Research ArticleLocal Fractional Variational Iteration andDecomposition Methods for Wave Equation onCantor Sets within Local Fractional Operators

Dumitru Baleanu123 J A Tenreiro Machado4 Carlo Cattani5

Mihaela Cristina Baleanu6 and Xiao-Jun Yang7

1 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz UniversityPO Box 80204 Jeddah 21589 Saudi Arabia

2Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University06530 Ankara Turkey

3 Institute of Space Sciences Magurele Bucharest Romania4Department of Electrical Engineering Institute of Engineering Polytechnic of Porto Rua Dr AntonioBernardino de Almeida 431 4200-072 Porto Portugal

5 Department of Mathematics University of Salerno Via Ponte don Melillo Fisciano 84084 Salerno Italy6Mihail SadoveanuTheoretical High School District 2 Street Popa Lazar No 8 021586 Bucharest Romania7Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou CampusXuzhou Jiangsu 221008 China

Correspondence should be addressed to Xiao-Jun Yang dyangxiaojun163com

Received 9 November 2013 Accepted 8 December 2013 Published 2 January 2014

Academic Editor Ming Li

Copyright copy 2014 Dumitru Baleanu et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantorset The operators are taken in the local sense The results illustrate the significant features of the two methods which are both veryeffective and straightforward for solving the differential equations with local fractional derivative

1 Introduction

Many problems of physics and engineering are expressed byordinary and partial differential equations which are termedboundary value problems We can mention for examplethe wave the Laplace the Klein-Gordon the Schrodingerrsquosthe telegraph the Advection the Burgers the KdV theBoussinesq and the Fisher equations and others [1]

Recently the fractional calculus theory was recognized tobe a good tool for modeling complex problems demonstrat-ing its applicability in numerical scientific disciplines Bound-ary value problems for the fractional differential equationshave been the focus of several studies due to their frequentappearance in various areas such as fractional diffusion

and wave [2] fractional telegraph [3] fractional KdV [4]fractional Schrodinger [5] fractional evolution [6] fractionalNavier-Stokes [7] fractionalHeisenberg [8] fractional Klein-Gordon [9] and fractional Fisher equations [10]

Several analytical and numerical techniques were suc-cessfully applied to deal with differential equations frac-tional differential equations and local fractional differentialequations (see eg [1ndash36] and the references therein) Thetechniques include the heat-balance integral [11] the frac-tional Fourier [12] the fractional Laplace transform [12] theharmonic wavelet [13 14] the local fractional Fourier andLaplace transform [15] local fractional variational iteration[16 17] the local fractional decomposition [18] and thegeneralized local fractional Fourier transform [19] methods

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 535048 6 pageshttpdxdoiorg1011552014535048

2 Abstract and Applied Analysis

Recently the wave equation on Cantor sets (local frac-tional wave equation) was given by [35]

1205972120572119906 (119909 119905)

1205971199052120572minus 1198862120572 1205972120572119906 (119909 119905)

1205971199092120572= 0 (1)

where the operators are local fractional ones [16ndash19 35 36]Following (1) a wave equation on Cantor sets was

proposed as follows [36]

1205972120572119906 (119909 119905)

1205971199052120572minus

1199092120572

Γ (1 + 2120572)

1205972120572119906 (119909 119905)

1205971199092120572= 0 (2)

where 119906(119909 119905) is a fractal wave functionIn this paper our purpose is to compare the local

fractional variational iteration and decomposition methodsfor solving the local fractional differential equations Forillustrating the concepts we adopt one example for solving thewave equation on Cantor sets with local fractional operator

Bearing these ideas in mind the paper is organized asfollows In Section 2 we present basic definitions and providesome properties of local fractional derivative and integrationIn Section 3 we introduce the local fractional variationaliteration and the decomposition methods In Section 4 wediscuss one application Finally in Section 5 we outline themain conclusions

2 Mathematical Tools

We recall in this section the notations and some properties ofthe local fractional operators [15ndash19 35 36]

Definition 1 (see [15ndash19 35 36]) The function 119891(119909) is localfractional continuous if it is valid for

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (3)

where |119909 minus 1199090| lt 120575 for 120576 gt 0 and 120576 isin 119877

We notice that there are existence conditions of localfractional continuities that operating functions are right-hand and left-hand local fractional continuity Meanwhilethe right-hand local fractional continuity is equal to its left-hand local fractional continuity For more details see [35]

Following (4) we have [15ndash19 35 36]

1205881205721003816100381610038161003816119909 minus 119909

0

1003816100381610038161003816120572

le1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 le 1205811205721003816100381610038161003816119909 minus 119909

0

1003816100381610038161003816120572 (4)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 120581 120588 isin 119877

For a fractal set 119865 there is a fractal measure [35]

119867120572(119865 cap (119909 119909

0)) = (119909 minus 119909

0)120572

(5)

where 119891(119909) presents a bi-Lipschitz mapping with fractaldimension 120572 and119867

120572 denotes a Hausdorff dimensionWe verify that there is a measure

1198671(119865 cap (119909 119909

0)) = 119909 minus 119909

0(6)

in the case of 120572 = 1 and 119891(119909) is a Lipschitz mapping If 119865 isa Cantor set we have 119867ln 2 ln 3

(119865 cap (119909 1199090)) = (119909 minus 119909

0)ln 2 ln 3

with 120572 = ln 2 ln 3

Definition 2 (see [15ndash19 35 36]) The local fractional deriva-tive of 119891(119909) at 119909 = 119909

