Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 632309 6 pageshttpdxdoiorg1011552013632309

Research ArticleAnalysis of Fractal Wave Equations by Local FractionalFourier Series Method

Yong-Ju Yang1 Dumitru Baleanu234 and Xiao-Jun Yang5

1 School of Mathematics and Statistics Nanyang Normal University Nanyang 473061 China2Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University 06530 Ankara Turkey3 Department of Chemical andMaterials Engineering Faculty of Engineering KingAbdulazizUniversity PO Box 80204 Jeddah 21589Saudi Arabia

4 Institute of Space Sciences Magurele 077125 Bucharest Romania5 Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Jiangsu 221008 China

Correspondence should be addressed to Xiao-Jun Yang dyangxiaojun163com

Received 12 May 2013 Accepted 13 June 2013

Academic Editor H Srivastava

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 fractal wave equations with local fractional derivatives are investigated in this paper The analytical solutions are obtained byusing local fractional Fourier series method The present method is very efficient and accurate to process a class of local fractionaldifferential equations

1 Introduction

Fractional calculus deals with derivative and integrals ofarbitrary orders [1] During the last four decades fractionalcalculus has been applied to almost every field of science andengineering [2ndash6] In recent years there has been a greatdeal of interest in fractional differential equations [7] As aresult various kinds of analytical methods were developed[8ndash18] For example there are the exp-function method[8] the variational iteration method [9 10] the homotopyperturbation method [11] the homotopy analysis method[12] the heat-balance integral method [13] the fractionalvariational iteration method [14 15] the fractional differencemethod [16] the finite element method [17] the fractionalFourier and Laplace transforms [18] and so on

Recently local fractional calculus was applied to deal withproblems for nondifferentiable functions see [19ndash26] andthe references therein There are also analytical methods forsolving the local fractional differential equations which arereferred to in [27ndash34]The local fractional seriesmethod [32ndash34] was applied to process the local fractional wave equationin fractal vibrating [32] and local fractional heat-conductionequation [33]

More recently the wave equation on the Cantor sets wasconsidered as [21 28]

1205972120572119906 (119909 119905)

1205971199052120572=1205972120572119906 (119909 119905)

1205971199092120572 (1)

Local damped wave equation was written in the form [30]

1205972120572119906 (119909 119905)

1205971199052120572minus120597120572119906 (119909 119905)

120597119905120572minus1205972120572119906 (119909 119905)

1205971199092120572= 119891 (119909 119905) (2)

and local fractional dissipative wave equation in fractalstrings was [31]

1205972120572119906 (119909 119905)

1205971199052120572minus120597120572119906 (119909 119905)

120597119905120572

minus1205972120572119906 (119909 119905)

1205971199092120572minus120597120572119906 (119909 119905)

120597119909120572= 119891 (119909 119905)

(3)

In this paper we investigate the application of local frac-tional series method for solving the following local fractionalwave equation

1205972120572119906 (119909 119905)

1205971199052120572minus120597120572119906 (119909 119905)

120597119905120572minus1205972120572119906 (119909 119905)

1205971199092120572= 0 (4)

2 Advances in Mathematical Physics

where initial and boundary conditions are presented as

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909)

(5)

The organization of the paper is as follows In Section 2 thebasic concepts of local fractional calculus and local fractionalFourier series are introduced In Section 3 we present a localfractional Fourier series solution of wave equation with localfractional derivative Two examples are shown in Section 4Finally Section 5 is devoted to our conclusions

2 Mathematical Tools

In this section we present some concepts of local fractionalcontinuity local fractional derivative and local fractionalFourier series

Definition 1 (see [21 28 30ndash32]) Suppose that there is

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (6)

with |119909minus1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 Then 119891(119909) is called

local fractional continuous at 119909 = 1199090 where 120588120572|119909 minus 119909

0|120572le

|119891(1199091) minus 119891(119909

2)| le 120591120572|119909 minus 119909

0|120572 with 120588 120591 gt 0

Suppose that the function119891(119909) satisfies the above proper-ties of the local fractional continuity Then the condition (6)for 119909 isin (119886 119887) is denoted as

119891 (119909) isin 119862120572(119886 119887) (7)

where dim119867119891(119909) = 120572

Definition 2 (see [19ndash21]) Let 119891(119909) isin 119862120572(119886 119887) Local frac-

tional derivative of 119891(119909) of order 120572 at 119909 = 1199090is given by

119863119909

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

=119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(8)

where Δ120572(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

Definition 3 (see [19 20 32ndash34]) Let 119891(119909) isin 119862120572(minusinfin +infin)

and let 119891(119909) be 2119897-periodic For 119896 isin 119885 local fraction Fourierseries of 119891(119909) is defined as

119891 (119909) =1198860

2+

infin

sum

119896=1

(119886119899cos120572

120587120572(119896119909)120572

119897120572

+ 119887119899sin120572

120587120572(119896119909)120572

119897120572)

