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
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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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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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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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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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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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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
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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
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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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
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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
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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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
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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
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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
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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