120572(119909120572

) (24)

A corrected local fractional functional for (24) reads as

119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(25)

Taking into account the properties of the local fractionalderivative we obtain

120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905)

+ 120575120572

0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(26)

Hence from (25)-(26) we get

120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

(119909 119905)

10038161003816100381610038161003816100381610038161003816120591=119905

minus [120582120572

Γ (1 + 120572)]

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

minus (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

10038161003816100381610038161003816100381610038161003816120591=119905

minus (120582120572

Γ (1 + 120572))

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

+ (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 0

(27)

As a result from (27) we can derive

(120582120572

Γ (1 + 120572))

(2120572)

= 0120582120572

Γ (1 + 120572)

10038161003816100381610038161003816100381610038161003816120591=119905

= 0

(120582120572

Γ (1 + 120572))

(120572)

= 1

(28)

We have 120582 = 120591 minus 119905 such that the fractal Lagrange multiplierreads as

120582120572

Γ (1 + 120572)=

(120591 minus 119905)120572

Γ (1 + 120572) (29)

From (24) we take the initial value which reads as

1199060

(119909 119905) = minus119864120572

(119909120572

) (30)

By using (25) we structure a local fractional iteration proce-dure as

119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(31)


Hence we can derive the first approximation term as

1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119864120572

(119909120572

) +0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus119864120572

(119909120572

))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

(32)

The second approximation can be calculated in the similarway which is

1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199052120572

119864120572

(119909120572

)

Γ (1 + 2120572))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572))

(33)

Proceeding in this manner we get

119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572)) (34)

Thus the final solution reads as

119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572))

= minus119864120572

(119909120572

) cos120572

(119905120572

)

(35)

Case 2 Consider the local fractional Laplace equation as

1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (36)

subject to fractal value conditions given by

120597120572

120597119905120572119879 (119909 0) = minus119864

120572(119909120572

) 119879 (119909 0) = 0 (37)

Now we can structure the same local fractional iterationprocedure (31)

By using (36)-(37) we take an initial value as

1199060

(119909 119905) = minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572) (38)

The first approximation term reads as

1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus

119905120572

119864120572

(119909120572

)

Γ (1 + 120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

(39)

In the same manner the second approximation is given by

1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)minus

1199055120572

119864120572

(119909120572

)

Γ (1 + 5120572)

(40)

Finally we can obtain the local fractional series solution asfollows

119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572)) (41)

Thus the expression of the final solution is given by

119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119894=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572))

= minus119864120572

(119909120572

) sin120572

(119905120572

)

(42)

As is known the Mittag-Leffler function in fractal spacecan be written in the form

1003816100381610038161003816119864120572 (119909120572

) minus 119864120572

(119909120572

0)1003816100381610038161003816 le 119864120572

(119909120572

0)

1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816

120572

lt 120576120572

1003816100381610038161003816sin120572 (119905120572

) minus sin120572

(119905120572

0)1003816100381610038161003816 lt

1003816100381610038161003816cos120572 (119909120572

0)1003816100381610038161003816

1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816

120572

lt 120576120572

(43)

Hence the fractal dimensions of both 119864120572

(119909120572

) andcos120572

(119905120572

) are equal to 120572


5 Conclusions

Local fractional calculus is set up on fractals and the localfractional variational iteration method is derived from localfractional calculus This new technique is efficient for theapplied scientists to process these differential and integralequations with the local fractional operators The variationaliteration method [9ndash19 27] is derived from fractional calcu-lus and classical calculus the fractional variational iterationmethod [20ndash22 27] is derived from the modified fractionalderivative while the local fractional variational iterationmethod [37] is derived from the local fractional calculus [37ndash43] Other methods for local fractional ordinary and partialdifferential equations were considered in [27]

In this paper two outstanding examples of applications ofthe local fractional variational iteration method to the localfractional Laplace equations are investigated in detail Thereliable obtained results are complementary with the onespresented in the literature

References

[1] R D Driver Ordinary and Delay Differential EquationsSpringer Berlin Germany 1977

[2] A M Wazwaz Partial Differential Equations Methods andApplications Balkema Publishers Leiden The Netherlands2002

[3] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[4] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Publishing Singapore 2000

[5] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

[6] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Publishers Boston MassUSA 1994

[7] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Publish-ing 2012

[8] A-M Wazwaz A First Course in Integral Equations WorldScientific Publishing Singapore 1997