(9)

where the local fraction Fourier coefficients are

119886119899=1

119897120572int

119897

minus119897

119891 (119909) cos120572

120587120572(119896119909)120572

119897120572(119889119909)120572

119887119899=1

119897120572int

119897

minus119897

119891 (119909) sin120572

120587120572(119896119909)120572

119897120572(119889119909)120572

(10)

with local fractional integral given by [21 29ndash34]

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

(11)

where Δ119905119895= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1] 119895 = 0 119873 minus 1 119905

0= 119886 119905

119873= 119887 is a partition of

the interval [119886 119887]In view of (10) theweights of the fractional trigonometric

functions are expressed as follows

119886119899=1 (Γ (1 + 120572)) int

119897

minus119897119891 (119909) cos

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

minus119897cos2120572119899120572(120587119909119897)

120572(119889119909)120572

119887119899=1 (Γ (1 + 120572)) int

119897

minus119897119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

minus119897sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(12)

Lemma 4 (see [21]) If119898 and ℎ are constant coefficients thenlocal fractional differential equation with constant coefficients

1198892120572119910

1198891199092120572+ 119898

119889120572119910

119889119909120572+ ℎ119910 = 0 (119898

2minus 4ℎ lt 0) (13)

has a family of solution

119910 (119909) = 119860119864120572(minus119898 minus 119894

120572radic4ℎ minus 1198982

2119909120572)

+ 119861119864120572(minus119898 + 119894

120572radic4ℎ minus 1198982

2119909120572)

(14)

with two constants 119860 and 119861

Proof See [21]

3 Solution to Wave Equation withLocal Fractional Derivative

If we have the particular solution of (4) in the form

119906 (119909 119905) = 120601 (119909) 119879 (119905) (15)

then we get the equations

120601(2120572)

+ 1205822120572120601 = 0 (16)

119879(2120572)

+ 119879(120572)+ 1205822120572119879 = 0 (17)

Advances in Mathematical Physics 3

with the boundary conditions

120601 (0) = 120601(120572)(119897) = 0 (18)

Equation (4) has the solution

120601 (119909) = 1198621cos120572120582120572119909120572+ 1198622sin120572120582120572119909120572 (19)

where 1198621and 119862

2are all constant numbers

According to (19) for 119909 = 0 and 119909 = 119897 we get

120601 (0) = 1198621= 0

120601 (119897) = 120601 (119909)1003816100381610038161003816119909=119897 = 1198622sin120572120582

120572119897120572= 0

(20)

Obviously 1198622= 0 since otherwise 120601(119909) equiv 0

Hence we arrive at

120582120572

119899119897120572= 119899120572120587120572 (21)

where 119899 is an integerWe notice

120582120572

119899= (

119899120587

119897)

120572

(119899 = 0 1 2 )

120601119899(119909) = sin

120572120582120572

119899119909120572

= sin120572119899120572(120587119909

119897)

120572

(119899 = 0 1 2 )

(22)

For 120582120572 = 120582120572119899and 0 lt 120588 following (17) implies that

infin

sum

119899=1

119879119899(119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572)

(23)

where

120588 =

radic4(119899120587119897)2120572minus 1

2

(24)

Therefore

119906119899(119909 119905) = 120601

119899(119909) 119879119899(119905)

= 119860119899cos120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

+ 119861119899sin120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

(25)

We now suppose a local fractional Fourier series solution of(4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905)

=

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(26)

Therefore

120597120572119906 (119909 119905)

120597119905120572=

infin

sum

119899=1

120597120572119906119899(119909 119905)

120597119905120572 (27)

where

120597119906119899(119909 119905)

120597119905120572

= minus1

2119864120572(minus

119905120572

2) (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

+ 120588119864120572(minus

119905120572

2) (minus119860

119899sin120572120588119905120572+ 119861119899cos120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(28)

with 120588 = radic(4(119899120587119897)2120572 minus 1)2Submitting (26) to (5) we have

119906 (119909 0) =

infin

sum

119899=1

119906119899(119909 0)

=

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

(29)

120597120572119906 (119909 119905)

120597119905120572

=

infin

sum

119899=1

(minus1

2119860119899+ 120588119861119899) sin120572119899120572(120587119909

119897)

120572

= 119892 (119909)

(30)

So

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +

infin

sum

119899=1

1

2119860119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +1

2119891 (119909)

(31)

Let

119866 (119909) = 119892 (119909) +1

2119891 (119909) (32)

In view of (30) and (31) we rewrite

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119866 (119909)

(33)

4 Advances in Mathematical Physics

We now find the local fractional Fourier coefficients of 119891(119909)and 119866(119909) respectively

119860119899=1 (Γ (1 + 120572)) int

119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

120588119861119899=1 (Γ (1 + 120572)) int

119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

(34)

