Research ArticleLocal Fractional Variational Iteration Method for LocalFractional Poisson Equations in Two Independent Variables

Li Chen1 Yang Zhao2 Hossein Jafari34 J A Tenreiro Machado5 and Xiao-Jun Yang6

1 School of Mathematics and Statistics Zhengzhou Normal University Zhengzhou 450044 China2 Electronic and Information Technology Department Jiangmen Polytechnic Jiangmen 529090 China3Department of Mathematics University of Mazandaran Babolsar 47416-95447 Iran4Department of Mathematical Sciences University of South Africa Pretoria 7945 South Africa5 Department of Electrical Engineering Institute of Engineering Polytechnic of Porto Rua Dr Antonio Bernardino de Almeida 4314200-072 Porto Portugal

6Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Campus XuzhouJiangsu 221008 China

Correspondence should be addressed to Xiao-Jun Yang dyangxiaojun163com

Received 13 March 2014 Accepted 25 March 2014 Published 10 April 2014

Academic Editor Jordan Hristov

Copyright copy 2014 Li Chen et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the localfractional derivatives are investigated in this paperThe approximate solutions with the nondifferentiable functions are obtained byusing the local fractional variational iteration method

1 Introduction

As it is known the Poisson equation plays an importantrole in mathematical physics [1 2] that is it describes theelectrodynamics and intersecting interface (see eg [3 4]and the cited references therein)The solution of this equationwas discussed by using different methods [5ndash9] We noticethat recently fractional Poisson equations based on fractionalderivatives were analyzed in [10] and the existence andapproximations of its solutions can be found in [11] The Leg-endre wavelet method was used to find the fractional Poissonequation with Dirichlet boundary conditions [12] In [13] theDirichlet problem for the fractional Poissonrsquos equation withCaputo derivatives was reported Furthermore the fractionalPoisson equation based on the shiftedGrunwald estimate wasobtained in [14]

The variational iteration method structured in [15ndash17]was applied to deal with the following type of equationsHelmholtz [18] Burgerrsquos and coupled Burgerrsquos [19] Klein-Gordon [20] KdV [21] the oscillation [22] Schrodinger [23]reaction-diffusion [24] diffusion equation [25] Bernoulli

equation [26] and others The extended variational iterationmethod called the fractional variational iteration methodwas developed and applied to handle some fractional dif-ferential equations within the modified Riemann-Liouvillederivative [27ndash31] More recently the local fractional varia-tional iteration method initiated in [32] was used to findthe nondifferentiable solutions for the heat-conduction [32]Laplace [33] damped and dissipative wave [34] Helmholtz[35] and Fokker-Planck [36] equations the wave equation onCantor sets [37] and the fractal heat transfer in silk cocoonhierarchy [38] with local fractional derivative

We mention that developing a numerical algorithm forlocal fractional differential equations on Cantor set is notstraightforward Thus in this paper we deal with the localfractional Poisson equation in two independent variablesnamely

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= 119891 (119909 119910)

(119909 119910) isin [0 +infin] times [0 119897]

(1)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 484323 7 pageshttpdxdoiorg1011552014484323

2 Abstract and Applied Analysis

where the nondifferentiable functions 119906(119909 119910) and 119891(119909 119910)

are adopted the local fractional differential operators and120572 denotes the fractal dimension subject to the initial andboundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 120593 (119910)

120597120572

120597119909120572119906 (0 119910) = 120601 (119910)

(2)

We recall that the local fractional Laplace equation presentedin [33] is a special case of the local fractional Poisson equationwith source term119891(119909 119910) = 0 Taking all the above thinks intoaccount the aim of this paper is to find the nondifferentiablesolutions for (1) with different conditions by utilizing the localfractional variational iteration algorithm

The paper has the following organization In Section 2the concepts of local fractional complex derivatives andintegrals are briefly reviewed In Section 3 the local frac-tional variational iteration method is recalled In Section 4the nondifferentiable solutions for local fractional Poissonequations are presented Finally Section 5 outlines the mainconclusions

