Research ArticleLocal Fractional Adomian Decomposition and FunctionDecomposition Methods for Laplace Equation within LocalFractional Operators

Sheng-Ping Yan1 Hossein Jafari2 and Hassan Kamil Jassim3

1 School of Mechanics and Civil Engineering China University of Mining and Technology Xuzhou 221116 China2Department of Mathematical Sciences University of South Africa Pretoria South Africa3 Department of Mathematics University of Mazandaran Babolsar 47415-416 Iran

Correspondence should be addressed to Sheng-Ping Yan spyancumteducn

Received 26 May 2014 Accepted 9 June 2014 Published 30 June 2014

Academic Editor Xiao-Jun Yang

Copyright copy 2014 Sheng-Ping Yan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We perform a comparison between the local fractional Adomian decomposition and local fractional function decompositionmethods applied to the Laplace equation The operators are taken in the local sense The results illustrate the significant featuresof the two methods which are both very effective and straightforward for solving the differential equations with local fractionalderivative

1 Introduction

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

Several analytical and numerical techniques were suc-cessfully applied to deal with differential equations fractionaldifferential equations and local fractional differential equa-tions [1ndash10]The techniques include the heat-balance integral[11] the fractional Fourier [12] the fractional Laplace trans-form [12] the harmonic wavelet [13 14] the local fractionalFourier and Laplace transform [15] local fractional varia-tional iteration [16ndash18] the local fractional decomposition[19] and the generalized local fractional Fourier transform[20] methods

In this paper we investigate the application of localfractional Adomian decomposition method and local frac-tional function decomposition method for solving the localfractional Laplace equation [21 22] with the different fractalconditions

This paper is organized as follows In Section 2 the basicmathematical tools are reviewed Section 3 presents brieflythe local fractional Adomian decomposition method and thelocal fractional function decomposition method Section 4presents solutions to the local fractional Laplace equationwith differential fractal conditions

2 Mathematical Fundamentals

We recall in this section the notations and some properties ofthe local fractional operators [15ndash20 23 24]

Definition 1 (see [15ndash20 23 24]) The function 119891(119909) is localfractional continuous at 119909 = 119909

0 if it is valid for

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 0 lt 120572 le 1 (1)

with |119909 minus 1199090| lt 120575 for 120576 gt 0 and 120576 isin 119877 For 119909 isin (119886 119887) it

is so called local fractional continuous on the interval (119886 119887)denoted by 119891(119909) isin 119862

120572(119886 119887)

We notice that there are existence conditions of local frac-tional continuities that operating functions are right-hand

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 161580 7 pageshttpdxdoiorg1011552014161580

2 Advances in Mathematical Physics

and left-hand local fractional continuities Meanwhile theright-hand local fractional continuity is equal to its left-handlocal fractional continuity For more details see [20]

Definition 2 (see [15ndash20 23 24]) The local fractional deriva-tive of 119891(119909) at 119909 = 119909

0is defined as

119863120572119909119891 (1199090) =

119889120572

119889119909120572119891 (119909)

10038161003816100381610038161003816100381610038161003816119909=1199090

= 119891(120572) (119909) = lim119909rarr1199090

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(2)

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

0))

Local fractional derivative of high order is written in theform

119891(119896120572) (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863120572119909119863120572119909sdot sdot sdot 119863120572119909119891 (119909)

(3)

And local fractional partial derivative of high order is writtenin the form

120597119896120572119891 (119909 119910)

119909119896120572=

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

120597119909120572120597120572

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909 119910)

(4)

Definition 3 (see [15ndash20 23 24]) A partition of the interval[119886 119887] is denoted by (119905

119895 119905119895+1) 119895 = 0 119873 minus 1 119905

0= 119886 and

119905119873= 119887 with Δ119905

119895= 119905119895+1minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Local

fractional integral of 119891(119909) in the interval [119886 119887] is given by

119886119868(120572)119887119891 (119909) =

1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(5)

If the functions are local fractional continuous thenthe local fractional derivatives and integrals exist Someproperties of local fractional derivative and integrals are givenin [20]

Definition 4 Let 119891(119909) be 2119897-periodic For 119896 isin 119885 and 119891(119909) isin119862120572(119886 119887) the local fraction Fourier series of 119891(119909) is defined as

(see [15 25])

119891 (119909) =1198860

2+infin

sum119896=1

(119886119896cos120572

120587120572(119896119909)120572

119897120572+ 119887119896sin120572

120587120572(119896119909)120572

119897120572)

(6)

where 119886119896= (1119897120572) int

1

minus1119891(119909)cos

120572(120587120572(119896119909)120572119897120572)(119889119909)120572

119887119896=1

119897120572int1

minus1

119891 (119909) sin120572120587120572(119896119909)120572

119897120572(119889119909)120572 (7)

are local fractional Fourier coefficients

Definition 5 Let (1Γ(1 + 120572)) intinfin0|119891(119909)|(119889119909)120572 lt 119896 lt infin The

Yang-Laplace transforms of 119891(119909) are given by [15 22]

119871120572119891 (119909) = 119891

119871120572

119904(119904) =

1

Γ (1 + 120572)intinfin

0

119864120572(minus119904120572119909120572) 119891 (119909) (119889119909)

120572

0 lt 120572 le 1

(8)

where the latter integral converges and 119904120572 isin 119877120572

Definition 6 The inverse formula of the Yang-Laplace trans-forms of 119891(119909) is given by [15 22]

119871minus1120572119891119871120572119904(119904)

= 119891 (119905) =1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572) 119891119871120572

119904(119904) (119889119904)

120572

0 lt 120572 le 1

(9)

where 119904120572 = 120573120572 + 119894120572120596120572 fractal imaginary unit 119894120572and Re(119904) =120573 gt 0