0is defined as [16ndash20]

119891(120572)

(1199090) =

119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(7)

where

Δ120572(119891 (119909) minus 119891 (119909

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

0)) (8)

We find that the existence condition for local fractionalderivative of 119891(119909) is that the right-hand local fractionalderivative is equal to the left-hand local fractional derivative(see eg [16 35] and the references therein)

Definition 3 (see [15ndash19 35 36]) A partition of the interval[119886 119887] is denoted as (119905

119895 119905119895+1

) 119895 = 0 119873 minus 1 1199050= 119886 and

119905119873

= 119887 with Δ119905119895= 119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

Local fractional integral of 119891(119909) in the interval [119886 119887] is givenby

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

(9)

If the functions are local fractional continuous then thelocal fractional derivatives and integrals exist That is tosay operating functions have nondifferentiable and fractalproperties (see [35] and the references therein)

Some properties of local fractional derivative and inte-grals are given in [35]

3 Analytical Methods

In order to illustrate two analytical methods we investigatethe nonlinear local fractional equation as follows

119871(119899)

120572119906 + 119877120572119906 = 0 (10)

where 119871(119899)

120572is linear local fractional operators respectively

with 119899 = 1 2 and 119877120572is linear local fractional operators of

order less than 119871(119899)

120572

31 Local Fractional Variational Iteration Method The localfractional variational iteration algorithm is given by [16 17]on the line of the formalism suggested in [35]

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

120582120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119906119899(119904) (119889119904)

120572

(11)

Here we can construct a correction functional as follows [1617]

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

120582120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119899(119904) (119889119904)

120572

(12)

Abstract and Applied Analysis 3

where 119899is considered as a restricted local fractional varia-

tion that is 120575120572119899= 0 (for more details see [35])

For 119899 = 2 we have

120582120572=

(119904 minus 119905)120572

Γ (1 + 120572) (13)

so that iteration is expressed as

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119906119899(119904) (119889119904)

120572

(14)

Finally the solution is

119906 (119909) = lim119899rarrinfin

119906119899(119909) (15)

32 Local Fractional Decomposition Method When 119871(119899)

120572in

(10) is a local fractional differential operator of order 2120572 wedenote it as

119871(119899)

120572= 119871(2120572)

119909119909=

1205972120572

1205971199092120572

119877120572119906 (119905) =

120597120572

120597119909120572119906 (119905) + 119891 (119905)

(16)

By defining the 119899-fold local fractional integral operator

119871(minus2)

120572119898(119904) =

0119868119909(120572)

0119868119909(120572)

119898(119904) (17)

we get

119871(minus2)

120572119871(2)

120572119906 (119904) = 119871

(minus2)

120572119877120572119906 (119904) (18)

Thus

119906 (119904) = 119903 (119909) + 119871(minus2)

120572119877120572119906 (119904) (19)

where the term 119903(119909) is to be determined from the fractalinitial conditions

Therefore we get the iterative formula as follows

119906 (119909) = 1199060(119909) + 119871

(minus2)

120572119877120572119906 (119904) (20)

with 1199060(119909) = 119903(119909)

Hence for 119899 ge 0 we have the following recurrencerelationship

119906119899+1

(119909) = 119871(minus2)

120572119877120572119906119899(119904)

1199060(119909) = 119903 (119909)

(21)

Finally the solution can be constructed as

119906 (119909) = lim119899rarrinfin

120601119899(119909) = lim

119899rarrinfin

infin

sum

119899=0

119906119899(119909) (22)

For more details see [18]

4 An Illustrative Example

In this section one example for wave equation is presented inorder to demonstrate the simplicity and the efficiency of theabove methods

In (2) we consider the following initial and boundaryconditions

120597120572119906 (119909 0)

120597119905120572= 0 119906 (119909 0) =

1199092120572

Γ (1 + 2120572) (23)

Using (14) we have the iterative formula

119906119899+1

(119909 119905)

= 119906119899(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572(119889119904)120572

(24)

where the initial value is given by

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572) (25)

Thus after computing (23) we obtain

1199061(119909 119905)

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199060(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199060(119909 119904)

1205971199092120572(119889119904)120572

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572))

1199062(119909 119905)

= 1199061(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199061(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199061(119909 119904)

1205971199092120572(119889119904)120572

= 1199061(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199042120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572))

4 Abstract and Applied Analysis

1199063(119909 119905)

= 1199062(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199062(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199062(119909 119904)

1205971199092120572(119889119904)120572

= 1199062(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

3

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

1199064(119909 119905)

= 1199063(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199063(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

1199092120572

Γ (1 + 2120572)

12059721205721199063(119909 119904)

1205971199092120572(119889119904)120572

= 1199063(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199056120572

Γ (1 + 6120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

4

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(26)

119906119899(119909 119905)

= 119906119899minus1

(119909 119905) +1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899minus1

(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899minus1

(119909 119904)

1205971199092120572(119889119904)120572

=1199092120572

Γ (1 + 2120572)

119899

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(27)

Hence from (27) we obtain the solution of (3) as

119906 (119909) = lim119899rarrinfin

119906119899(119909) =

1199092120572

Γ (1 + 2120572)cosh120572(119905120572) (28)

Here from (21) we get

119906119899+1

(119909 119905) =0119868119905(120572)

0119868119905(120572) 119909

2120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

(29)

Therefore from (29) we give the components as follows

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052120572

Γ (1 + 2120572)

1199062(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572)