Following (34) we have

1

Γ (1 + 120572)int

119897

0

sin2120572119899120572(120587119909

119897)

120572

(119889119909)120572=

119897120572

2Γ (1 + 120572) (35)

such that

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

(36)

Thus we get the solution of (4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905) (37)

where

119906119899(119909 119905) = 119864

120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(38)

with

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572(119899 = 0 1 2 )

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572(119899 = 0 1 2 )

(39)

with

119866 (119909) = 119892 (119909) +1

2119891 (119909) (40)

4 Illustrative Examples

In order to illustrate the above result in this section we givetwo examples

Let us consider (4) subject to initial and boundaryconditions

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) =

119909120572

Γ (1 + 120572)

(41)

In view of (40) we have

119866 (119909) = 119892 (119909) +1

2119891 (119909) =

3

2

119909120572

Γ (1 + 120572) (42)

such that

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

(43)

119861119899=3 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=3Γ (1 + 120572)

120588119897120572 0

119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(44)

Hence

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(45)

where

119860119899= minus

2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899= minus

3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(46)

Advances in Mathematical Physics 5

In view of (4) our second example is initial and boundaryconditions as follows

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) = 0

(47)

Following (40) we get

119866 (119909) =1

2

119909120572

Γ (1 + 120572) (48)

Hence we obtain

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899=int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=Γ (1 + 120572)

1205881198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minusΓ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(49)

So

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(50)

with

119860119899=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

119861119899= minus

Γ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(51)

10908070605040302010

1

09

08

07

06

05

04

03

02

01

0

f(x)

x

Figure 1 For 120572 = ln 2 ln 3 graph of a Lebesgue-Cantor staircasefunction shown at 119909 isin [0 1]

We notice that fraction boundary condition is expressedas a Lebesgue-Cantor staircase function [21 32] namely

119906 (119909 0) = 119891 (119909) = 119867120572(119862 cap (0 119909))

=0119868119909

(120572)1 =

119909120572

Γ (1 + 120572)

(52)

where 119862 is any fractal set and the fractal dimension of119909120572Γ(1 + 120572) is 120572 For 119909 isin [0 1] the graph of the Lebesgue-

Cantor staircase function (52) is shown in Figure 1 whenfractal dimension is 120572 = ln 2 ln 3

5 Conclusions

The present work expresses the local fractional Fourier seriessolution to wave equations with local fractional derivativeTwo examples are given to illustrat approximate solutions forwave equations with local fractional derivative resulting fromlocal fractional Fourier series method The results obtainedfrom the local fractional analysis seem to be general sincethe obtained solutions go back to the classical one whenfractal dimension 120572 = 1 namely it is a process from fractalgeometry to Euclidean geometry Local fractional Fourierseriesmethod is one of very efficient and powerful techniquesfor finding the solutions of the local fractional differentialequations It is also worth noting that the advantage of thelocal fractional differential equations displays the nondiffer-ential solutions which show the fractal and local behaviorsof moments However the classical Fourier series is used tohandle the continuous functions

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204

6 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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

2 Advances in Mathematical Physics

where initial and boundary conditions are presented as

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909)

(5)

The organization of the paper is as follows In Section 2 thebasic concepts of local fractional calculus and local fractionalFourier series are introduced In Section 3 we present a localfractional Fourier series solution of wave equation with localfractional derivative Two examples are shown in Section 4Finally Section 5 is devoted to our conclusions

2 Mathematical Tools

In this section we present some concepts of local fractionalcontinuity local fractional derivative and local fractionalFourier series

Definition 1 (see [21 28 30ndash32]) Suppose that there is

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 (6)

with |119909minus1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 Then 119891(119909) is called

local fractional continuous at 119909 = 1199090 where 120588120572|119909 minus 119909

0|120572le

|119891(1199091) minus 119891(119909

2)| le 120591120572|119909 minus 119909

0|120572 with 120588 120591 gt 0

Suppose that the function119891(119909) satisfies the above proper-ties of the local fractional continuity Then the condition (6)for 119909 isin (119886 119887) is denoted as

119891 (119909) isin 119862120572(119886 119887) (7)

where dim119867119891(119909) = 120572

Definition 2 (see [19ndash21]) Let 119891(119909) isin 119862120572(119886 119887) Local frac-

tional derivative of 119891(119909) of order 120572 at 119909 = 1199090is given by

119863119909

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

=119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(8)

where Δ120572(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

Definition 3 (see [19 20 32ndash34]) Let 119891(119909) isin 119862120572(minusinfin +infin)

and let 119891(119909) be 2119897-periodic For 119896 isin 119885 local fraction Fourierseries of 119891(119909) is defined as

119891 (119909) =1198860

2+

infin

sum

119896=1

(119886119899cos120572

120587120572(119896119909)120572

119897120572

+ 119887119899sin120572

120587120572(119896119909)120572

119897120572)