3 Analytical Methods

In order to illustrate two analytical methods we investigatethe nonlinear local fractional equation of order 2120572 as follows

1205972120572119906 (119909 119905)

1205971199052120572+ 1198961

120597120572119906 (119909 119905)

120597119905120572+ 1198962

1205972120572119906 (119909 119905)

1205971199092120572+ 1198963

120597120572119906 (119909 119905)

120597119909120572

= 119891 (119909 119905)

(10)

with constants 1198961 1198962 1198963 0 lt 120572 le 1 and with boundary and

initial conditions

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

119906 (119909 0) = 120593 (119909)

120597120572119906 (119909 0)

120597119905120572= 120595 (119909)

(11)

31 Local Fractional Adomian Decomposition Method Werewrite (10) in the following form

119871(2120572)119905119905119906 (119909 119905) + 119896

1119871(120572)119905119906 (119909 119905) + 119896

2119871(2120572)119909119909119906 (119909 119905) + 119896

3119871(120572)119909119906 (119909 119905)

= 119891 (119909 119905)

(12)

Advances in Mathematical Physics 3

Applying the inverse operator 119871(minus2120572)119905119905

to both sides of (12)yields

119871(minus2120572)119905119905

119871(2120572)119905119905119906 (119909 119905)

= 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 1198962119871

(2120572)

119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904) + 119891 (119909 119904))

119906 (119909 119905) = 119903 (119909 119905) + 119871(minus2120572)

119905119905(119891 (119909 119904))

+ 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 119896

2119871(2120572)119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904))

(13)

where the term 119903(119909 119905) is to be determined from the fractalinitial conditions

Now we decompose the unknown function 119906(119909 119905) as asum of components defined by the series

119906 (119909 119905) =infin

sum119899=0

119906119899(119909 119905) (14)

The components 119906119899(119909 119905) are obtained by the recursive for-

mula

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899(119909 119904)) 119899 ge 0

(15)

32 Local Fractional Function Decomposition MethodAccording to the decomposition of the local fractionalfunction with respect to the system sin

120572119899120572(120587119909119897)120572 the

following functions coefficients can be given by

119906 (119909 119905) =infin

sum119899=1

V119899(119905) sin

120572119899120572(

120587119909

119897)120572

119891 (119909 119905) =infin

sum119899=1

119891119899(119905) sin

120572119899120572(

120587119909

119897)120572

120593 (119909) =infin

sum119899=1

119862119899sin120572119899120572(

120587119909

119897)120572

120595 (119909) =infin

sum119899=1

119863119899sin120572119899120572(

120587119909

119897)120572

(16)

where

119891119899(119905) =

2

119897120572int1

0

119891 (119909 119905) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119862119899=2

119897120572int1

0

120593 (119909) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119863119899=2

119897120572int1

0

120593 (119909) sin120572119899120572 (

120587119909

119897)

120572

(119889119909)120572

(17)

Substituting (16) into (10) implies that

1205972120572V119899 (119905)

1205971199052120572+ 1198961

120597120572V119899 (119905)

120597119905120572+ 1198962(119899120587

119897)2120572

V119899 (119905) + 1198963(

119899120587

119897)120572

V119899 (119905)

= 119891119899(119905)

V119899 (0) = 119862119899 V1015840

119899(0) = 119863119899

(18)

Suppose that the Yang-Laplace transforms of functions V119899(119905)

and 119891119899(119905) are 119881

119899(119904) and 119865

119899(119904) respectively Then we obtain

1199042120572119881119899(119904) minus 119862

119899119904120572 minus 119863

119899+ 1198961(119904120572119881119899(119904) minus 119862

119899) + 1198962(119899120587

119897)2120572

119881119899(119904)

+ 1198963(119899120587

119897)120572

119881119899 (119904) = 119865119899 (119904)

(19)

That is

119881119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

+119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(20)

Hence we have

V119899(119905)

= 119871minus1120572119881119899(119904)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572) 119881

119899(119904) (119889119904)

120572

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

+1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

(21)

4 Advances in Mathematical Physics

Let V119899(119905) = V

1119899(119905) + V

2119899(119905)

V1119899 (119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899 (119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(22)

V2119899(119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(23)

Hence we get

1198811119899(119904) =

119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

1198812119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(24)

Then making use of (8) and (9) and rearranging integrationsequence we have the following several formulas about V

1119899(119905)

and V2119899(119905)

If minus(14)11989621+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 gt 0 then

1199042120572 + 1198961119904 + 1198962(119899120587

119897)2120572

+ 1198963(119899120587

119897)120572

= (119904120572 +1198961

2)2

+ 1198631015840119899

(25)

where1198631015840119899= radicminus(14)1198962

1+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572

Then we get

V1119899(119905) =

1

Γ (1 + 120572)1198631015840119899

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(1198631015840119899120591120572) 119891119899(119905 minus 120591) (119889120591)

120572

V2119899(119905) = 119862

119899119864120572(minus1198961119905120572

2120572) cos120572(1198631015840119899119905120572)

+ (119863119899+ 1198961119862119899minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(1198631015840119899119905120572)

(26)

In case minus(14)11989621+1198962(119899120587119897)2120572 +119896

3(119899120587119897)120572 lt 0 and minus(14)1198962

1+

1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 = 0 see [26]

4 Solutions of Local Fractional LaplaceEquation in Fractal Time-Space

In this section two examples for Laplace equation arepresented in order to demonstrate the simplicity and theefficiency of the above methods

The local fractional Laplace equation (see [21]) is oneof the important differential equations with local fractionalderivatives In the following we consider solutions to localfractional Laplace equations in fractal time-space

Example 7 Consider the following local fractional Laplaceequation

1205972120572119906 (119909 119905)