1199063(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199053120572

Γ (1 + 6120572)

1199064(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199058120572

Γ (1 + 8120572)

119906119899(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

(30)

Consequently the exact solution is given by

119906 (119909 119905) = lim119899rarrinfin

infin

sum

119899=0

119906119899(119909 119905)

= lim119899rarrinfin

infin

sum

119899=0

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

=1199092120572

Γ (1 + 2120572)cosh120572(119909120572)

(31)

where

cosh120572(119909120572) =

infin

sum

119899=0

1199052119899120572

Γ (1 + 2119899120572) (32)

The solution of (2) for 120572 = ln 2 ln 3 is depicted in Figure 1

5 Conclusions

In this work we developed a comparison between thevariational iteration method and the decomposition methodwithin local fractional operators The two approaches con-stitute efficient tools to handle the approximation solutionsfor differential equations on Cantor sets with local fractionalderivative We notice that the fractional variational iterationmethod gives the several successive approximate formulasusing the iteration of the correction local fractional func-tional However the local fractional decomposition method

Abstract and Applied Analysis 5

25

2

15

1

05

01

05

0 002

0406

081

x

t

u(xt)

Figure 1 Graph of 119906(119909 119905) for 120572 = ln 2 ln 3

provides the components of the exact solution which is localfractional continuous function where these componentsare also local fractional continuous functions Both thevariational iteration method and the decomposition methodwithin local fractional operators provide the solution insuccessive componentsThemethods are structured to get thelocal fractional series solution which is a nondifferentiablefunction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

References

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[2] W R Schneider and W Wyss ldquoFractional diffusion and waveequationsrdquo Journal of Mathematical Physics vol 30 no 1 pp134ndash144 1989

[3] Z G Zhao and C P Li ldquoFractional differencefinite elementapproximations for the time-space fractional telegraph equa-tionrdquo Applied Mathematics and Computation vol 219 no 6 pp2975ndash2988 2012

[4] S Momani Z Odibat and A Alawneh ldquoVariational iterationmethod for solving the space- and time-fractional KdV equa-tionrdquo Numerical Methods for Partial Differential Equations vol24 no 1 pp 262ndash271 2008

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE vol 66 Article ID 056108 2002

[6] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[8] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[9] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoOn nonlinear fractional KleinGordon equationrdquo Signal Pro-cessing vol 91 no 3 pp 446ndash451 2011

[10] Z B Li W H Zhu and L L Huang ldquoApplication of fractionalvariational iteration method to time-fractional Fisher equa-tionrdquo Advanced Science Letters vol 10 no 1 pp 610ndash614 2012

[11] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[12] 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 BostonMass USA 2012

[13] C Cattani ldquoHarmonic wavelet solution of Poissonrsquos problemrdquoBalkan Journal of Geometry and Its Applications vol 13 no 1pp 27ndash37 2008

[14] C Cattani ldquoHarmonic wavelets towards the solution of nonlin-ear PDErdquo Computers amp Mathematics with Applications vol 50no 8-9 pp 1191ndash1210 2005

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

[16] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 article 89 2013

[18] X J Yang D Baleanu and W P Zhong ldquoApproximation solu-tions for diffusion equation on Cantor time-spacerdquo Proceedingof the Romanian Academy A vol 14 no 2 pp 127ndash133 2013

[19] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013

[20] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific Singapore 2012

[21] J Sabatier O P Agrawal and J T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer 2007

[22] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[23] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[24] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

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

[26] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[27] Y Chen Y Yan and K Zhang ldquoOn the local fractionalderivativerdquo Journal of Mathematical Analysis and Applicationsvol 362 no 1 pp 17ndash33 2010

[28] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Recently the wave equation on Cantor sets (local frac-tional wave equation) was given by [35]

1205972120572119906 (119909 119905)

1205971199052120572minus 1198862120572 1205972120572119906 (119909 119905)

1205971199092120572= 0 (1)

where the operators are local fractional ones [16ndash19 35 36]Following (1) a wave equation on Cantor sets was

proposed as follows [36]

1205972120572119906 (119909 119905)

1205971199052120572minus

1199092120572

Γ (1 + 2120572)

1205972120572119906 (119909 119905)

1205971199092120572= 0 (2)

where 119906(119909 119905) is a fractal wave functionIn this paper our purpose is to compare the local

fractional variational iteration and decomposition methodsfor solving the local fractional differential equations Forillustrating the concepts we adopt one example for solving thewave equation on Cantor sets with local fractional operator

Bearing these ideas in mind the paper is organized asfollows In Section 2 we present basic definitions and providesome properties of local fractional derivative and integrationIn Section 3 we introduce the local fractional variationaliteration and the decomposition methods In Section 4 wediscuss one application Finally in Section 5 we outline themain conclusions

2 Mathematical Tools

We recall in this section the notations and some properties ofthe local fractional operators [15ndash19 35 36]

Definition 1 (see [15ndash19 35 36]) The function 119891(119909) is localfractional continuous if it is valid for

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (3)

where |119909 minus 1199090| lt 120575 for 120576 gt 0 and 120576 isin 119877

We notice that there are existence conditions of localfractional continuities that operating functions are right-hand and left-hand local fractional continuity Meanwhilethe right-hand local fractional continuity is equal to its left-hand local fractional continuity For more details see [35]

Following (4) we have [15ndash19 35 36]

1205881205721003816100381610038161003816119909 minus 119909

0

1003816100381610038161003816120572

le1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 le 1205811205721003816100381610038161003816119909 minus 119909