[9] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[10] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[11] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[12] J H He ldquoComment on variational iteration method for frac-tional calculus using Hersquos polynomialsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 964974 2 pages 2012

[13] S T Mohyud-Din M A Noor K I Noor and M MHosseini ldquoSolution of singular equation by Hersquos variationaliteration methodrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 2 pp 81ndash86 2010

[14] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[15] J H He G C Wu and F Austin ldquoThe variational iterationmethod which should be followedrdquoNonlinear Science Letters Avol 1 no 1 pp 1ndash30 2010

[16] M Tatari and M Dehghan ldquoHersquos variational iteration methodfor computing a control parameter in a semi-linear inverseparabolic equationrdquo Chaos Solitons amp Fractals vol 33 no 2pp 671ndash677 2007

[17] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[18] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[19] L Xu ldquoVariational iteration method for solving integral equa-tionsrdquo Computers amp Mathematics with Applications vol 54 no7-8 pp 1071ndash1078 2007

[20] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A vol 375 no 38 pp 3362ndash3364 2011

[21] G-c Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A vol 374 no 25pp 2506ndash2509 2010

[22] Y Khan N Faraz A Yildirim and Q Wu ldquoFractionalvariational iteration method for fractional initial-boundaryvalue problems arising in the application of nonlinear sciencerdquoComputers amp Mathematics with Applications vol 62 no 5 pp2273ndash2278 2011

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

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

[25] Z B Li W H Zhu and J H He ldquoExact solutions of time-fractional heat conduction equation by the fractional complextransformrdquoThermal Science vol 16 no 2 pp 335ndash338 2012

[26] Q L Wang J H He and Z B Li ldquoFractional model for heatconduction in polar bear hairsrdquo Thermal Science vol 16 no 2pp 339ndash342 2012

[27] J H He ldquoAsymptotic methods for solitary solutions andcompactonsrdquo Abstract and Applied Analysis vol 2012 ArticleID 916793 130 pages 2012

[28] S T Mohyud-Din A Yildirim and G Demirli ldquoAnalyticalsolution of wave system in 119877

119899 with coupling controllersrdquoInternational Journal of Numerical Methods for Heat amp FluidFlow vol 21 no 2 pp 198ndash205 2011

[29] S T Mohyud-Din A Yıldırım and S Sarıaydın ldquoNumericalsoliton solution of the Kaup-Kupershmidt equationrdquo Interna-tional Journal of Numerical Methods for Heat amp Fluid Flow vol21 no 3-4 pp 272ndash281 2011

[30] S TMohyud-Din A Yıldırım and S A Sezer ldquoNumerical soli-ton solutions of improved Boussinesq equationrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 21 no6-7 pp 822ndash827 2011

[31] M A Abdou A A Soliman and S T El-Basyony ldquoNewapplication of Exp-function method for improved Boussinesqequationrdquo Physics Letters A vol 369 no 5-6 pp 469ndash475 2007

[32] S A El-Wakil M A Madkour and M A Abdou ldquoApplicationof Exp-functionmethod for nonlinear evolution equations withvariable coefficientsrdquo Physics Letters A vol 369 no 1-2 pp 62ndash69 2007


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

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

[35] W-X Ma H Wu and J He ldquoPartial differential equations pos-sessing Frobenius integrable decompositionsrdquo Physics Letters Avol 364 no 1 pp 29ndash32 2007

[36] W X Ma and Y You ldquoRational solutions of the Toda latticeequation in Casoratian formrdquo Chaos Solitons and Fractals vol22 no 2 pp 395ndash406 2004

[37] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience 2012

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

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

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

[41] W P Zhong X J Yang and F Gao ldquoA Cauchy problem forsome local fractional abstract differential equation with fractalconditionsrdquo Journal of Applied Functional Analysis vol 8 no 1pp 92ndash99 2013

[42] M S Hu R P Agarwal and X J Yang ldquoLocal fractional Fourierseries with application to wave equation in fractal vibratingstringrdquo Abstract and Applied Analysis vol 2012 Article ID567401 15 pages 2012

[43] M S Hu D Baleanu and X J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

[44] K M Kolwankar and A D Gangal ldquoFractional differentiabilityof nowhere differentiable functions and dimensionsrdquo Chaosvol 6 no 4 pp 505ndash513 1996

[45] J H He ldquoA new fractal derivationrdquoThermal Science vol 15 ppS145ndashS147 2011