1205971199052120572+1205972119906 (119909 119905)

1205971199092120572= 0 (27)

subject to the fractal value conditions

119906 (119909 0) = minus119864120572(119909120572)

120597120572119906 (119909 0)

120597119905120572= 0 (28)

According to formula (15) we have

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899 (119909 119904))

(29)

where

1199060(119909 119905) = minus119864

120572(119909120572) (30)

Hence from (29) we obtain

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905minus119871(2120572)119909119909119906119899(119909 119904)

=0119868(120572)119905 0

119868(120572)119905minus

1205972120572119906119899 (119909 119904)

1205971199092120572

119899 ge 1

(31)

where

1199060(119909 119905) = minus119864

120572(119909120572) (32)

Making use of (31) we present

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905119864120572(119909120572)

=1199052120572

Γ (1 + 2120572)119864120572(119909120572)

Advances in Mathematical Physics 5

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199052120572

Γ (1 + 2120572)119864120572(119909120572)

= minus1199054120572

Γ (1 + 4120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199054120572

Γ (1 + 4120572)119864120572(119909120572)

=1199056120572

Γ (1 + 6120572)119864120572(119909120572)

(33)

Proceeding in this manner we get

119906119899(119909 119905) = 119864

120572(119909120572) (minus1)

119899+1 1199052119899120572

Γ (1 + 2119899120572) (34)

Thus the final solution reads as follows

119906 (119909 119905) =infin

sum119899=0

119906119899 (119909 119905) = 1199060 (119909 119905) + 1199061 (119909 119905) + 1199062 (119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus1 +

1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572)

+1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) [1 minus

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572)

minus1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) cos

120572(119905120572)

(35)

Now we solve Example 7 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) =infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(36)

002

0406

081

0

05

10

05

1

15

2

25

3

t

x

u(xt)

Figure 1 Exact solution for local fractional Laplace equation withfractal dimension 120572 = ln 2 ln 3

which leads to

119891119899 (119905) = 0 forall119899 119862

119899= 0 119899 = 1 119862

1= minus1

119863119899= 0 forall119899

(37)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1

and 1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899(119905) = 0 119899 = 1

(38)

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911(119905 minus 120591) (119889120591)

120572 = 0

(39)

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(11986310158401119905120572)

= minus cos120572(119905120572)

(40)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minuscos

120572(119905120572) (41)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) cos

120572(119905120572) (42)

and its graph is shown in Figure 1

Example 8 We consider the following local fractionalLaplace equation

1205972120572119906 (119909 119905)

1205971199052120572+1205972120572119906 (119909 119905)

1205971199092120572= 0 (43)

6 Advances in Mathematical Physics

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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

and left-hand local fractional continuities Meanwhile theright-hand local fractional continuity is equal to its left-handlocal fractional continuity For more details see [20]

Definition 2 (see [15ndash20 23 24]) The local fractional deriva-tive of 119891(119909) at 119909 = 119909

0is defined as

119863120572119909119891 (1199090) =

119889120572

119889119909120572119891 (119909)

10038161003816100381610038161003816100381610038161003816119909=1199090

= 119891(120572) (119909) = lim119909rarr1199090

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(2)

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

0))

Local fractional derivative of high order is written in theform

119891(119896120572) (119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863120572119909119863120572119909sdot sdot sdot 119863120572119909119891 (119909)

(3)

And local fractional partial derivative of high order is writtenin the form

120597119896120572119891 (119909 119910)

119909119896120572=

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

120597119909120572120597120572

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909 119910)

(4)

Definition 3 (see [15ndash20 23 24]) A partition of the interval[119886 119887] is denoted by (119905

119895 119905119895+1) 119895 = 0 119873 minus 1 119905

0= 119886 and

119905119873= 119887 with Δ119905

119895= 119905119895+1minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Local

fractional integral of 119891(119909) in the interval [119886 119887] is given by

119886119868(120572)119887119891 (119909) =

1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(5)

If the functions are local fractional continuous thenthe local fractional derivatives and integrals exist Someproperties of local fractional derivative and integrals are givenin [20]

Definition 4 Let 119891(119909) be 2119897-periodic For 119896 isin 119885 and 119891(119909) isin119862120572(119886 119887) the local fraction Fourier series of 119891(119909) is defined as

(see [15 25])

119891 (119909) =1198860

2+infin

sum119896=1

(119886119896cos120572

120587120572(119896119909)120572

119897120572+ 119887119896sin120572

120587120572(119896119909)120572

119897120572)

(6)

where 119886119896= (1119897120572) int

1

minus1119891(119909)cos

120572(120587120572(119896119909)120572119897120572)(119889119909)120572

119887119896=1

119897120572int1

minus1

119891 (119909) sin120572120587120572(119896119909)120572

119897120572(119889119909)120572 (7)

are local fractional Fourier coefficients

Definition 5 Let (1Γ(1 + 120572)) intinfin0|119891(119909)|(119889119909)120572 lt 119896 lt infin The

Yang-Laplace transforms of 119891(119909) are given by [15 22]

119871120572119891 (119909) = 119891

119871120572

119904(119904) =

1

Γ (1 + 120572)intinfin

0

119864120572(minus119904120572119909120572) 119891 (119909) (119889119909)

120572

0 lt 120572 le 1

(8)

where the latter integral converges and 119904120572 isin 119877120572

Definition 6 The inverse formula of the Yang-Laplace trans-forms of 119891(119909) is given by [15 22]

119871minus1120572119891119871120572119904(119904)

= 119891 (119905) =1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572) 119891119871120572

119904(119904) (119889119904)

120572

0 lt 120572 le 1

(9)

where 119904120572 = 120573120572 + 119894120572120596120572 fractal imaginary unit 119894120572and Re(119904) =120573 gt 0