2 A Brief Review of the LocalFractional Calculus

Definition 1 (see [32ndash38]) Let the function 119891(119909) isin 119862120572(119886 119887)

if it satisfies the condition1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 lt 120576120572 (3)

where |119909 minus 1199090| lt 120575 for 120576 gt 0 0 lt 120572 lt 1 and 120576 isin 119877

Definition 2 (see [32ndash38]) Let 119891(119909) isin 119862120572(119886 119887) The local

fractional derivative of 119891(119909) of order 120572 is defined as

119889120572119891 (1199090)

119889119909120572=

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(4)

where

Δ120572(119891 (119909) minus 119891 (119909

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

0)] (5)

The formulas of local fractional derivatives of special func-tions [37] used in the paper are as follows

119889120572

119889119909120572

119909119899120572

Γ (1 + 119899120572)=

119909(119899minus1)120572

Γ (1 + (119899 minus 1) 120572) 119899 isin 119873

119863(120572)

119909119886119892 (119909) = 119886119863

(120572)

119909119892 (119909)

119863(120572)

119909[119863(120572)

119909119891 (119909)] = 119863

(2120572)

119909119891 (119909)

(6)

where 119892(119909) is a local fractional continuous function 119886 is aconstant and119873 is a set of positive integers

Definition 3 (see [32ndash38]) Let 119891(119909) isin 119862120572[119886 119887] The local

fractional integral of 119891(119909) of order 120572 in the interval [119886 119887] isdefined as

119886119868(120572)

119887119891 (119909) =

1

Γ (1 + 120572)int

119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) (Δ119905119895)120572

(7)

where the partitions of the interval [119886 119887] are denoted by(119905119895 119905119895+1

) 119895 = 0 119873 minus 1 1199050

= 119886 and 119905119873

= 119887 with Δ119905119895

=

119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

The formulas of local fractional integrals of special func-tions used in the paper are presented as follows [37]

0119868(120572)

119909119886119892 (119909) = 119886

0119868(120572)

119909119892 (119909)

0119868(120572)

119905(

(119905 minus 119904)120572119905119899120572

Γ (1 + 120572) Γ (1 + 119899120572)) =

119905(119899+2)120572

Γ (1 + (119899 + 2) 120572)

0119868(120572)

119909

119909(119899minus1)120572

Γ (1 + (119899 minus 1) 120572)=

119909119899120572

Γ (1 + 119899120572) 119899 isin 119873

(8)

where 119892(119909) is a local fractional continuous function 119886 is aconstant and119873 is a set of positive integers

3 Analysis of the Method

The local fractional variational iterationmethod structured in[32] was applied to deal with the local fractional differentialequations arising in mathematical physics (see eg [33ndash38])In this section we introduce the idea of the local fractionalvariational iteration method

Let us consider the local fractional operator equation inthe form

119871120572119906 + 119873

120572119906 = 119892 (119905) (9)

where 119871120572and 119873

120572are linear and nonlinear local fractional

operators respectively and 119892(119905) is the source term within thenondifferentiable function

Local fractional variational iteration algorithm reads as

119906119899+1

(119905) = 119906119899(119905) +0119868(120572)

119905120578 [119871120572119906119899(119904) + 119873

120572119906119899(119904) minus 119892 (119904)]

(10)

where 120578 is a fractal Lagrange multiplier and 119871120572= 120597119899120572120597119906119899120572

Therefore a local fractional correction functional wasstructured as follows

119906119899+1

(119905) = 119906119899(119905) +0119868(120572)

119905120585120572[119871120572119906119899(119904) + 119873

120572119899(119904) minus 119892 (119904)]