0

1003816100381610038161003816120572 (4)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 120581 120588 isin 119877

For a fractal set 119865 there is a fractal measure [35]

119867120572(119865 cap (119909 119909

0)) = (119909 minus 119909

0)120572

(5)

where 119891(119909) presents a bi-Lipschitz mapping with fractaldimension 120572 and119867

120572 denotes a Hausdorff dimensionWe verify that there is a measure

1198671(119865 cap (119909 119909

0)) = 119909 minus 119909

0(6)

in the case of 120572 = 1 and 119891(119909) is a Lipschitz mapping If 119865 isa Cantor set we have 119867ln 2 ln 3

(119865 cap (119909 1199090)) = (119909 minus 119909

0)ln 2 ln 3

with 120572 = ln 2 ln 3

Definition 2 (see [15ndash19 35 36]) The local fractional deriva-tive of 119891(119909) at 119909 = 119909

0is defined as [16ndash20]

119891(120572)

(1199090) =

119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(7)

where

Δ120572(119891 (119909) minus 119891 (119909

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

0)) (8)

We find that the existence condition for local fractionalderivative of 119891(119909) is that the right-hand local fractionalderivative is equal to the left-hand local fractional derivative(see eg [16 35] and the references therein)

Definition 3 (see [15ndash19 35 36]) A partition of the interval[119886 119887] is denoted as (119905

119895 119905119895+1

) 119895 = 0 119873 minus 1 1199050= 119886 and

119905119873

= 119887 with Δ119905119895= 119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

Local fractional integral of 119891(119909) in the interval [119886 119887] is givenby

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

(9)

If the functions are local fractional continuous then thelocal fractional derivatives and integrals exist That is tosay operating functions have nondifferentiable and fractalproperties (see [35] and the references therein)

Some properties of local fractional derivative and inte-grals are given in [35]

3 Analytical Methods

In order to illustrate two analytical methods we investigatethe nonlinear local fractional equation as follows

119871(119899)

120572119906 + 119877120572119906 = 0 (10)

where 119871(119899)

120572is linear local fractional operators respectively

with 119899 = 1 2 and 119877120572is linear local fractional operators of

order less than 119871(119899)

120572

31 Local Fractional Variational Iteration Method The localfractional variational iteration algorithm is given by [16 17]on the line of the formalism suggested in [35]

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

120582120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119906119899(119904) (119889119904)

120572

(11)

Here we can construct a correction functional as follows [1617]

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

120582120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119899(119904) (119889119904)

120572

(12)

Abstract and Applied Analysis 3

where 119899is considered as a restricted local fractional varia-

tion that is 120575120572119899= 0 (for more details see [35])

For 119899 = 2 we have

120582120572=

(119904 minus 119905)120572

Γ (1 + 120572) (13)

so that iteration is expressed as

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119906119899(119904) (119889119904)

120572

(14)

Finally the solution is

119906 (119909) = lim119899rarrinfin

119906119899(119909) (15)

32 Local Fractional Decomposition Method When 119871(119899)

120572in

(10) is a local fractional differential operator of order 2120572 wedenote it as

119871(119899)

120572= 119871(2120572)

119909119909=

1205972120572

1205971199092120572

119877120572119906 (119905) =

120597120572

120597119909120572119906 (119905) + 119891 (119905)

(16)

By defining the 119899-fold local fractional integral operator

119871(minus2)

120572119898(119904) =

0119868119909(120572)

0119868119909(120572)

119898(119904) (17)

we get

119871(minus2)

120572119871(2)

120572119906 (119904) = 119871

(minus2)

120572119877120572119906 (119904) (18)

Thus

119906 (119904) = 119903 (119909) + 119871(minus2)

120572119877120572119906 (119904) (19)

where the term 119903(119909) is to be determined from the fractalinitial conditions

Therefore we get the iterative formula as follows

119906 (119909) = 1199060(119909) + 119871

(minus2)

120572119877120572119906 (119904) (20)

with 1199060(119909) = 119903(119909)

Hence for 119899 ge 0 we have the following recurrencerelationship

119906119899+1

(119909) = 119871(minus2)

120572119877120572119906119899(119904)

1199060(119909) = 119903 (119909)

(21)

Finally the solution can be constructed as

119906 (119909) = lim119899rarrinfin

120601119899(119909) = lim

119899rarrinfin

infin

sum

119899=0

119906119899(119909) (22)

For more details see [18]

4 An Illustrative Example

In this section one example for wave equation is presented inorder to demonstrate the simplicity and the efficiency of theabove methods

In (2) we consider the following initial and boundaryconditions

120597120572119906 (119909 0)

120597119905120572= 0 119906 (119909 0) =

1199092120572

Γ (1 + 2120572) (23)

Using (14) we have the iterative formula

119906119899+1

(119909 119905)

= 119906119899(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572(119889119904)120572

(24)

where the initial value is given by

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572) (25)

Thus after computing (23) we obtain

1199061(119909 119905)

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199060(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199060(119909 119904)

1205971199092120572(119889119904)120572

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572))

1199062(119909 119905)

= 1199061(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199061(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199061(119909 119904)

1205971199092120572(119889119904)120572

= 1199061(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199042120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572))

4 Abstract and Applied Analysis

1199063(119909 119905)

= 1199062(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199062(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199062(119909 119904)

1205971199092120572(119889119904)120572

= 1199062(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

3

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

1199064(119909 119905)