3 Analytical Methods

In order to illustrate two analytical methods we investigatethe nonlinear local fractional equation of order 2120572 as follows

1205972120572119906 (119909 119905)

1205971199052120572+ 1198961

120597120572119906 (119909 119905)

120597119905120572+ 1198962

1205972120572119906 (119909 119905)

1205971199092120572+ 1198963

120597120572119906 (119909 119905)

120597119909120572

= 119891 (119909 119905)

(10)

with constants 1198961 1198962 1198963 0 lt 120572 le 1 and with boundary and

initial conditions

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

119906 (119909 0) = 120593 (119909)

120597120572119906 (119909 0)

120597119905120572= 120595 (119909)

(11)

31 Local Fractional Adomian Decomposition Method Werewrite (10) in the following form

119871(2120572)119905119905119906 (119909 119905) + 119896

1119871(120572)119905119906 (119909 119905) + 119896

2119871(2120572)119909119909119906 (119909 119905) + 119896

3119871(120572)119909119906 (119909 119905)

= 119891 (119909 119905)

(12)

Advances in Mathematical Physics 3

Applying the inverse operator 119871(minus2120572)119905119905

to both sides of (12)yields

119871(minus2120572)119905119905

119871(2120572)119905119905119906 (119909 119905)

= 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 1198962119871

(2120572)

119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904) + 119891 (119909 119904))

119906 (119909 119905) = 119903 (119909 119905) + 119871(minus2120572)

119905119905(119891 (119909 119904))

+ 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 119896

2119871(2120572)119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904))

(13)

where the term 119903(119909 119905) is to be determined from the fractalinitial conditions

Now we decompose the unknown function 119906(119909 119905) as asum of components defined by the series

119906 (119909 119905) =infin

sum119899=0

119906119899(119909 119905) (14)

The components 119906119899(119909 119905) are obtained by the recursive for-

mula

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899(119909 119904)) 119899 ge 0

(15)

32 Local Fractional Function Decomposition MethodAccording to the decomposition of the local fractionalfunction with respect to the system sin

120572119899120572(120587119909119897)120572 the

following functions coefficients can be given by

119906 (119909 119905) =infin

sum119899=1

V119899(119905) sin

120572119899120572(

120587119909

119897)120572

119891 (119909 119905) =infin

sum119899=1

119891119899(119905) sin

120572119899120572(

120587119909

119897)120572

120593 (119909) =infin

sum119899=1

119862119899sin120572119899120572(

120587119909

119897)120572

120595 (119909) =infin

sum119899=1

119863119899sin120572119899120572(

120587119909

119897)120572

(16)

where

119891119899(119905) =

2

119897120572int1

0

119891 (119909 119905) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119862119899=2

119897120572int1

0

120593 (119909) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119863119899=2

119897120572int1

0

120593 (119909) sin120572119899120572 (

120587119909

119897)

120572

(119889119909)120572

(17)

Substituting (16) into (10) implies that

1205972120572V119899 (119905)

1205971199052120572+ 1198961

120597120572V119899 (119905)

120597119905120572+ 1198962(119899120587

119897)2120572

V119899 (119905) + 1198963(

119899120587

119897)120572

V119899 (119905)

= 119891119899(119905)

V119899 (0) = 119862119899 V1015840

119899(0) = 119863119899

(18)

Suppose that the Yang-Laplace transforms of functions V119899(119905)

and 119891119899(119905) are 119881

119899(119904) and 119865

119899(119904) respectively Then we obtain

1199042120572119881119899(119904) minus 119862

119899119904120572 minus 119863

119899+ 1198961(119904120572119881119899(119904) minus 119862

119899) + 1198962(119899120587

119897)2120572

119881119899(119904)

+ 1198963(119899120587

119897)120572

119881119899 (119904) = 119865119899 (119904)

(19)

That is

119881119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

+119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(20)

Hence we have

V119899(119905)

= 119871minus1120572119881119899(119904)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572) 119881

119899(119904) (119889119904)

120572

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

+1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

(21)

4 Advances in Mathematical Physics

Let V119899(119905) = V

1119899(119905) + V

2119899(119905)

V1119899 (119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899 (119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(22)

V2119899(119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(23)

Hence we get

1198811119899(119904) =

119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

1198812119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(24)

Then making use of (8) and (9) and rearranging integrationsequence we have the following several formulas about V

1119899(119905)

and V2119899(119905)

If minus(14)11989621+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 gt 0 then

1199042120572 + 1198961119904 + 1198962(119899120587

119897)2120572

+ 1198963(119899120587

119897)120572

= (119904120572 +1198961

2)2

+ 1198631015840119899

(25)

where1198631015840119899= radicminus(14)1198962

1+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572

Then we get

V1119899(119905) =

1

Γ (1 + 120572)1198631015840119899

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(1198631015840119899120591120572) 119891119899(119905 minus 120591) (119889120591)

120572

V2119899(119905) = 119862

119899119864120572(minus1198961119905120572

2120572) cos120572(1198631015840119899119905120572)

+ (119863119899+ 1198961119862119899minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(1198631015840119899119905120572)

(26)

In case minus(14)11989621+1198962(119899120587119897)2120572 +119896

3(119899120587119897)120572 lt 0 and minus(14)1198962

1+

1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 = 0 see [26]

4 Solutions of Local Fractional LaplaceEquation in Fractal Time-Space

In this section two examples for Laplace equation arepresented in order to demonstrate the simplicity and theefficiency of the above methods

The local fractional Laplace equation (see [21]) is oneof the important differential equations with local fractionalderivatives In the following we consider solutions to localfractional Laplace equations in fractal time-space

Example 7 Consider the following local fractional Laplaceequation

1205972120572119906 (119909 119905)