(11)

where 119899is considered as a restricted local fractional varia-

tion and 120578 is a fractal Lagrange multiplier That is 120575120572119899= 0

[27 30]After the fractal Lagrangian multiplier is determined for

119899 ge 0 the successive approximations 119906119899+1

of the solution 119906

Abstract and Applied Analysis 3

can be readily given by using any selective local fractionalfunction 119906

0 Consequently we obtain the solution in the

following form

119906 = lim119899rarrinfin

119906119899 (12)

The local fractional variational method was comparedwith the fractional series expansion and decompositiontechnologies

If 119871120572

= 12059721205721205971199062120572 then we have the local fractional

variational iteration formula [32ndash34 36 37] as follows

119906119899+1

(119905) = 119906119899(119905) minus0119868(120572)

119905

times (119904 minus 119905)

120572

Γ (1 + 120572)[119871120572119906119899(119904) + 119873

120572119906119899(119904) minus 119892 (119904)]

(13)

The above formula plays an important role in dealing withthe 2120572-order local fractional differential equation with eitherlinearity or nonlinearity

4 The Nondifferentiable Solutions for LocalFractional Poisson Equations

In this section we investigate the nondifferentiable solutionsfor the local fractional Poisson equations in two independentvariables with different initial-boundary conditions

Example 1 We analyze the local fractional Poisson equationin the following form

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572=

119910120572

Γ (1 + 120572) (14)

subject to the initial and boundary conditions namely

119906 (119909 0) = 0 (15)

119906 (119909 119897) = 0 (16)

119906 (0 119910) =1199103120572

Γ (1 + 3120572) (17)

120597120572

120597119909120572119906 (0 119910) = sin

120572(119910120572) (18)

In view of (17) and (18) we take the initial value given by

1199060(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

119909120572

Γ (1 + 120572) (19)

From (13) the local fractional iteration procedure is given by

119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906119899(119904 119910)

1205971199042120572+1205972120572119906119899(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

(20)

Making use of (19) and (20) we get the first approximation asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(21)

The second approximation can be written as

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(22)

The third approximation reads as

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(23)

The fourth approximation is as follows

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(24)

and so on

4 Abstract and Applied Analysis

08

06

04

02

0

1

0

05

1

002

04

0608

xy

u(xy)

Figure 1 Plot of the nondifferentiable solution of (14) with theparameter 120572 = ln 2 ln 3

Finally by direct calculations we obtain

119906119899(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

times

119899

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(25)

Hence we report the nondifferentiable solution of (14)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910)

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572) sin120572(119909120572)

(26)

and its graph is shown in Figure 1

Example 2 Next we discuss the local fractional Poissonequations as

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= 119864120572(119910120572) (27)

with the initial and boundary conditions given as follows

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 119864120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = cos

120572(119910120572)

(28)

In view of (13) the local fractional iteration procedurebecomes119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus 119864120572(119910120572))

(29)

where the initial value is given by

1199060(119909 119910) = 119864

120572(119910120572) +

119909120572

Γ (1 + 120572)cos120572(119910120572) (30)

Making use of (29) and (30) the first approximation reads asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(31)

The expression of the second approximation is as follows

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(32)

The third approximation becomes

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(33)

The fourth approximation is given by

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(34)

Abstract and Applied Analysis 5

5

4

3

2

1

01

0

05

1

002

0406

08

x

y

u(xy)

Figure 2 Plot of the nondifferentiable solution of (27) with theparameter 120572 = ln 2 ln 3

Therefore we get the nondifferentiable solution of (27)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = 119864

120572(119910120572) + cos

120572(119910120572) sin120572(119909120572)

(35)

and the corresponding graph is depicted in Figure 2

Example 3 The next particular case is the local fractionalPoisson equations as follows

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= sin120572(119909120572) (36)

subject to the initial and boundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = sin120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = 119864

120572(minus119910120572)

(37)

We start with the initial value as follows

1199060(119909 119905) = sin

120572(119910120572) +

119909120572

Γ (1 + 120572)119864120572(minus119910120572) (38)