(9)

where the local fraction Fourier coefficients are

119886119899=1

119897120572int

119897

minus119897

119891 (119909) cos120572

120587120572(119896119909)120572

119897120572(119889119909)120572

119887119899=1

119897120572int

119897

minus119897

119891 (119909) sin120572

120587120572(119896119909)120572

119897120572(119889119909)120572

(10)

with local fractional integral given by [21 29ndash34]

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

(11)

where Δ119905119895= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1] 119895 = 0 119873 minus 1 119905

0= 119886 119905

119873= 119887 is a partition of

the interval [119886 119887]In view of (10) theweights of the fractional trigonometric

functions are expressed as follows

119886119899=1 (Γ (1 + 120572)) int

119897

minus119897119891 (119909) cos

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

minus119897cos2120572119899120572(120587119909119897)

120572(119889119909)120572

119887119899=1 (Γ (1 + 120572)) int

119897

minus119897119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

minus119897sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(12)

Lemma 4 (see [21]) If119898 and ℎ are constant coefficients thenlocal fractional differential equation with constant coefficients

1198892120572119910

1198891199092120572+ 119898

119889120572119910

119889119909120572+ ℎ119910 = 0 (119898

2minus 4ℎ lt 0) (13)

has a family of solution

119910 (119909) = 119860119864120572(minus119898 minus 119894

120572radic4ℎ minus 1198982

2119909120572)

+ 119861119864120572(minus119898 + 119894

120572radic4ℎ minus 1198982

2119909120572)

(14)

with two constants 119860 and 119861

Proof See [21]

3 Solution to Wave Equation withLocal Fractional Derivative

If we have the particular solution of (4) in the form

119906 (119909 119905) = 120601 (119909) 119879 (119905) (15)

then we get the equations

120601(2120572)

+ 1205822120572120601 = 0 (16)

119879(2120572)

+ 119879(120572)+ 1205822120572119879 = 0 (17)

Advances in Mathematical Physics 3

with the boundary conditions

120601 (0) = 120601(120572)(119897) = 0 (18)

Equation (4) has the solution

120601 (119909) = 1198621cos120572120582120572119909120572+ 1198622sin120572120582120572119909120572 (19)

where 1198621and 119862

2are all constant numbers

According to (19) for 119909 = 0 and 119909 = 119897 we get

120601 (0) = 1198621= 0

120601 (119897) = 120601 (119909)1003816100381610038161003816119909=119897 = 1198622sin120572120582

120572119897120572= 0

(20)

Obviously 1198622= 0 since otherwise 120601(119909) equiv 0

Hence we arrive at

120582120572

119899119897120572= 119899120572120587120572 (21)

where 119899 is an integerWe notice

120582120572

119899= (

119899120587

119897)

120572

(119899 = 0 1 2 )

120601119899(119909) = sin

120572120582120572

119899119909120572

= sin120572119899120572(120587119909

119897)

120572

(119899 = 0 1 2 )

(22)

For 120582120572 = 120582120572119899and 0 lt 120588 following (17) implies that

infin

sum

119899=1

119879119899(119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572)

(23)

where

120588 =

radic4(119899120587119897)2120572minus 1

2

(24)

Therefore

119906119899(119909 119905) = 120601

119899(119909) 119879119899(119905)

= 119860119899cos120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

+ 119861119899sin120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

(25)

We now suppose a local fractional Fourier series solution of(4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905)

=

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(26)

Therefore

120597120572119906 (119909 119905)

120597119905120572=

infin

sum

119899=1

120597120572119906119899(119909 119905)

120597119905120572 (27)

where

120597119906119899(119909 119905)

120597119905120572

= minus1

2119864120572(minus

119905120572

2) (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

+ 120588119864120572(minus

119905120572

2) (minus119860

119899sin120572120588119905120572+ 119861119899cos120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(28)

with 120588 = radic(4(119899120587119897)2120572 minus 1)2Submitting (26) to (5) we have

119906 (119909 0) =

infin

sum

119899=1

119906119899(119909 0)

=

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

(29)

120597120572119906 (119909 119905)

120597119905120572

=

infin

sum

119899=1

(minus1

2119860119899+ 120588119861119899) sin120572119899120572(120587119909

119897)

120572

= 119892 (119909)

(30)

So

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +

infin

sum

119899=1

1

2119860119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +1

2119891 (119909)

(31)

Let

119866 (119909) = 119892 (119909) +1

2119891 (119909) (32)

In view of (30) and (31) we rewrite

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119866 (119909)

(33)

4 Advances in Mathematical Physics

We now find the local fractional Fourier coefficients of 119891(119909)and 119866(119909) respectively

119860119899=1 (Γ (1 + 120572)) int

119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

120588119861119899=1 (Γ (1 + 120572)) int

119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

(34)

Following (34) we have

1

Γ (1 + 120572)int

119897

0

sin2120572119899120572(120587119909

119897)