1205971199052120572+1205972119906 (119909 119905)

1205971199092120572= 0 (27)

subject to the fractal value conditions

119906 (119909 0) = minus119864120572(119909120572)

120597120572119906 (119909 0)

120597119905120572= 0 (28)

According to formula (15) we have

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899 (119909 119904))

(29)

where

1199060(119909 119905) = minus119864

120572(119909120572) (30)

Hence from (29) we obtain

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905minus119871(2120572)119909119909119906119899(119909 119904)

=0119868(120572)119905 0

119868(120572)119905minus

1205972120572119906119899 (119909 119904)

1205971199092120572

119899 ge 1

(31)

where

1199060(119909 119905) = minus119864

120572(119909120572) (32)

Making use of (31) we present

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905119864120572(119909120572)

=1199052120572

Γ (1 + 2120572)119864120572(119909120572)

Advances in Mathematical Physics 5

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199052120572

Γ (1 + 2120572)119864120572(119909120572)

= minus1199054120572

Γ (1 + 4120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199054120572

Γ (1 + 4120572)119864120572(119909120572)

=1199056120572

Γ (1 + 6120572)119864120572(119909120572)

(33)

Proceeding in this manner we get

119906119899(119909 119905) = 119864

120572(119909120572) (minus1)

119899+1 1199052119899120572

Γ (1 + 2119899120572) (34)

Thus the final solution reads as follows

119906 (119909 119905) =infin

sum119899=0

119906119899 (119909 119905) = 1199060 (119909 119905) + 1199061 (119909 119905) + 1199062 (119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus1 +

1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572)

+1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) [1 minus

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572)

minus1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) cos

120572(119905120572)

(35)

Now we solve Example 7 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) =infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(36)

002

0406

081

0

05

10

05

1

15

2

25

3

t

x

u(xt)

Figure 1 Exact solution for local fractional Laplace equation withfractal dimension 120572 = ln 2 ln 3

which leads to

119891119899 (119905) = 0 forall119899 119862

119899= 0 119899 = 1 119862

1= minus1

119863119899= 0 forall119899

(37)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1

and 1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899(119905) = 0 119899 = 1

(38)

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911(119905 minus 120591) (119889120591)

120572 = 0

(39)

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(11986310158401119905120572)

= minus cos120572(119905120572)

(40)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minuscos

120572(119905120572) (41)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) cos

120572(119905120572) (42)

and its graph is shown in Figure 1

Example 8 We consider the following local fractionalLaplace equation

1205972120572119906 (119909 119905)

1205971199052120572+1205972120572119906 (119909 119905)

1205971199092120572= 0 (43)

6 Advances in Mathematical Physics

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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

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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

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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

Applying the inverse operator 119871(minus2120572)119905119905

to both sides of (12)yields

119871(minus2120572)119905119905

119871(2120572)119905119905119906 (119909 119905)

= 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 1198962119871

(2120572)

119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904) + 119891 (119909 119904))

119906 (119909 119905) = 119903 (119909 119905) + 119871(minus2120572)

119905119905(119891 (119909 119904))

+ 119871(minus2120572)119905119905

(minus1198961119871(120572)119904119906 (119909 119904) minus 119896

2119871(2120572)119909119909119906 (119909 119904)

minus 1198963119871(120572)119909119906 (119909 119904))

(13)

where the term 119903(119909 119905) is to be determined from the fractalinitial conditions

Now we decompose the unknown function 119906(119909 119905) as asum of components defined by the series

119906 (119909 119905) =infin

sum119899=0

119906119899(119909 119905) (14)

The components 119906119899(119909 119905) are obtained by the recursive for-

mula

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899(119909 119904)) 119899 ge 0

(15)

32 Local Fractional Function Decomposition MethodAccording to the decomposition of the local fractionalfunction with respect to the system sin

120572119899120572(120587119909119897)120572 the

following functions coefficients can be given by

119906 (119909 119905) =infin

sum119899=1

V119899(119905) sin

120572119899120572(

120587119909

119897)120572

119891 (119909 119905) =infin

sum119899=1

119891119899(119905) sin

120572119899120572(

120587119909

119897)120572

120593 (119909) =infin

sum119899=1

119862119899sin120572119899120572(

120587119909

119897)120572

120595 (119909) =infin

sum119899=1

119863119899sin120572119899120572(

120587119909

119897)120572

(16)

where

119891119899(119905) =

2

119897120572int1

0

119891 (119909 119905) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119862119899=2

119897120572int1

0

120593 (119909) sin120572119899120572(

120587119909

119897)120572

(119889119909)120572

119863119899=2

119897120572int1

0

120593 (119909) sin120572119899120572 (

120587119909

119897)

120572

(119889119909)120572

(17)

Substituting (16) into (10) implies that

1205972120572V119899 (119905)

1205971199052120572+ 1198961

120597120572V119899 (119905)

120597119905120572+ 1198962(119899120587

119897)2120572

V119899 (119905) + 1198963(

119899120587

119897)120572

V119899 (119905)

= 119891119899(119905)

V119899 (0) = 119862119899 V1015840

119899(0) = 119863119899

(18)

Suppose that the Yang-Laplace transforms of functions V119899(119905)

and 119891119899(119905) are 119881

119899(119904) and 119865

119899(119904) respectively Then we obtain

1199042120572119881119899(119904) minus 119862

119899119904120572 minus 119863

119899+ 1198961(119904120572119881119899(119904) minus 119862

119899) + 1198962(119899120587

119897)2120572

119881119899(119904)

+ 1198963(119899120587

119897)120572

119881119899 (119904) = 119865119899 (119904)

(19)

That is

119881119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

+119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(20)

Hence we have

V119899(119905)