The local fractional iteration procedure leads us to

119906119899+1

(119909 119910)

= 119906119899(119909 119910)

+0119868(120572)

119909

(119904 minus 119909)120572

Γ (1 + 120572)

times (1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus sin120572(119910120572))

(39)

In view of (38) and (39) we obtain the following successiveapproximations

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus sin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+12059721205721199061(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(40)

and so onThus the nondifferentiable solution of (36) has the form

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = sin

120572(119910120572) + 119864120572(minus119910120572) sin120572(119909120572)

(41)

and its graph is shown in Figure 3

5 Conclusions

The local fractional operators started to be deeply investigatedduring the last few years One of the major problems is

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

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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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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

where the nondifferentiable functions 119906(119909 119910) and 119891(119909 119910)

are adopted the local fractional differential operators and120572 denotes the fractal dimension subject to the initial andboundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 120593 (119910)

120597120572

120597119909120572119906 (0 119910) = 120601 (119910)

(2)

We recall that the local fractional Laplace equation presentedin [33] is a special case of the local fractional Poisson equationwith source term119891(119909 119910) = 0 Taking all the above thinks intoaccount the aim of this paper is to find the nondifferentiablesolutions for (1) with different conditions by utilizing the localfractional variational iteration algorithm

The paper has the following organization In Section 2the concepts of local fractional complex derivatives andintegrals are briefly reviewed In Section 3 the local frac-tional variational iteration method is recalled In Section 4the nondifferentiable solutions for local fractional Poissonequations are presented Finally Section 5 outlines the mainconclusions

2 A Brief Review of the LocalFractional Calculus

Definition 1 (see [32ndash38]) Let the function 119891(119909) isin 119862120572(119886 119887)

if it satisfies the condition1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 lt 120576120572 (3)

where |119909 minus 1199090| lt 120575 for 120576 gt 0 0 lt 120572 lt 1 and 120576 isin 119877

Definition 2 (see [32ndash38]) Let 119891(119909) isin 119862120572(119886 119887) The local

fractional derivative of 119891(119909) of order 120572 is defined as

119889120572119891 (1199090)

119889119909120572=

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(4)

where

Δ120572(119891 (119909) minus 119891 (119909

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

0)] (5)

The formulas of local fractional derivatives of special func-tions [37] used in the paper are as follows

119889120572

119889119909120572

119909119899120572

Γ (1 + 119899120572)=

119909(119899minus1)120572

Γ (1 + (119899 minus 1) 120572) 119899 isin 119873

119863(120572)

119909119886119892 (119909) = 119886119863

(120572)

119909119892 (119909)

119863(120572)

119909[119863(120572)

119909119891 (119909)] = 119863

(2120572)

119909119891 (119909)

(6)

where 119892(119909) is a local fractional continuous function 119886 is aconstant and119873 is a set of positive integers

Definition 3 (see [32ndash38]) Let 119891(119909) isin 119862120572[119886 119887] The local

fractional integral of 119891(119909) of order 120572 in the interval [119886 119887] isdefined as

119886119868(120572)

119887119891 (119909) =

1

Γ (1 + 120572)int

119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) (Δ119905119895)120572

(7)

where the partitions of the interval [119886 119887] are denoted by(119905119895 119905119895+1

) 119895 = 0 119873 minus 1 1199050

= 119886 and 119905119873

= 119887 with Δ119905119895

=

119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

The formulas of local fractional integrals of special func-tions used in the paper are presented as follows [37]

0119868(120572)

119909119886119892 (119909) = 119886

0119868(120572)

119909119892 (119909)

0119868(120572)

119905(

(119905 minus 119904)120572119905119899120572

Γ (1 + 120572) Γ (1 + 119899120572)) =

119905(119899+2)120572

Γ (1 + (119899 + 2) 120572)

0119868(120572)