120572

(119889119909)120572=

119897120572

2Γ (1 + 120572) (35)

such that

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

(36)

Thus we get the solution of (4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905) (37)

where

119906119899(119909 119905) = 119864

120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(38)

with

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572(119899 = 0 1 2 )

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572(119899 = 0 1 2 )

(39)

with

119866 (119909) = 119892 (119909) +1

2119891 (119909) (40)

4 Illustrative Examples

In order to illustrate the above result in this section we givetwo examples

Let us consider (4) subject to initial and boundaryconditions

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) =

119909120572

Γ (1 + 120572)

(41)

In view of (40) we have

119866 (119909) = 119892 (119909) +1

2119891 (119909) =

3

2

119909120572

Γ (1 + 120572) (42)

such that

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

(43)

119861119899=3 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=3Γ (1 + 120572)

120588119897120572 0

119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(44)

Hence

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(45)

where

119860119899= minus

2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899= minus

3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(46)

Advances in Mathematical Physics 5

In view of (4) our second example is initial and boundaryconditions as follows

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) = 0

(47)

Following (40) we get

119866 (119909) =1

2

119909120572

Γ (1 + 120572) (48)

Hence we obtain

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899=int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=Γ (1 + 120572)

1205881198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minusΓ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(49)

So

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(50)

with

119860119899=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

119861119899= minus

Γ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(51)

10908070605040302010

1

09

08

07

06

05

04

03

02

01

0

f(x)

x

Figure 1 For 120572 = ln 2 ln 3 graph of a Lebesgue-Cantor staircasefunction shown at 119909 isin [0 1]

We notice that fraction boundary condition is expressedas a Lebesgue-Cantor staircase function [21 32] namely

119906 (119909 0) = 119891 (119909) = 119867120572(119862 cap (0 119909))

=0119868119909

(120572)1 =

119909120572

Γ (1 + 120572)

(52)

where 119862 is any fractal set and the fractal dimension of119909120572Γ(1 + 120572) is 120572 For 119909 isin [0 1] the graph of the Lebesgue-

Cantor staircase function (52) is shown in Figure 1 whenfractal dimension is 120572 = ln 2 ln 3

5 Conclusions

The present work expresses the local fractional Fourier seriessolution to wave equations with local fractional derivativeTwo examples are given to illustrat approximate solutions forwave equations with local fractional derivative resulting fromlocal fractional Fourier series method The results obtainedfrom the local fractional analysis seem to be general sincethe obtained solutions go back to the classical one whenfractal dimension 120572 = 1 namely it is a process from fractalgeometry to Euclidean geometry Local fractional Fourierseriesmethod is one of very efficient and powerful techniquesfor finding the solutions of the local fractional differentialequations It is also worth noting that the advantage of thelocal fractional differential equations displays the nondiffer-ential solutions which show the fractal and local behaviorsof moments However the classical Fourier series is used tohandle the continuous functions

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204

6 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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

Advances in Mathematical Physics 3

with the boundary conditions

120601 (0) = 120601(120572)(119897) = 0 (18)

Equation (4) has the solution

120601 (119909) = 1198621cos120572120582120572119909120572+ 1198622sin120572120582120572119909120572 (19)

where 1198621and 119862

2are all constant numbers

According to (19) for 119909 = 0 and 119909 = 119897 we get

120601 (0) = 1198621= 0

120601 (119897) = 120601 (119909)1003816100381610038161003816119909=119897 = 1198622sin120572120582

120572119897120572= 0

(20)

Obviously 1198622= 0 since otherwise 120601(119909) equiv 0

Hence we arrive at

120582120572

119899119897120572= 119899120572120587120572 (21)

where 119899 is an integerWe notice

120582120572

119899= (

119899120587

119897)

120572

(119899 = 0 1 2 )

120601119899(119909) = sin

120572120582120572

119899119909120572

= sin120572119899120572(120587119909

119897)

120572

(119899 = 0 1 2 )

(22)

For 120582120572 = 120582120572119899and 0 lt 120588 following (17) implies that

infin

sum

119899=1

119879119899(119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572)

(23)

where

120588 =

radic4(119899120587119897)2120572minus 1

2

(24)

Therefore

119906119899(119909 119905) = 120601

119899(119909) 119879119899(119905)

= 119860119899cos120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

+ 119861119899sin120572120588119905120572sin120572119899120572(120587119909

119897)

120572

119864120572(minus

1

2119905120572)

(25)

We now suppose a local fractional Fourier series solution of(4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905)

=

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(26)

Therefore

120597120572119906 (119909 119905)

120597119905120572=

infin

sum

119899=1

120597120572119906119899(119909 119905)

120597119905120572 (27)

where

120597119906119899(119909 119905)