= 119871minus1120572119881119899(119904)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572) 119881

119899(119904) (119889119904)

120572

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

+1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(119889119904)120572

(21)

4 Advances in Mathematical Physics

Let V119899(119905) = V

1119899(119905) + V

2119899(119905)

V1119899 (119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899 (119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(22)

V2119899(119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(23)

Hence we get

1198811119899(119904) =

119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

1198812119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(24)

Then making use of (8) and (9) and rearranging integrationsequence we have the following several formulas about V

1119899(119905)

and V2119899(119905)

If minus(14)11989621+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 gt 0 then

1199042120572 + 1198961119904 + 1198962(119899120587

119897)2120572

+ 1198963(119899120587

119897)120572

= (119904120572 +1198961

2)2

+ 1198631015840119899

(25)

where1198631015840119899= radicminus(14)1198962

1+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572

Then we get

V1119899(119905) =

1

Γ (1 + 120572)1198631015840119899

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(1198631015840119899120591120572) 119891119899(119905 minus 120591) (119889120591)

120572

V2119899(119905) = 119862

119899119864120572(minus1198961119905120572

2120572) cos120572(1198631015840119899119905120572)

+ (119863119899+ 1198961119862119899minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(1198631015840119899119905120572)

(26)

In case minus(14)11989621+1198962(119899120587119897)2120572 +119896

3(119899120587119897)120572 lt 0 and minus(14)1198962

1+

1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 = 0 see [26]

4 Solutions of Local Fractional LaplaceEquation in Fractal Time-Space

In this section two examples for Laplace equation arepresented in order to demonstrate the simplicity and theefficiency of the above methods

The local fractional Laplace equation (see [21]) is oneof the important differential equations with local fractionalderivatives In the following we consider solutions to localfractional Laplace equations in fractal time-space

Example 7 Consider the following local fractional Laplaceequation

1205972120572119906 (119909 119905)

1205971199052120572+1205972119906 (119909 119905)

1205971199092120572= 0 (27)

subject to the fractal value conditions

119906 (119909 0) = minus119864120572(119909120572)

120597120572119906 (119909 0)

120597119905120572= 0 (28)

According to formula (15) we have

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899 (119909 119904))

(29)

where

1199060(119909 119905) = minus119864

120572(119909120572) (30)

Hence from (29) we obtain

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905minus119871(2120572)119909119909119906119899(119909 119904)

=0119868(120572)119905 0

119868(120572)119905minus

1205972120572119906119899 (119909 119904)

1205971199092120572

119899 ge 1

(31)

where

1199060(119909 119905) = minus119864

120572(119909120572) (32)

Making use of (31) we present

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905119864120572(119909120572)

=1199052120572

Γ (1 + 2120572)119864120572(119909120572)

Advances in Mathematical Physics 5

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199052120572

Γ (1 + 2120572)119864120572(119909120572)

= minus1199054120572

Γ (1 + 4120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199054120572

Γ (1 + 4120572)119864120572(119909120572)

=1199056120572

Γ (1 + 6120572)119864120572(119909120572)

(33)

Proceeding in this manner we get

119906119899(119909 119905) = 119864

120572(119909120572) (minus1)

119899+1 1199052119899120572

Γ (1 + 2119899120572) (34)

Thus the final solution reads as follows

119906 (119909 119905) =infin

sum119899=0

119906119899 (119909 119905) = 1199060 (119909 119905) + 1199061 (119909 119905) + 1199062 (119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus1 +

1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572)

+1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) [1 minus

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572)

minus1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) cos

120572(119905120572)

(35)

Now we solve Example 7 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) =infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(36)

002

0406

081

0

05

10

05

1

15

2

25

3

t

x

u(xt)

Figure 1 Exact solution for local fractional Laplace equation withfractal dimension 120572 = ln 2 ln 3

which leads to

119891119899 (119905) = 0 forall119899 119862

119899= 0 119899 = 1 119862

1= minus1

119863119899= 0 forall119899

(37)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1

and 1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899(119905) = 0 119899 = 1

(38)

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911(119905 minus 120591) (119889120591)

120572 = 0

(39)

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(11986310158401119905120572)

= minus cos120572(119905120572)

(40)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minuscos

120572(119905120572) (41)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) cos

120572(119905120572) (42)

and its graph is shown in Figure 1

Example 8 We consider the following local fractionalLaplace equation

1205972120572119906 (119909 119905)

1205971199052120572+1205972120572119906 (119909 119905)

1205971199092120572= 0 (43)

6 Advances in Mathematical Physics

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Let V119899(119905) = V

1119899(119905) + V

2119899(119905)

V1119899 (119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119865119899 (119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(22)

V2119899(119905)

=1

(2120587)120572int120573+119894120596

120573minus119894120596

119864120572(119904120572119909120572)

times119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572(119889119904)120572

(23)

Hence we get

1198811119899(119904) =

119865119899(119904)

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

1198812119899 (119904) =

119863119899+ 1198961119862119899+ 119862119899119904120572

1199042120572 + 1198961119904120572 + 119896

2(119899120587119897)2120572 + 119896

3(119899120587119897)120572

(24)

Then making use of (8) and (9) and rearranging integrationsequence we have the following several formulas about V

1119899(119905)

and V2119899(119905)

If minus(14)11989621+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 gt 0 then

1199042120572 + 1198961119904 + 1198962(119899120587

119897)2120572

+ 1198963(119899120587

119897)120572

= (119904120572 +1198961

2)2

+ 1198631015840119899

(25)

where1198631015840119899= radicminus(14)1198962

1+ 1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572

Then we get

V1119899(119905) =

1

Γ (1 + 120572)1198631015840119899

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(1198631015840119899120591120572) 119891119899(119905 minus 120591) (119889120591)