= 1199063(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199063(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

1199092120572

Γ (1 + 2120572)

12059721205721199063(119909 119904)

1205971199092120572(119889119904)120572

= 1199063(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199056120572

Γ (1 + 6120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

4

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(26)

119906119899(119909 119905)

= 119906119899minus1

(119909 119905) +1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899minus1

(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899minus1

(119909 119904)

1205971199092120572(119889119904)120572

=1199092120572

Γ (1 + 2120572)

119899

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(27)

Hence from (27) we obtain the solution of (3) as

119906 (119909) = lim119899rarrinfin

119906119899(119909) =

1199092120572

Γ (1 + 2120572)cosh120572(119905120572) (28)

Here from (21) we get

119906119899+1

(119909 119905) =0119868119905(120572)

0119868119905(120572) 119909

2120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

(29)

Therefore from (29) we give the components as follows

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052120572

Γ (1 + 2120572)

1199062(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572)

1199063(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199053120572

Γ (1 + 6120572)

1199064(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199058120572

Γ (1 + 8120572)

119906119899(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

(30)

Consequently the exact solution is given by

119906 (119909 119905) = lim119899rarrinfin

infin

sum

119899=0

119906119899(119909 119905)

= lim119899rarrinfin

infin

sum

119899=0

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

=1199092120572

Γ (1 + 2120572)cosh120572(119909120572)

(31)

where

cosh120572(119909120572) =

infin

sum

119899=0

1199052119899120572

Γ (1 + 2119899120572) (32)

The solution of (2) for 120572 = ln 2 ln 3 is depicted in Figure 1

5 Conclusions

In this work we developed a comparison between thevariational iteration method and the decomposition methodwithin local fractional operators The two approaches con-stitute efficient tools to handle the approximation solutionsfor differential equations on Cantor sets with local fractionalderivative We notice that the fractional variational iterationmethod gives the several successive approximate formulasusing the iteration of the correction local fractional func-tional However the local fractional decomposition method

Abstract and Applied Analysis 5

25

2

15

1

05

01

05

0 002

0406

081

x

t

u(xt)

Figure 1 Graph of 119906(119909 119905) for 120572 = ln 2 ln 3

provides the components of the exact solution which is localfractional continuous function where these componentsare also local fractional continuous functions Both thevariational iteration method and the decomposition methodwithin local fractional operators provide the solution insuccessive componentsThemethods are structured to get thelocal fractional series solution which is a nondifferentiablefunction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

References

[1] A M Wazwaz Partial Differential Equations Methods andApplications Elsevier Balkema The Netherlands 2002

[2] W R Schneider and W Wyss ldquoFractional diffusion and waveequationsrdquo Journal of Mathematical Physics vol 30 no 1 pp134ndash144 1989

[3] Z G Zhao and C P Li ldquoFractional differencefinite elementapproximations for the time-space fractional telegraph equa-tionrdquo Applied Mathematics and Computation vol 219 no 6 pp2975ndash2988 2012

[4] S Momani Z Odibat and A Alawneh ldquoVariational iterationmethod for solving the space- and time-fractional KdV equa-tionrdquo Numerical Methods for Partial Differential Equations vol24 no 1 pp 262ndash271 2008

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE vol 66 Article ID 056108 2002

[6] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[8] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[9] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoOn nonlinear fractional KleinGordon equationrdquo Signal Pro-cessing vol 91 no 3 pp 446ndash451 2011

[10] Z B Li W H Zhu and L L Huang ldquoApplication of fractionalvariational iteration method to time-fractional Fisher equa-tionrdquo Advanced Science Letters vol 10 no 1 pp 610ndash614 2012

[11] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[12] 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 BostonMass USA 2012

[13] C Cattani ldquoHarmonic wavelet solution of Poissonrsquos problemrdquoBalkan Journal of Geometry and Its Applications vol 13 no 1pp 27ndash37 2008

[14] C Cattani ldquoHarmonic wavelets towards the solution of nonlin-ear PDErdquo Computers amp Mathematics with Applications vol 50no 8-9 pp 1191ndash1210 2005

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

[16] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 article 89 2013

[18] X J Yang D Baleanu and W P Zhong ldquoApproximation solu-tions for diffusion equation on Cantor time-spacerdquo Proceedingof the Romanian Academy A vol 14 no 2 pp 127ndash133 2013

[19] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013

[20] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific Singapore 2012

[21] J Sabatier O P Agrawal and J T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer 2007

[22] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[23] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[24] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

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

[26] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[27] Y Chen Y Yan and K Zhang ldquoOn the local fractionalderivativerdquo Journal of Mathematical Analysis and Applicationsvol 362 no 1 pp 17ndash33 2010

[28] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

where 119899is considered as a restricted local fractional varia-

tion that is 120575120572119899= 0 (for more details see [35])

For 119899 = 2 we have

120582120572=

(119904 minus 119905)120572

Γ (1 + 120572) (13)

so that iteration is expressed as

119906119899+1

(119905) = 119906119899(119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)119871(119899)

120572119906119899(119904) + 119877

120572119906119899(119904) (119889119904)

120572

(14)

Finally the solution is

119906 (119909) = lim119899rarrinfin

119906119899(119909) (15)

32 Local Fractional Decomposition Method When 119871(119899)

120572in

(10) is a local fractional differential operator of order 2120572 wedenote it as

119871(119899)

120572= 119871(2120572)

119909119909=

1205972120572

1205971199092120572

119877120572119906 (119905) =

120597120572

120597119909120572119906 (119905) + 119891 (119905)