119909

119909(119899minus1)120572

Γ (1 + (119899 minus 1) 120572)=

119909119899120572

Γ (1 + 119899120572) 119899 isin 119873

(8)

where 119892(119909) is a local fractional continuous function 119886 is aconstant and119873 is a set of positive integers

3 Analysis of the Method

The local fractional variational iterationmethod structured in[32] was applied to deal with the local fractional differentialequations arising in mathematical physics (see eg [33ndash38])In this section we introduce the idea of the local fractionalvariational iteration method

Let us consider the local fractional operator equation inthe form

119871120572119906 + 119873

120572119906 = 119892 (119905) (9)

where 119871120572and 119873

120572are linear and nonlinear local fractional

operators respectively and 119892(119905) is the source term within thenondifferentiable function

Local fractional variational iteration algorithm reads as

119906119899+1

(119905) = 119906119899(119905) +0119868(120572)

119905120578 [119871120572119906119899(119904) + 119873

120572119906119899(119904) minus 119892 (119904)]

(10)

where 120578 is a fractal Lagrange multiplier and 119871120572= 120597119899120572120597119906119899120572

Therefore a local fractional correction functional wasstructured as follows

119906119899+1

(119905) = 119906119899(119905) +0119868(120572)

119905120585120572[119871120572119906119899(119904) + 119873

120572119899(119904) minus 119892 (119904)]

(11)

where 119899is considered as a restricted local fractional varia-

tion and 120578 is a fractal Lagrange multiplier That is 120575120572119899= 0

[27 30]After the fractal Lagrangian multiplier is determined for

119899 ge 0 the successive approximations 119906119899+1

of the solution 119906

Abstract and Applied Analysis 3

can be readily given by using any selective local fractionalfunction 119906

0 Consequently we obtain the solution in the

following form

119906 = lim119899rarrinfin

119906119899 (12)

The local fractional variational method was comparedwith the fractional series expansion and decompositiontechnologies

If 119871120572

= 12059721205721205971199062120572 then we have the local fractional

variational iteration formula [32ndash34 36 37] as follows

119906119899+1

(119905) = 119906119899(119905) minus0119868(120572)

119905

times (119904 minus 119905)

120572

Γ (1 + 120572)[119871120572119906119899(119904) + 119873

120572119906119899(119904) minus 119892 (119904)]

(13)

The above formula plays an important role in dealing withthe 2120572-order local fractional differential equation with eitherlinearity or nonlinearity

4 The Nondifferentiable Solutions for LocalFractional Poisson Equations

In this section we investigate the nondifferentiable solutionsfor the local fractional Poisson equations in two independentvariables with different initial-boundary conditions

Example 1 We analyze the local fractional Poisson equationin the following form

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572=

119910120572

Γ (1 + 120572) (14)

subject to the initial and boundary conditions namely

119906 (119909 0) = 0 (15)

119906 (119909 119897) = 0 (16)

119906 (0 119910) =1199103120572

Γ (1 + 3120572) (17)

120597120572

120597119909120572119906 (0 119910) = sin

120572(119910120572) (18)

In view of (17) and (18) we take the initial value given by

1199060(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

119909120572

Γ (1 + 120572) (19)

From (13) the local fractional iteration procedure is given by

119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906119899(119904 119910)

1205971199042120572+1205972120572119906119899(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

(20)

Making use of (19) and (20) we get the first approximation asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(21)

The second approximation can be written as

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(22)

The third approximation reads as

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(23)

The fourth approximation is as follows

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(24)

and so on

4 Abstract and Applied Analysis

08

06

04

02

0

1

0

05

1

002

04

0608

xy

u(xy)

Figure 1 Plot of the nondifferentiable solution of (14) with theparameter 120572 = ln 2 ln 3

Finally by direct calculations we obtain

119906119899(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

times

119899

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(25)

Hence we report the nondifferentiable solution of (14)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910)

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572) sin120572(119909120572)

(26)

and its graph is shown in Figure 1

Example 2 Next we discuss the local fractional Poissonequations as

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= 119864120572(119910120572) (27)

with the initial and boundary conditions given as follows

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 119864120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = cos