120597119905120572

= minus1

2119864120572(minus

119905120572

2) (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

+ 120588119864120572(minus

119905120572

2) (minus119860

119899sin120572120588119905120572+ 119861119899cos120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(28)

with 120588 = radic(4(119899120587119897)2120572 minus 1)2Submitting (26) to (5) we have

119906 (119909 0) =

infin

sum

119899=1

119906119899(119909 0)

=

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

(29)

120597120572119906 (119909 119905)

120597119905120572

=

infin

sum

119899=1

(minus1

2119860119899+ 120588119861119899) sin120572119899120572(120587119909

119897)

120572

= 119892 (119909)

(30)

So

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +

infin

sum

119899=1

1

2119860119899sin120572119899120572(120587119909

119897)

120572

= 119892 (119909) +1

2119891 (119909)

(31)

Let

119866 (119909) = 119892 (119909) +1

2119891 (119909) (32)

In view of (30) and (31) we rewrite

infin

sum

119899=1

119860119899sin120572119899120572(120587119909

119897)

120572

= 119891 (119909)

infin

sum

119899=1

120588119861119899sin120572119899120572(120587119909

119897)

120572

= 119866 (119909)

(33)

4 Advances in Mathematical Physics

We now find the local fractional Fourier coefficients of 119891(119909)and 119866(119909) respectively

119860119899=1 (Γ (1 + 120572)) int

119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

120588119861119899=1 (Γ (1 + 120572)) int

119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

(34)

Following (34) we have

1

Γ (1 + 120572)int

119897

0

sin2120572119899120572(120587119909

119897)

120572

(119889119909)120572=

119897120572

2Γ (1 + 120572) (35)

such that

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

(36)

Thus we get the solution of (4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905) (37)

where

119906119899(119909 119905) = 119864

120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(38)

with

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572(119899 = 0 1 2 )

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572(119899 = 0 1 2 )

(39)

with

119866 (119909) = 119892 (119909) +1

2119891 (119909) (40)

4 Illustrative Examples

In order to illustrate the above result in this section we givetwo examples

Let us consider (4) subject to initial and boundaryconditions

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) =

119909120572

Γ (1 + 120572)

(41)

In view of (40) we have

119866 (119909) = 119892 (119909) +1

2119891 (119909) =

3

2

119909120572

Γ (1 + 120572) (42)

such that

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

(43)

119861119899=3 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=3Γ (1 + 120572)

120588119897120572 0

119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(44)

Hence

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(45)

where

119860119899= minus

2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899= minus

3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(46)

Advances in Mathematical Physics 5

In view of (4) our second example is initial and boundaryconditions as follows

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) = 0

(47)

Following (40) we get

119866 (119909) =1

2

119909120572

Γ (1 + 120572) (48)

Hence we obtain

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899=int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=Γ (1 + 120572)

1205881198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minusΓ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(49)

So

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(50)

with

119860119899=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

119861119899= minus

Γ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(51)

10908070605040302010

1

09

08

07

06

05

04

03

02

01

0

f(x)

x

Figure 1 For 120572 = ln 2 ln 3 graph of a Lebesgue-Cantor staircasefunction shown at 119909 isin [0 1]

We notice that fraction boundary condition is expressedas a Lebesgue-Cantor staircase function [21 32] namely

119906 (119909 0) = 119891 (119909) = 119867120572(119862 cap (0 119909))

=0119868119909

(120572)1 =

119909120572

Γ (1 + 120572)

(52)

where 119862 is any fractal set and the fractal dimension of119909120572Γ(1 + 120572) is 120572 For 119909 isin [0 1] the graph of the Lebesgue-

Cantor staircase function (52) is shown in Figure 1 whenfractal dimension is 120572 = ln 2 ln 3

5 Conclusions

The present work expresses the local fractional Fourier seriessolution to wave equations with local fractional derivativeTwo examples are given to illustrat approximate solutions forwave equations with local fractional derivative resulting fromlocal fractional Fourier series method The results obtainedfrom the local fractional analysis seem to be general sincethe obtained solutions go back to the classical one whenfractal dimension 120572 = 1 namely it is a process from fractalgeometry to Euclidean geometry Local fractional Fourierseriesmethod is one of very efficient and powerful techniquesfor finding the solutions of the local fractional differentialequations It is also worth noting that the advantage of thelocal fractional differential equations displays the nondiffer-ential solutions which show the fractal and local behaviorsof moments However the classical Fourier series is used tohandle the continuous functions

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204

6 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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 Advances in Mathematical Physics

We now find the local fractional Fourier coefficients of 119891(119909)and 119866(119909) respectively

119860119899=1 (Γ (1 + 120572)) int

119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

120588119861119899=1 (Γ (1 + 120572)) int

119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

1 (Γ (1 + 120572)) int119897

0sin2120572119899120572(120587119909119897)

120572(119889119909)120572

(119899 = 0 1 2 )

(34)