[46] W Chen ldquoTime-space fabric underlying anomalous diffusionrdquoChaos Solitons amp Fractals vol 28 no 4 pp 923ndash929 2006

[47] J Fan and J H He ldquoBiomimic design of multi-scale fabric withefficient heat transfer propertyrdquo Thermal Science vol 16 no 5pp 1349ndash1352 2012

[48] J Fan and JHHe ldquoFractal derivativemodel for air permeabilityin hierarchic porous mediardquo Abstract and Applied Analysis vol2012 Article ID 354701 7 pages 2012

[49] A LiangpromandKNonlaopon ldquoOn the convolution equationrelated to the diamond Klein-Gordon operatorrdquo Abstract andApplied Analysis vol 2011 Article ID 908491 16 pages 2011

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


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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


21 Local Fractional Continuity

Lemma 1 (see [42]) Let 119865 be a subset of the real line and afractal If 119891 (119865 119889) rarr (Ω

1015840 1198891015840) is a bi-Lipschitz mapping then

there is for constants 120588 120591 gt 0 and 119865 sub 119877

120588119904119867119904

(119865) le 119867119904

(119891 (119865)) le 120591119904119867119904

(119865) (1)

such that for all 1199091 1199092

isin 119865

12058812057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572

le1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572

(2)

As a direct result of Lemma 1 one has [42]1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816

120572 (3)

such that1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 lt 120576120572

(4)

where 120572 is fractal dimension of 119865Suppose that there is [38ndash43]

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (5)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then 119891(119909) is called

local fractional continuous at 119909 = 1199090and it is denoted by

lim119909rarr1199090

119891 (119909) = 119891 (1199090) (6)

Suppose that the function 119891(119909) is satisfied the condition (5)for119909 isin (119886 119887) and hence it is called a local fractional continuouson the interval (119886 119887) denoted by

119891 (119909) isin 119862120572

(119886 119887) (7)

22 Local Fractional Derivatives and Integrals Suppose that119891(119909) isin 119862

120572(119886 119887) then the local fractional derivative of 119891(119909)

of order 120572 at 119909 = 1199090is given by [37ndash43]

119863119909

(120572)119891 (1199090) = 119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(8)

where Δ120572

(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

There is [38ndash40]

119891 (119909) isin 119863119909

(120572)

(119886 119887) (9)

if

119891(120572)

(119909) = 119863119909

(120572)119891 (119909) (10)

for any 119909 isin (119886 119887)Local fractional derivative of high order is written in the

form [38ndash40]

119891(119896120572)

(119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

119863119909

(120572)sdot sdot sdot 119863119909

(120572)119891 (119909)

(11)

and local fractional partial derivative of high order is [38ndash40]

120597119896120572

120597119909119896120572119891 (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120597120572

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909)

(12)

Let a function 119891(119909) satisfy the condition (7) Localfractional integral of 119891(119909) of order 120572 in the interval [119886 119887] isgiven by [37ndash43]

119886119868119887

(120572)119891 (119909) =

1

Γ (1 + 120572)int

119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) (Δ119905119895)120572

(13)

where Δ119905119895

= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1

] 119895 = 0 119873 minus 1 1199050

= 119886 119905119873

= 119887 is a partitionof the interval [119886 119887] For other definition of local fractionalderivative see [44ndash48]

There exists [38ndash40]

119891 (119909) isin 119868119909

(120572)

(119886 119887) (14)

if

119891(120572)

(119909) =119886119868119909

(120572)119891 (119909) (15)

for any 119909 isin (119886 119887)Local fractionalmultiple integrals of119891(119909) is written in the

form [40]

1199090119868119909

(119896120572)119891 (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

1199090119868119909

(120572)sdot sdot sdot1199090

119868119909

(120572)119891 (119909)

(16)

if (7) is valid for 119909 isin (119886 119887)

3 Local Fractional VariationalIteration Method

In this section we introduce the local fractional variationaliterationmethod derived from the local fractional variationalapproach for fractal Lagrange multipliers [40]

Let us consider the local fractional variational approachin the one-dimensional case through the following localfractional functional which reads [40]

119868 (119910) =119886119868119887

(120572)119891 (119909 119910 (119909) 119910

(120572)

(119909)) (17)

where 119910(120572)

(119909) is taken in local fractional differential operatorand 119886 le 119909 le 119887

The local fractional variational derivative is given by [40]