120572(119910120572)

(28)

In view of (13) the local fractional iteration procedurebecomes119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus 119864120572(119910120572))

(29)

where the initial value is given by

1199060(119909 119910) = 119864

120572(119910120572) +

119909120572

Γ (1 + 120572)cos120572(119910120572) (30)

Making use of (29) and (30) the first approximation reads asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(31)

The expression of the second approximation is as follows

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(32)

The third approximation becomes

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(33)

The fourth approximation is given by

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(34)

Abstract and Applied Analysis 5

5

4

3

2

1

01

0

05

1

002

0406

08

x

y

u(xy)

Figure 2 Plot of the nondifferentiable solution of (27) with theparameter 120572 = ln 2 ln 3

Therefore we get the nondifferentiable solution of (27)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = 119864

120572(119910120572) + cos

120572(119910120572) sin120572(119909120572)

(35)

and the corresponding graph is depicted in Figure 2

Example 3 The next particular case is the local fractionalPoisson equations as follows

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= sin120572(119909120572) (36)

subject to the initial and boundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = sin120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = 119864

120572(minus119910120572)

(37)

We start with the initial value as follows

1199060(119909 119905) = sin

120572(119910120572) +

119909120572

Γ (1 + 120572)119864120572(minus119910120572) (38)

The local fractional iteration procedure leads us to

119906119899+1

(119909 119910)

= 119906119899(119909 119910)

+0119868(120572)

119909

(119904 minus 119909)120572

Γ (1 + 120572)

times (1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus sin120572(119910120572))

(39)

In view of (38) and (39) we obtain the following successiveapproximations

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus sin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+12059721205721199061(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(40)

and so onThus the nondifferentiable solution of (36) has the form

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = sin

120572(119910120572) + 119864120572(minus119910120572) sin120572(119909120572)

(41)

and its graph is shown in Figure 3

5 Conclusions

The local fractional operators started to be deeply investigatedduring the last few years One of the major problems is

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

can be readily given by using any selective local fractionalfunction 119906

0 Consequently we obtain the solution in the

following form

119906 = lim119899rarrinfin

119906119899 (12)

The local fractional variational method was comparedwith the fractional series expansion and decompositiontechnologies

If 119871120572

= 12059721205721205971199062120572 then we have the local fractional

variational iteration formula [32ndash34 36 37] as follows

119906119899+1

(119905) = 119906119899(119905) minus0119868(120572)

119905

times (119904 minus 119905)

120572

Γ (1 + 120572)[119871120572119906119899(119904) + 119873

120572119906119899(119904) minus 119892 (119904)]

(13)

The above formula plays an important role in dealing withthe 2120572-order local fractional differential equation with eitherlinearity or nonlinearity

4 The Nondifferentiable Solutions for LocalFractional Poisson Equations

In this section we investigate the nondifferentiable solutionsfor the local fractional Poisson equations in two independentvariables with different initial-boundary conditions

Example 1 We analyze the local fractional Poisson equationin the following form

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572=

119910120572

Γ (1 + 120572) (14)

subject to the initial and boundary conditions namely

119906 (119909 0) = 0 (15)

119906 (119909 119897) = 0 (16)

119906 (0 119910) =1199103120572

Γ (1 + 3120572) (17)

120597120572

120597119909120572119906 (0 119910) = sin

120572(119910120572) (18)

In view of (17) and (18) we take the initial value given by

1199060(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

119909120572

Γ (1 + 120572) (19)

From (13) the local fractional iteration procedure is given by

119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906119899(119904 119910)

1205971199042120572+1205972120572119906119899(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

(20)

Making use of (19) and (20) we get the first approximation asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(21)

The second approximation can be written as

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(22)

The third approximation reads as

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(23)

The fourth approximation is as follows

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus

119910120572

Γ (1 + 120572))