Following (34) we have

1

Γ (1 + 120572)int

119897

0

sin2120572119899120572(120587119909

119897)

120572

(119889119909)120572=

119897120572

2Γ (1 + 120572) (35)

such that

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

(36)

Thus we get the solution of (4)

119906 (119909 119905) =

infin

sum

119899=1

119906119899(119909 119905) (37)

where

119906119899(119909 119905) = 119864

120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(38)

with

119860119899=2 int119897

0119891 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572(119899 = 0 1 2 )

119861119899=2 int119897

0119866 (119909) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572(119899 = 0 1 2 )

(39)

with

119866 (119909) = 119892 (119909) +1

2119891 (119909) (40)

4 Illustrative Examples

In order to illustrate the above result in this section we givetwo examples

Let us consider (4) subject to initial and boundaryconditions

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) =

119909120572

Γ (1 + 120572)

(41)

In view of (40) we have

119866 (119909) = 119892 (119909) +1

2119891 (119909) =

3

2

119909120572

Γ (1 + 120572) (42)

such that

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

(43)

119861119899=3 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=3Γ (1 + 120572)

120588119897120572 0

119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus (119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(44)

Hence

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(45)

where

119860119899= minus

2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899= minus

3Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(46)

Advances in Mathematical Physics 5

In view of (4) our second example is initial and boundaryconditions as follows

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) = 0

(47)

Following (40) we get

119866 (119909) =1

2

119909120572

Γ (1 + 120572) (48)

Hence we obtain

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899=int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=Γ (1 + 120572)

1205881198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minusΓ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(49)

So

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(50)

with

119860119899=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

119861119899= minus

Γ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(51)

10908070605040302010

1

09

08

07

06

05

04

03

02

01

0

f(x)

x

Figure 1 For 120572 = ln 2 ln 3 graph of a Lebesgue-Cantor staircasefunction shown at 119909 isin [0 1]

We notice that fraction boundary condition is expressedas a Lebesgue-Cantor staircase function [21 32] namely

119906 (119909 0) = 119891 (119909) = 119867120572(119862 cap (0 119909))

=0119868119909

(120572)1 =

119909120572

Γ (1 + 120572)

(52)

where 119862 is any fractal set and the fractal dimension of119909120572Γ(1 + 120572) is 120572 For 119909 isin [0 1] the graph of the Lebesgue-

Cantor staircase function (52) is shown in Figure 1 whenfractal dimension is 120572 = ln 2 ln 3

5 Conclusions

The present work expresses the local fractional Fourier seriessolution to wave equations with local fractional derivativeTwo examples are given to illustrat approximate solutions forwave equations with local fractional derivative resulting fromlocal fractional Fourier series method The results obtainedfrom the local fractional analysis seem to be general sincethe obtained solutions go back to the classical one whenfractal dimension 120572 = 1 namely it is a process from fractalgeometry to Euclidean geometry Local fractional Fourierseriesmethod is one of very efficient and powerful techniquesfor finding the solutions of the local fractional differentialequations It is also worth noting that the advantage of thelocal fractional differential equations displays the nondiffer-ential solutions which show the fractal and local behaviorsof moments However the classical Fourier series is used tohandle the continuous functions

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204

6 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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

Advances in Mathematical Physics 5

In view of (4) our second example is initial and boundaryconditions as follows

119906 (0 119905) = 119906 (119897 119905) =120597120572119906 (119897 0)

120597119909120572= 0

119906 (119909 0) = 119891 (119909) =119909120572

Γ (1 + 120572)

120597120572119906 (119909 0)

120597119905120572= 119892 (119909) = 0

(47)

Following (40) we get

119866 (119909) =1

2

119909120572

Γ (1 + 120572) (48)

Hence we obtain

119860119899=2 int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

119897120572

=2Γ (1 + 120572)

1198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minus2Γ (1 + 120572)

120588(119899120587)120572

[119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

119861119899=int119897

0(119909120572Γ (1 + 120572)) sin

120572119899120572(120587119909119897)

120572(119889119909)120572

120588119897120572

=Γ (1 + 120572)

1205881198971205720119868119897

(120572) 119909120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

= minusΓ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(49)

So

119906 (119909 119905) =

infin

sum

119899=1

119864120572(minus

119905120572

2)

times (119860119899cos120572120588119905120572+ 119861119899sin120572120588119905120572) sin120572119899120572(120587119909

119897)

120572

(50)

with

119860119899=2Γ (1 + 120572)

(119899120587)120572

119897120572

Γ (1 + 120572)sin120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

[cos120572119899120572(120587119909

119897)

120572

minus 1]

119861119899= minus

Γ (1 + 120572)

120588(119899120587)120572[

119897120572

Γ (1 + 120572)cos120572119899120572(120587119909

119897)

120572

minus(119897

119899120587)

120572

sin120572119899120572(120587119909

119897)

120572

]

(51)