120575120572

119868 =119886119868119887

(120572)(

120597119891

120597119910minus

119889120572

119889119909120572(

120597119891

120597119910(120572))) 120578 (119909) (18)

where 120575120572 is local fractional variation signal and 120578(119886) = 120578(119887) =

0



119871120572

119906 + 119873120572

119906 = 0 (19)

where 119871120572and 119873



written as [37]

119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119906119899

(119904)] (20)


119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119899

(119904)] (21)



120572119899

= 0






119906119899 (22)





1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (23)


120597120572

120597119905120572119879 (119909 0) = 0 119879 (119909 0) = minus119864

120572(119909120572

) (24)


119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(25)


120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905)

+ 120575120572

0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(26)


120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

(119909 119905)

10038161003816100381610038161003816100381610038161003816120591=119905

minus [120582120572

Γ (1 + 120572)]

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

minus (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

10038161003816100381610038161003816100381610038161003816120591=119905

minus (120582120572

Γ (1 + 120572))

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

+ (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 0

(27)


(120582120572

Γ (1 + 120572))

(2120572)

= 0120582120572

Γ (1 + 120572)

10038161003816100381610038161003816100381610038161003816120591=119905

= 0

(120582120572

Γ (1 + 120572))

(120572)

= 1

(28)


120582120572

Γ (1 + 120572)=

(120591 minus 119905)120572

Γ (1 + 120572) (29)


1199060

(119909 119905) = minus119864120572

(119909120572

) (30)


119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(31)



1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119864120572

(119909120572

) +0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus119864120572

(119909120572

))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

(32)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199052120572

119864120572

(119909120572

)

Γ (1 + 2120572))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572))

(33)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572)) (34)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572))

= minus119864120572

(119909120572

) cos120572

(119905120572

)

(35)


1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (36)


120597120572

120597119905120572119879 (119909 0) = minus119864

120572(119909120572

) 119879 (119909 0) = 0 (37)



1199060

(119909 119905) = minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572) (38)


1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus

119905120572

119864120572

(119909120572

)

Γ (1 + 120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

(39)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)minus

1199055120572

119864120572

(119909120572

)

Γ (1 + 5120572)

(40)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572)) (41)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119894=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572))

= minus119864120572

(119909120572

) sin120572

(119905120572

)

(42)


1003816100381610038161003816119864120572 (119909120572

) minus 119864120572

(119909120572

0)1003816100381610038161003816 le 119864120572

(119909120572

0)

1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816

120572

lt 120576120572

1003816100381610038161003816sin120572 (119905120572

) minus sin120572

(119905120572

0)1003816100381610038161003816 lt

1003816100381610038161003816cos120572 (119909120572

0)1003816100381610038161003816

1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816

120572

lt 120576120572

(43)


(119909120572

) andcos120572

(119905120572



5 Conclusions



References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871120572

119906 + 119873120572

119906 = 0 (19)

where 119871120572and 119873



written as [37]

119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119906119899

(119904)] (20)


119906119899+1

(119905) = 119906119899

(119905) +1199050

119868119905

(120572)120585120572

[119871120572

119906119899

(119904) + 119873120572

119899

(119904)] (21)



120572119899

= 0






119906119899 (22)





1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (23)


120597120572

120597119905120572119879 (119909 0) = 0 119879 (119909 0) = minus119864

120572(119909120572

) (24)


119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(25)


120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905)

+ 120575120572

0119868119905

(120572)

120582120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(26)


120575120572

119906119899+1

(119909 119905)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

(119909 119905)

10038161003816100381610038161003816100381610038161003816120591=119905

minus [120582120572

Γ (1 + 120572)]

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

minus (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 120575120572

119906119899

(119909 119905) +120582120572

Γ (1 + 120572)120575120572

119906119899

(120572)

10038161003816100381610038161003816100381610038161003816120591=119905

minus (120582120572

Γ (1 + 120572))

(120572)

120575120572

119906119899

(119909 119905)

1003816100381610038161003816100381610038161003816100381610038161003816120591=119905

+ (120575120572

119906119899

(119909 120591))0119868119905

(120572)(

120582120572

Γ (1 + 120572))

(2120572)

= 0

(27)


(120582120572

Γ (1 + 120572))

(2120572)

= 0120582120572

Γ (1 + 120572)

10038161003816100381610038161003816100381610038161003816120591=119905

= 0

(120582120572

Γ (1 + 120572))

(120572)

= 1

(28)