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(24)

and so on

4 Abstract and Applied Analysis

08

06

04

02

0

1

0

05

1

002

04

0608

xy

u(xy)

Figure 1 Plot of the nondifferentiable solution of (14) with theparameter 120572 = ln 2 ln 3

Finally by direct calculations we obtain

119906119899(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

times

119899

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(25)

Hence we report the nondifferentiable solution of (14)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910)

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572) sin120572(119909120572)

(26)

and its graph is shown in Figure 1

Example 2 Next we discuss the local fractional Poissonequations as

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= 119864120572(119910120572) (27)

with the initial and boundary conditions given as follows

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 119864120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = cos

120572(119910120572)

(28)

In view of (13) the local fractional iteration procedurebecomes119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus 119864120572(119910120572))

(29)

where the initial value is given by

1199060(119909 119910) = 119864

120572(119910120572) +

119909120572

Γ (1 + 120572)cos120572(119910120572) (30)

Making use of (29) and (30) the first approximation reads asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(31)

The expression of the second approximation is as follows

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(32)

The third approximation becomes

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(33)

The fourth approximation is given by

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(34)

Abstract and Applied Analysis 5

5

4

3

2

1

01

0

05

1

002

0406

08

x

y

u(xy)

Figure 2 Plot of the nondifferentiable solution of (27) with theparameter 120572 = ln 2 ln 3

Therefore we get the nondifferentiable solution of (27)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = 119864

120572(119910120572) + cos

120572(119910120572) sin120572(119909120572)

(35)

and the corresponding graph is depicted in Figure 2

Example 3 The next particular case is the local fractionalPoisson equations as follows

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= sin120572(119909120572) (36)

subject to the initial and boundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = sin120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = 119864

120572(minus119910120572)

(37)

We start with the initial value as follows

1199060(119909 119905) = sin

120572(119910120572) +

119909120572

Γ (1 + 120572)119864120572(minus119910120572) (38)

The local fractional iteration procedure leads us to

119906119899+1

(119909 119910)

= 119906119899(119909 119910)

+0119868(120572)

119909

(119904 minus 119909)120572

Γ (1 + 120572)

times (1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus sin120572(119910120572))

(39)

In view of (38) and (39) we obtain the following successiveapproximations

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus sin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+12059721205721199061(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(40)

and so onThus the nondifferentiable solution of (36) has the form

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = sin

120572(119910120572) + 119864120572(minus119910120572) sin120572(119909120572)

(41)

and its graph is shown in Figure 3

5 Conclusions

The local fractional operators started to be deeply investigatedduring the last few years One of the major problems is

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

08

06

04

02

0

1

0

05

1

002

04

0608

xy

u(xy)

Figure 1 Plot of the nondifferentiable solution of (14) with theparameter 120572 = ln 2 ln 3

Finally by direct calculations we obtain

119906119899(119909 119910) =

1199103120572

Γ (1 + 3120572)+ sin120572(119910120572)

times

119899

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(25)

Hence we report the nondifferentiable solution of (14)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910)

=1199103120572

Γ (1 + 3120572)+ sin120572(119910120572) sin120572(119909120572)

(26)

and its graph is shown in Figure 1

Example 2 Next we discuss the local fractional Poissonequations as

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= 119864120572(119910120572) (27)

with the initial and boundary conditions given as follows

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = 119864120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = cos

120572(119910120572)

(28)

In view of (13) the local fractional iteration procedurebecomes119906119899+1

(119909 119910)

= 119906119899(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus 119864120572(119910120572))

(29)

where the initial value is given by

1199060(119909 119910) = 119864

120572(119910120572) +

119909120572

Γ (1 + 120572)cos120572(119910120572) (30)

Making use of (29) and (30) the first approximation reads asfollows

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(31)

The expression of the second approximation is as follows

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+

12059721205721199061(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(32)

The third approximation becomes

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(33)

The fourth approximation is given by

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minus 119864120572(119910120572))