(16)

By defining the 119899-fold local fractional integral operator

119871(minus2)

120572119898(119904) =

0119868119909(120572)

0119868119909(120572)

119898(119904) (17)

we get

119871(minus2)

120572119871(2)

120572119906 (119904) = 119871

(minus2)

120572119877120572119906 (119904) (18)

Thus

119906 (119904) = 119903 (119909) + 119871(minus2)

120572119877120572119906 (119904) (19)

where the term 119903(119909) is to be determined from the fractalinitial conditions

Therefore we get the iterative formula as follows

119906 (119909) = 1199060(119909) + 119871

(minus2)

120572119877120572119906 (119904) (20)

with 1199060(119909) = 119903(119909)

Hence for 119899 ge 0 we have the following recurrencerelationship

119906119899+1

(119909) = 119871(minus2)

120572119877120572119906119899(119904)

1199060(119909) = 119903 (119909)

(21)

Finally the solution can be constructed as

119906 (119909) = lim119899rarrinfin

120601119899(119909) = lim

119899rarrinfin

infin

sum

119899=0

119906119899(119909) (22)

For more details see [18]

4 An Illustrative Example

In this section one example for wave equation is presented inorder to demonstrate the simplicity and the efficiency of theabove methods

In (2) we consider the following initial and boundaryconditions

120597120572119906 (119909 0)

120597119905120572= 0 119906 (119909 0) =

1199092120572

Γ (1 + 2120572) (23)

Using (14) we have the iterative formula

119906119899+1

(119909 119905)

= 119906119899(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572(119889119904)120572

(24)

where the initial value is given by

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572) (25)

Thus after computing (23) we obtain

1199061(119909 119905)

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199060(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199060(119909 119904)

1205971199092120572(119889119904)120572

= 1199060(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572))

1199062(119909 119905)

= 1199061(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199061(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199061(119909 119904)

1205971199092120572(119889119904)120572

= 1199061(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199042120572

Γ (1 + 2120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)(1 +

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572))

4 Abstract and Applied Analysis

1199063(119909 119905)

= 1199062(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199062(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199062(119909 119904)

1205971199092120572(119889119904)120572

= 1199062(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

3

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

1199064(119909 119905)

= 1199063(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199063(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

1199092120572

Γ (1 + 2120572)

12059721205721199063(119909 119904)

1205971199092120572(119889119904)120572

= 1199063(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199056120572

Γ (1 + 6120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

4

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(26)

119906119899(119909 119905)

= 119906119899minus1

(119909 119905) +1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899minus1

(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899minus1

(119909 119904)

1205971199092120572(119889119904)120572

=1199092120572

Γ (1 + 2120572)

119899

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(27)

Hence from (27) we obtain the solution of (3) as

119906 (119909) = lim119899rarrinfin

119906119899(119909) =

1199092120572

Γ (1 + 2120572)cosh120572(119905120572) (28)

Here from (21) we get

119906119899+1

(119909 119905) =0119868119905(120572)

0119868119905(120572) 119909

2120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

(29)

Therefore from (29) we give the components as follows

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052120572

Γ (1 + 2120572)

1199062(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572)

1199063(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199053120572

Γ (1 + 6120572)

1199064(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199058120572

Γ (1 + 8120572)

119906119899(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

(30)

Consequently the exact solution is given by

119906 (119909 119905) = lim119899rarrinfin

infin

sum

119899=0

119906119899(119909 119905)

= lim119899rarrinfin

infin

sum

119899=0

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

=1199092120572

Γ (1 + 2120572)cosh120572(119909120572)

(31)

where

cosh120572(119909120572) =

infin

sum

119899=0

1199052119899120572

Γ (1 + 2119899120572) (32)

The solution of (2) for 120572 = ln 2 ln 3 is depicted in Figure 1

5 Conclusions

In this work we developed a comparison between thevariational iteration method and the decomposition methodwithin local fractional operators The two approaches con-stitute efficient tools to handle the approximation solutionsfor differential equations on Cantor sets with local fractionalderivative We notice that the fractional variational iterationmethod gives the several successive approximate formulasusing the iteration of the correction local fractional func-tional However the local fractional decomposition method

Abstract and Applied Analysis 5

25

2

15

1

05

01

05

0 002

0406

081

x

t

u(xt)

Figure 1 Graph of 119906(119909 119905) for 120572 = ln 2 ln 3

provides the components of the exact solution which is localfractional continuous function where these componentsare also local fractional continuous functions Both thevariational iteration method and the decomposition methodwithin local fractional operators provide the solution insuccessive componentsThemethods are structured to get thelocal fractional series solution which is a nondifferentiablefunction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

References

[1] A M Wazwaz Partial Differential Equations Methods andApplications Elsevier Balkema The Netherlands 2002

[2] W R Schneider and W Wyss ldquoFractional diffusion and waveequationsrdquo Journal of Mathematical Physics vol 30 no 1 pp134ndash144 1989

[3] Z G Zhao and C P Li ldquoFractional differencefinite elementapproximations for the time-space fractional telegraph equa-tionrdquo Applied Mathematics and Computation vol 219 no 6 pp2975ndash2988 2012

[4] S Momani Z Odibat and A Alawneh ldquoVariational iterationmethod for solving the space- and time-fractional KdV equa-tionrdquo Numerical Methods for Partial Differential Equations vol24 no 1 pp 262ndash271 2008