120582120572

Γ (1 + 120572)=

(120591 minus 119905)120572

Γ (1 + 120572) (29)


1199060

(119909 119905) = minus119864120572

(119909120572

) (30)


119906119899+1

(119909 119905)

= 119906119899

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

119879119899

(119909 120591)

1205971205912120572+

1205972120572

119879119899

(119909 120591)

1205971199092120572)

(31)



1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119864120572

(119909120572

) +0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus119864120572

(119909120572

))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

(32)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199052120572

119864120572

(119909120572

)

Γ (1 + 2120572))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572))

(33)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572)) (34)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572))

= minus119864120572

(119909120572

) cos120572

(119905120572

)

(35)


1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (36)


120597120572

120597119905120572119879 (119909 0) = minus119864

120572(119909120572

) 119879 (119909 0) = 0 (37)



1199060

(119909 119905) = minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572) (38)


1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus

119905120572

119864120572

(119909120572

)

Γ (1 + 120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

(39)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)minus

1199055120572

119864120572

(119909120572

)

Γ (1 + 5120572)

(40)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572)) (41)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119894=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572))

= minus119864120572

(119909120572

) sin120572

(119905120572

)

(42)


1003816100381610038161003816119864120572 (119909120572

) minus 119864120572

(119909120572

0)1003816100381610038161003816 le 119864120572

(119909120572

0)

1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816

120572

lt 120576120572

1003816100381610038161003816sin120572 (119905120572

) minus sin120572

(119905120572

0)1003816100381610038161003816 lt

1003816100381610038161003816cos120572 (119909120572

0)1003816100381610038161003816

1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816

120572

lt 120576120572

(43)


(119909120572

) andcos120572

(119905120572



5 Conclusions



References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119864120572

(119909120572

) +0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus119864120572

(119909120572

))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

(32)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572))

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199052120572

119864120572

(119909120572

)

Γ (1 + 2120572))

= 119864120572

(119909120572

) (minus1 +1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572))

(33)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572)) (34)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119896=0

(minus1)119896 119905

2119896120572

Γ (1 + 2119896120572))

= minus119864120572

(119909120572

) cos120572

(119905120572

)

(35)


1205972120572

119879 (119909 119905)

1205971199052120572+

1205972120572

119879 (119909 119905)

1205971199092120572= 0 (36)


120597120572

120597119905120572119879 (119909 0) = minus119864

120572(119909120572

) 119879 (119909 0) = 0 (37)



1199060

(119909 119905) = minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572) (38)


1199061

(119909 119905)

= 1199060

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198790

(119909 120591)

1205971205912120572+

1205972120572

1198790

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(minus

119905120572

119864120572

(119909120572

)

Γ (1 + 120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

(39)


1199062

(119909 119905)

= 1199061

(119909 119905)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1205972120572

1198791

(119909 120591)

1205971205912120572+

1205972120572

1198791

(119909 120591)

1205971199092120572)

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)

+0119868119905

(120572)

(120591 minus 119905)120572

Γ (1 + 120572)(

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572))

= minus119905120572

119864120572

(119909120572

)

Γ (1 + 120572)+

1199053120572

119864120572

(119909120572

)

Γ (1 + 3120572)minus

1199055120572

119864120572

(119909120572

)

Γ (1 + 5120572)

(40)


119906119899

(119909 119905) = 119864120572

(119909120572

) (

119899

sum

119896=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572)) (41)


119906 (119909 119905) = lim119899rarrinfin

119906119899

(119909 119905)

= 119864120572

(119909120572

) (

infin

sum

119894=0

(minus1)119896 119905

(2119896+1)120572

Γ (1 + (2119896 + 1) 120572))

= minus119864120572

(119909120572

) sin120572

(119905120572

)

(42)


1003816100381610038161003816119864120572 (119909120572

) minus 119864120572

(119909120572

0)1003816100381610038161003816 le 119864120572

(119909120572

0)

1003816100381610038161003816119909 minus 1199090

1003816100381610038161003816

120572

lt 120576120572

1003816100381610038161003816sin120572 (119905120572

) minus sin120572

(119905120572

0)1003816100381610038161003816 lt

1003816100381610038161003816cos120572 (119909120572

0)1003816100381610038161003816

1003816100381610038161003816119905 minus 1199050

1003816100381610038161003816

120572

lt 120576120572

(43)


(119909120572

) andcos120572

(119905120572



5 Conclusions



References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusions



References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a local fractional variational iteration...

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