10908070605040302010

1

09

08

07

06

05

04

03

02

01

0

f(x)

x

Figure 1 For 120572 = ln 2 ln 3 graph of a Lebesgue-Cantor staircasefunction shown at 119909 isin [0 1]

We notice that fraction boundary condition is expressedas a Lebesgue-Cantor staircase function [21 32] namely

119906 (119909 0) = 119891 (119909) = 119867120572(119862 cap (0 119909))

=0119868119909

(120572)1 =

119909120572

Γ (1 + 120572)

(52)

where 119862 is any fractal set and the fractal dimension of119909120572Γ(1 + 120572) is 120572 For 119909 isin [0 1] the graph of the Lebesgue-

Cantor staircase function (52) is shown in Figure 1 whenfractal dimension is 120572 = ln 2 ln 3

5 Conclusions

The present work expresses the local fractional Fourier seriessolution to wave equations with local fractional derivativeTwo examples are given to illustrat approximate solutions forwave equations with local fractional derivative resulting fromlocal fractional Fourier series method The results obtainedfrom the local fractional analysis seem to be general sincethe obtained solutions go back to the classical one whenfractal dimension 120572 = 1 namely it is a process from fractalgeometry to Euclidean geometry Local fractional Fourierseriesmethod is one of very efficient and powerful techniquesfor finding the solutions of the local fractional differentialequations It is also worth noting that the advantage of thelocal fractional differential equations displays the nondiffer-ential solutions which show the fractal and local behaviorsof moments However the classical Fourier series is used tohandle the continuous functions

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204

6 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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 Advances in Mathematical Physics

of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[2] F Mainardi Fractional Calculus and Waves in Linear Viscoelas-ticity Imperial College Press London UK 2010

[3] J Klafter S C Lim and R Metzler Fractional Dynamics inPhysics Recent Advances World Scientific Singapore 2012

[4] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[5] D Baleanu J A Tenreiro Machado and A C J Luo FractionalDynamics and Control Springer New York NY USA 2012

[6] J A Tenreiro Machado A C J Luo and D Baleanu NonlinearDynamics of Complex Systems Applications in Physical Biologi-cal and Financial Systems Springer New York NY USA 2011

[7] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press New YorkNY USA 1999

[8] 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

[9] S Das ldquoAnalytical solution of a fractional diffusion equation byvariational iteration methodrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 483ndash487 2009

[10] S Momani and Z Odibat ldquoComparison between the homotopyperturbation method and the variational iteration method forlinear fractional partial differential equationsrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 910ndash9192007

[11] 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

[12] H Jafari and S Seifi ldquoHomotopy analysis method for solvinglinear and nonlinear fractional diffusion-wave equationrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 5 pp 2006ndash2012 2009

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

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

[15] 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

[16] Z Zhao and C Li ldquoFractional differencefinite element approx-imations for the time-space fractional telegraph equationrdquoAppliedMathematics and Computation vol 219 no 6 pp 2975ndash2988 2012

[17] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[18] 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 Hacken-sack NJ USA 2012

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

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

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

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

[23] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo ZAMM Zeitschrift fur Ange-wandte Mathematik und Mechanik vol 90 no 3 pp 203ndash2102010

[24] G Jumarie ldquoProbability calculus of fractional order and frac-tional Taylorrsquos series application to Fokker-Planck equation andinformation of non-random functionsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1428ndash1448 2009

[25] Y Khan Q Wu N Faraz A Yildirim and M Madani ldquoAnew fractional analytical approach via a modified Riemann-Liouville derivativerdquo Applied Mathematics Letters vol 25 no10 pp 1340ndash1346 2012

[26] Y Khan N Faraz S Kumar and A Yildirim ldquoA couplingmethod of homotopy perturbation and Laplace transformationfor fractional modelsrdquo ldquoPolitehnicardquo University of Bucharest vol74 no 1 pp 57ndash68 2012

[27] 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

[28] W-H Su X-J Yang H Jafari and D Baleanu ldquoFractionalcomplex transform method for wave equations on Cantorsets within local fractional differential operatorrdquo Advances inDifference Equations vol 2013 article 97 2013

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

[30] 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

[31] Y J Yang D Baleanu and X J Yang ldquoA local fractionalvariational iteration method for Laplace equation within localfractional operatorsrdquo Abstract and Applied Analysis vol 2013Article ID 202650 6 pages 2013

[32] M-S Hu R P Agarwal and X-J Yang ldquoLocal fractionalFourier series with application to wave equation in fractalvibrating stringrdquo Abstract and Applied Analysis vol 2012Article ID 567401 15 pages 2012

[33] Y Zhang A Yang and X J Yang ldquo1-D heat conduction in afractal medium a solution by the local fractional Fourier seriesmethodrdquoThermal Science 2013

[34] G A Anastassiou and O Duman In Applied Mathematics andApproximation Theory Springer New York NY USA 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