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE vol 66 Article ID 056108 2002

[6] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[8] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[9] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoOn nonlinear fractional KleinGordon equationrdquo Signal Pro-cessing vol 91 no 3 pp 446ndash451 2011

[10] Z B Li W H Zhu and L L Huang ldquoApplication of fractionalvariational iteration method to time-fractional Fisher equa-tionrdquo Advanced Science Letters vol 10 no 1 pp 610ndash614 2012

[11] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[12] 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 BostonMass USA 2012

[13] C Cattani ldquoHarmonic wavelet solution of Poissonrsquos problemrdquoBalkan Journal of Geometry and Its Applications vol 13 no 1pp 27ndash37 2008

[14] C Cattani ldquoHarmonic wavelets towards the solution of nonlin-ear PDErdquo Computers amp Mathematics with Applications vol 50no 8-9 pp 1191ndash1210 2005

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

[16] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 article 89 2013

[18] X J Yang D Baleanu and W P Zhong ldquoApproximation solu-tions for diffusion equation on Cantor time-spacerdquo Proceedingof the Romanian Academy A vol 14 no 2 pp 127ndash133 2013

[19] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013

[20] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific Singapore 2012

[21] J Sabatier O P Agrawal and J T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer 2007

[22] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[23] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[24] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

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

[26] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[27] Y Chen Y Yan and K Zhang ldquoOn the local fractionalderivativerdquo Journal of Mathematical Analysis and Applicationsvol 362 no 1 pp 17ndash33 2010

[28] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

1199063(119909 119905)

= 1199062(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199062(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

12059721205721199062(119909 119904)

1205971199092120572(119889119904)120572

= 1199062(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

3

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

1199064(119909 119905)

= 1199063(119909 119905) +

1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

12059721205721199063(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)int

119905

0

1199092120572

Γ (1 + 2120572)

12059721205721199063(119909 119904)

1205971199092120572(119889119904)120572

= 1199063(119909 119905) +

1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)minus

1199092120572

Γ (1 + 2120572)

1199056120572

Γ (1 + 6120572) (119889119904)

120572

=1199092120572

Γ (1 + 2120572)

4

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(26)

119906119899(119909 119905)

= 119906119899minus1

(119909 119905) +1

Γ (1 + 120572)int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1205972120572119906119899minus1

(119909 119904)

1205971199042120572(119889119904)120572

minus1

Γ (1 + 120572)

times int

119905

0

(119904 minus 119905)120572

Γ (1 + 120572)

1199092120572

Γ (1 + 2120572)

1205972120572119906119899minus1

(119909 119904)

1205971199092120572(119889119904)120572

=1199092120572

Γ (1 + 2120572)

119899

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572)

(27)

Hence from (27) we obtain the solution of (3) as

119906 (119909) = lim119899rarrinfin

119906119899(119909) =

1199092120572

Γ (1 + 2120572)cosh120572(119905120572) (28)

Here from (21) we get

119906119899+1

(119909 119905) =0119868119905(120572)

0119868119905(120572) 119909

2120572

Γ (1 + 2120572)

1205972120572119906119899(119909 119904)

1205971199092120572

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

(29)

Therefore from (29) we give the components as follows

1199060(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052120572

Γ (1 + 2120572)

1199062(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199054120572

Γ (1 + 4120572)

1199063(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199053120572

Γ (1 + 6120572)

1199064(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199058120572

Γ (1 + 8120572)

119906119899(119909 119905) =

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

(30)

Consequently the exact solution is given by

119906 (119909 119905) = lim119899rarrinfin

infin

sum

119899=0

119906119899(119909 119905)

= lim119899rarrinfin

infin

sum

119899=0

1199092120572

Γ (1 + 2120572)

1199052119899120572

Γ (1 + 2119899120572)

=1199092120572

Γ (1 + 2120572)cosh120572(119909120572)

(31)

where

cosh120572(119909120572) =

infin

sum

119899=0

1199052119899120572

Γ (1 + 2119899120572) (32)

The solution of (2) for 120572 = ln 2 ln 3 is depicted in Figure 1

5 Conclusions

In this work we developed a comparison between thevariational iteration method and the decomposition methodwithin local fractional operators The two approaches con-stitute efficient tools to handle the approximation solutionsfor differential equations on Cantor sets with local fractionalderivative We notice that the fractional variational iterationmethod gives the several successive approximate formulasusing the iteration of the correction local fractional func-tional However the local fractional decomposition method

Abstract and Applied Analysis 5

25

2

15

1

05

01

05

0 002

0406

081

x

t

u(xt)

Figure 1 Graph of 119906(119909 119905) for 120572 = ln 2 ln 3

provides the components of the exact solution which is localfractional continuous function where these componentsare also local fractional continuous functions Both thevariational iteration method and the decomposition methodwithin local fractional operators provide the solution insuccessive componentsThemethods are structured to get thelocal fractional series solution which is a nondifferentiablefunction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

References

[1] A M Wazwaz Partial Differential Equations Methods andApplications Elsevier Balkema The Netherlands 2002

[2] W R Schneider and W Wyss ldquoFractional diffusion and waveequationsrdquo Journal of Mathematical Physics vol 30 no 1 pp134ndash144 1989

[3] Z G Zhao and C P Li ldquoFractional differencefinite elementapproximations for the time-space fractional telegraph equa-tionrdquo Applied Mathematics and Computation vol 219 no 6 pp2975ndash2988 2012