120572

V2119899(119905) = 119862

119899119864120572(minus1198961119905120572

2120572) cos120572(1198631015840119899119905120572)

+ (119863119899+ 1198961119862119899minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(1198631015840119899119905120572)

(26)

In case minus(14)11989621+1198962(119899120587119897)2120572 +119896

3(119899120587119897)120572 lt 0 and minus(14)1198962

1+

1198962(119899120587119897)2120572 + 119896

3(119899120587119897)120572 = 0 see [26]

4 Solutions of Local Fractional LaplaceEquation in Fractal Time-Space

In this section two examples for Laplace equation arepresented in order to demonstrate the simplicity and theefficiency of the above methods

The local fractional Laplace equation (see [21]) is oneof the important differential equations with local fractionalderivatives In the following we consider solutions to localfractional Laplace equations in fractal time-space

Example 7 Consider the following local fractional Laplaceequation

1205972120572119906 (119909 119905)

1205971199052120572+1205972119906 (119909 119905)

1205971199092120572= 0 (27)

subject to the fractal value conditions

119906 (119909 0) = minus119864120572(119909120572)

120597120572119906 (119909 0)

120597119905120572= 0 (28)

According to formula (15) we have

1199060(119909 119905) = 119903 (119909 119905) + 119871

(minus2120572)

119905119905(119891 (119909 119904))

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905(minus1198961119871(120572)119904119906119899(119909 119904) minus 119896

2119871(2120572)119909119909119906119899(119909 119904)

minus 1198963119871(120572)119909119906119899 (119909 119904))

(29)

where

1199060(119909 119905) = minus119864

120572(119909120572) (30)

Hence from (29) we obtain

119906119899+1

(119909 119905) = 119871(minus2120572)

119905119905minus119871(2120572)119909119909119906119899(119909 119904)

=0119868(120572)119905 0

119868(120572)119905minus

1205972120572119906119899 (119909 119904)

1205971199092120572

119899 ge 1

(31)

where

1199060(119909 119905) = minus119864

120572(119909120572) (32)

Making use of (31) we present

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905119864120572(119909120572)

=1199052120572

Γ (1 + 2120572)119864120572(119909120572)

Advances in Mathematical Physics 5

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199052120572

Γ (1 + 2120572)119864120572(119909120572)

= minus1199054120572

Γ (1 + 4120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199054120572

Γ (1 + 4120572)119864120572(119909120572)

=1199056120572

Γ (1 + 6120572)119864120572(119909120572)

(33)

Proceeding in this manner we get

119906119899(119909 119905) = 119864

120572(119909120572) (minus1)

119899+1 1199052119899120572

Γ (1 + 2119899120572) (34)

Thus the final solution reads as follows

119906 (119909 119905) =infin

sum119899=0

119906119899 (119909 119905) = 1199060 (119909 119905) + 1199061 (119909 119905) + 1199062 (119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus1 +

1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572)

+1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) [1 minus

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572)

minus1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) cos

120572(119905120572)

(35)

Now we solve Example 7 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) =infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(36)

002

0406

081

0

05

10

05

1

15

2

25

3

t

x

u(xt)

Figure 1 Exact solution for local fractional Laplace equation withfractal dimension 120572 = ln 2 ln 3

which leads to

119891119899 (119905) = 0 forall119899 119862

119899= 0 119899 = 1 119862

1= minus1

119863119899= 0 forall119899

(37)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1

and 1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899(119905) = 0 119899 = 1

(38)

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911(119905 minus 120591) (119889120591)

120572 = 0

(39)

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(11986310158401119905120572)

= minus cos120572(119905120572)

(40)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minuscos

120572(119905120572) (41)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) cos

120572(119905120572) (42)

and its graph is shown in Figure 1

Example 8 We consider the following local fractionalLaplace equation

1205972120572119906 (119909 119905)

1205971199052120572+1205972120572119906 (119909 119905)

1205971199092120572= 0 (43)

6 Advances in Mathematical Physics

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

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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

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199052120572

Γ (1 + 2120572)119864120572(119909120572)

= minus1199054120572

Γ (1 + 4120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199054120572

Γ (1 + 4120572)119864120572(119909120572)

=1199056120572

Γ (1 + 6120572)119864120572(119909120572)

(33)

Proceeding in this manner we get

119906119899(119909 119905) = 119864

120572(119909120572) (minus1)

119899+1 1199052119899120572

Γ (1 + 2119899120572) (34)

Thus the final solution reads as follows

119906 (119909 119905) =infin

sum119899=0

119906119899 (119909 119905) = 1199060 (119909 119905) + 1199061 (119909 119905) + 1199062 (119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus1 +

1199052120572

Γ (1 + 2120572)minus

1199054120572

Γ (1 + 4120572)

+1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) [1 minus

1199052120572

Γ (1 + 2120572)+

1199054120572

Γ (1 + 4120572)

minus1199056120572

Γ (1 + 6120572)sdot sdot sdot ]

= minus 119864120572(119909120572) cos

120572(119905120572)

(35)

Now we solve Example 7 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) =infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(36)

002

0406

081

0

05

10

05

1

15

2

25

3

t

x

u(xt)

Figure 1 Exact solution for local fractional Laplace equation withfractal dimension 120572 = ln 2 ln 3

which leads to

119891119899 (119905) = 0 forall119899 119862

119899= 0 119899 = 1 119862

1= minus1

119863119899= 0 forall119899

(37)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1

and 1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899(119905) = 0 119899 = 1

(38)

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911(119905 minus 120591) (119889120591)

120572 = 0

(39)

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(11986310158401119905120572)

= minus cos120572(119905120572)

(40)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minuscos

120572(119905120572) (41)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) cos

120572(119905120572) (42)

and its graph is shown in Figure 1

Example 8 We consider the following local fractionalLaplace equation

1205972120572119906 (119909 119905)

1205971199052120572+1205972120572119906 (119909 119905)

1205971199092120572= 0 (43)

6 Advances in Mathematical Physics

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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

subject to the fractal value conditions

119906 (119909 0) = 0120597120572119906 (119909 0)

120597119905120572= minus119864120572(119909120572) (44)

Now we can structure the same local fractional iterationprocedure (15) Hence we have

1199060 (119909 119905) = minus

119905120572

Γ (1 + 120572)119864120572(119909120572)

1199061(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199060(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

119905120572

Γ (1 + 120572)119864120572(119909120572)

=1199053120572

Γ (1 + 3120572)119864120572(119909120572)

1199062(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199061(119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905minus

1199053120572

Γ (1 + 3120572)119864120572(119909120572)

= minus1199055120572

Γ (1 + 5120572)119864120572(119909120572)

1199063(119909 119905) =

0119868(120572)119905 0

119868(120572)119905minus

12059721205721199062 (119909 119904)

1205971199092120572

=0119868(120572)119905 0

119868(120572)119905

1199055120572

Γ (1 + 5120572)119864120572(119909120572)

=1199057120572

Γ (1 + 7120572)119864120572(119909120572)

(45)

Finally we can obtain the local fractional series solution asfollows

119906119899(119909 119905) = (minus1)

119899+1 119905(2119899+1)120572

Γ (1 + (2119899 + 1) 120572)119864120572(119909120572) (46)

Thus the final solution reads as follows

119906 (119909 119905)

=infin

sum119899=0

119906119899(119909 119905) = 119906

0(119909 119905) + 119906

1(119909 119905) + 119906

2(119909 119905) + sdot sdot sdot

= 119864120572(119909120572) [minus

119905120572

Γ (1 + 120572)+

1199053120572

Γ (1 + 3120572)minus

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus119864120572(119909120572) [

119905120572

Γ (1 + 120572)minus

1199053120572

Γ (1 + 3120572)+

1199055120572

Γ (1 + 5120572)sdot sdot sdot ]

= minus 119864120572(119909120572) sin

120572(119905120572)

(47)

Now we solve Example 8 by using the local fractionalfunction decomposition method

We suppose that

119906 (119909 119905) =infin

sum119899=1

V119899(119905) 119864120572(119899120572119909120572)

119891 (119909 119905) = 0 =infin

sum119899=1

119891119899 (119905) 119864120572 (119899

120572119909120572)

120593 (119909) = 0 =infin

sum119899=1

119862119899119864120572(119899120572119909120572)

120595 (119909) = minus119864120572(119909120572) =

infin

sum119899=1

119863119899119864120572(119899120572119909120572)

(48)

which leads to

119891119899 (119905) = 0 forall119899 119863

119899= 0 119899 = 1 119863

1= minus1

119862119899= 0 forall119899

(49)

Contrasting (28) with (36) we directly get 1198961= 0 119896

2= 1 and

1198963= 0 and

1198631015840119899= 0 119899 = 1 1198631015840

1= 1

V119899 (119905) = 0 119899 = 1

V11(119905) =

1

Γ (1 + 120572)11986310158401

times int119905

0

119864120572(minus1198961120591120572

2120572) sin120572(11986310158401120591120572) 1198911 (119905 minus 120591) (119889120591)

120572

= 0

V21(119905) = 119862

1119864120572(minus1198961119905120572

2120572) cos120572(11986310158401119905120572)

+ (1198631+ 11989611198621minus1198961

2)119864120572(minus1198961119905120572

2120572) sin120572(119905120572)

= minus sin120572(119905120572)

(50)

Conclusively we get

V1(119905) = V

11(119905) + V

21(119905) = minussin

120572(119905120572) (51)

Thus we obtain

119906 (119909 119905) = minus119864120572(119909120572) sin

120572(119905120572) (52)

and its graph is given in Figure 2

5 Conclusions

In this work solving the Laplace equations using the localfractional function decomposition method with local frac-tional operators is discussed in detail Two examples of

Advances in Mathematical Physics 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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 7

002

0406

081

0

05

10

05

1

15

2

t

x

u(xt)

Figure 2 The plot of solution to local fractional Laplace equationwith fractal dimension 120572 = ln 2 ln 3

applications of the local fractional Adomian decompositionmethod and local fractional function decomposition methodto the local fractional Laplace equations are investigated indetail The reliable obtained results are complementary withthe ones presented in the literature

Conflict of Interests

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

References

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

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

[3] Z 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

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

[5] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 2002

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

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

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

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

[10] Z Li W Zhu and L Huang ldquoApplication of fractional vari-ational iteration method to time-fractional Fisher equationrdquoAdvanced Science Letters vol 10 pp 610ndash614 2012

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

[12] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

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

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

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

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

[17] W-H Su D Baleanu X-J Yang and H Jafari ldquoDamped waveequation and dissipative wave equation in fractal strings withinthe local fractional variational iteration methodrdquo Fixed PointTheory and Applications vol 2013 no 89 2013

[18] 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

[19] X Yang D Baleanu andW Zhong ldquoApproximate solutions fordiffusion equations on Cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

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

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

[22] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

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

[24] A Yang X Yang and Z Li ldquoLocal fractional series expansionmethod for solving wave and diffusion equations on Cantorsetsrdquo Abstract and Applied Analysis vol 2013 Article ID 3510575 pages 2013

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

[26] S Q Wang Y J Yang and H K Jassim ldquoLocal fractionalfunction decomposition method for solving inhomogeneouswave equations with local fractional derivativerdquo Abstract andApplied Analysis vol 2014 Article ID 176395 7 pages 2014

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