[4] S Momani Z Odibat and A Alawneh ldquoVariational iterationmethod for solving the space- and time-fractional KdV equa-tionrdquo Numerical Methods for Partial Differential Equations vol24 no 1 pp 262ndash271 2008

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE vol 66 Article ID 056108 2002

[6] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[8] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[9] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoOn nonlinear fractional KleinGordon equationrdquo Signal Pro-cessing vol 91 no 3 pp 446ndash451 2011

[10] Z B Li W H Zhu and L L Huang ldquoApplication of fractionalvariational iteration method to time-fractional Fisher equa-tionrdquo Advanced Science Letters vol 10 no 1 pp 610ndash614 2012

[11] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[12] 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 BostonMass USA 2012

[13] C Cattani ldquoHarmonic wavelet solution of Poissonrsquos problemrdquoBalkan Journal of Geometry and Its Applications vol 13 no 1pp 27ndash37 2008

[14] C Cattani ldquoHarmonic wavelets towards the solution of nonlin-ear PDErdquo Computers amp Mathematics with Applications vol 50no 8-9 pp 1191ndash1210 2005

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

[16] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 article 89 2013

[18] X J Yang D Baleanu and W P Zhong ldquoApproximation solu-tions for diffusion equation on Cantor time-spacerdquo Proceedingof the Romanian Academy A vol 14 no 2 pp 127ndash133 2013

[19] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013

[20] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific Singapore 2012

[21] J Sabatier O P Agrawal and J T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer 2007

[22] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[23] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[24] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

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

[26] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[27] Y Chen Y Yan and K Zhang ldquoOn the local fractionalderivativerdquo Journal of Mathematical Analysis and Applicationsvol 362 no 1 pp 17ndash33 2010

[28] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

25

2

15

1

05

01

05

0 002

0406

081

x

t

u(xt)

Figure 1 Graph of 119906(119909 119905) for 120572 = ln 2 ln 3

provides the components of the exact solution which is localfractional continuous function where these componentsare also local fractional continuous functions Both thevariational iteration method and the decomposition methodwithin local fractional operators provide the solution insuccessive componentsThemethods are structured to get thelocal fractional series solution which is a nondifferentiablefunction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

References

[1] A M Wazwaz Partial Differential Equations Methods andApplications Elsevier Balkema The Netherlands 2002

[2] W R Schneider and W Wyss ldquoFractional diffusion and waveequationsrdquo Journal of Mathematical Physics vol 30 no 1 pp134ndash144 1989

[3] Z G Zhao and C P Li ldquoFractional differencefinite elementapproximations for the time-space fractional telegraph equa-tionrdquo Applied Mathematics and Computation vol 219 no 6 pp2975ndash2988 2012

[4] S Momani Z Odibat and A Alawneh ldquoVariational iterationmethod for solving the space- and time-fractional KdV equa-tionrdquo Numerical Methods for Partial Differential Equations vol24 no 1 pp 262ndash271 2008

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE vol 66 Article ID 056108 2002

[6] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions vol 11 no 5 pp 4465ndash4475 2010

[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006

[8] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[9] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoOn nonlinear fractional KleinGordon equationrdquo Signal Pro-cessing vol 91 no 3 pp 446ndash451 2011

[10] Z B Li W H Zhu and L L Huang ldquoApplication of fractionalvariational iteration method to time-fractional Fisher equa-tionrdquo Advanced Science Letters vol 10 no 1 pp 610ndash614 2012

[11] J Hristov ldquoHeat-balance integral to fractional (half-time) heatdiffusion sub-modelrdquoThermal Science vol 14 no 2 pp 291ndash3162010

[12] 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 BostonMass USA 2012

[13] C Cattani ldquoHarmonic wavelet solution of Poissonrsquos problemrdquoBalkan Journal of Geometry and Its Applications vol 13 no 1pp 27ndash37 2008

[14] C Cattani ldquoHarmonic wavelets towards the solution of nonlin-ear PDErdquo Computers amp Mathematics with Applications vol 50no 8-9 pp 1191ndash1210 2005

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

[16] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 article 89 2013

[18] X J Yang D Baleanu and W P Zhong ldquoApproximation solu-tions for diffusion equation on Cantor time-spacerdquo Proceedingof the Romanian Academy A vol 14 no 2 pp 127ndash133 2013

[19] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013

[20] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific Singapore 2012

[21] J Sabatier O P Agrawal and J T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer 2007

[22] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[23] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[24] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

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

[26] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[27] Y Chen Y Yan and K Zhang ldquoOn the local fractionalderivativerdquo Journal of Mathematical Analysis and Applicationsvol 362 no 1 pp 17ndash33 2010

[28] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

fractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[29] A Carpinteri B Chiaia and P Cornetti ldquoThe elastic problemfor fractal media basic theory and finite element formulationrdquoComputers and Structures vol 82 no 6 pp 499ndash508 2004

[30] A Carpinteri and P Cornetti ldquoA fractional calculus approach tothe description of stress and strain localization in fractalmediardquoChaos Solitons amp Fractals vol 13 no 1 pp 85ndash94 2002

[31] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

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

[33] W Chen H G Sun X D Zhang and D Korosak ldquoAnoma-lous diffusion modeling by fractal and fractional derivativesrdquoComputers amp Mathematics with Applications vol 59 no 5 pp1754ndash1758 2010

[34] G Jumarie ldquoModified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresultsrdquoComputersampMathematics with Applications vol 51 no9-10 pp 1367ndash1376 2006

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

[36] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of