= 119864120572(119910120572) + cos

120572(119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(34)

Abstract and Applied Analysis 5

5

4

3

2

1

01

0

05

1

002

0406

08

x

y

u(xy)

Figure 2 Plot of the nondifferentiable solution of (27) with theparameter 120572 = ln 2 ln 3

Therefore we get the nondifferentiable solution of (27)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = 119864

120572(119910120572) + cos

120572(119910120572) sin120572(119909120572)

(35)

and the corresponding graph is depicted in Figure 2

Example 3 The next particular case is the local fractionalPoisson equations as follows

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= sin120572(119909120572) (36)

subject to the initial and boundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = sin120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = 119864

120572(minus119910120572)

(37)

We start with the initial value as follows

1199060(119909 119905) = sin

120572(119910120572) +

119909120572

Γ (1 + 120572)119864120572(minus119910120572) (38)

The local fractional iteration procedure leads us to

119906119899+1

(119909 119910)

= 119906119899(119909 119910)

+0119868(120572)

119909

(119904 minus 119909)120572

Γ (1 + 120572)

times (1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus sin120572(119910120572))

(39)

In view of (38) and (39) we obtain the following successiveapproximations

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus sin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+12059721205721199061(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(40)

and so onThus the nondifferentiable solution of (36) has the form

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = sin

120572(119910120572) + 119864120572(minus119910120572) sin120572(119909120572)

(41)

and its graph is shown in Figure 3

5 Conclusions

The local fractional operators started to be deeply investigatedduring the last few years One of the major problems is

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

5

4

3

2

1

01

0

05

1

002

0406

08

x

y

u(xy)

Figure 2 Plot of the nondifferentiable solution of (27) with theparameter 120572 = ln 2 ln 3

Therefore we get the nondifferentiable solution of (27)

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = 119864

120572(119910120572) + cos

120572(119910120572) sin120572(119909120572)

(35)

and the corresponding graph is depicted in Figure 2

Example 3 The next particular case is the local fractionalPoisson equations as follows

1205972120572119906 (119909 119910)

1205971199092120572+

1205972120572119906 (119909 119910)

1205971199102120572= sin120572(119909120572) (36)

subject to the initial and boundary conditions

119906 (119909 0) = 0

119906 (119909 119897) = 0

119906 (0 119910) = sin120572(119910120572)

120597120572

120597119909120572119906 (0 119910) = 119864

120572(minus119910120572)

(37)

We start with the initial value as follows

1199060(119909 119905) = sin

120572(119910120572) +

119909120572

Γ (1 + 120572)119864120572(minus119910120572) (38)

The local fractional iteration procedure leads us to

119906119899+1

(119909 119910)

= 119906119899(119909 119910)

+0119868(120572)

119909

(119904 minus 119909)120572

Γ (1 + 120572)

times (1205972120572119906 (119904 119910)

1205971199042120572+

1205972120572119906 (119904 119909)

1205971199102120572minus sin120572(119910120572))

(39)

In view of (38) and (39) we obtain the following successiveapproximations

1199061(119909 119910)

= 1199060(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199060(119904 119910)

1205971199042120572+

12059721205721199060(119904 119909)

1205971199102120572minus sin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

1

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199062(119909 119910)

= 1199061(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199061(119904 119910)

1205971199042120572+12059721205721199061(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

2

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199063(119909 119910)

= 1199062(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199062(119904 119910)

1205971199042120572+

12059721205721199062(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

3

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

1199064(119909 119910)

= 1199063(119909 119910) +

0119868(120572)

119909

times (119904 minus 119909)

120572

Γ (1 + 120572)(12059721205721199063(119904 119910)

1205971199042120572+

12059721205721199063(119904 119909)

1205971199102120572minussin120572(119910120572))

= sin120572(119910120572) + 119864120572(minus119910120572)

4

sum

119894=0

(minus1)119894 119909

(2119894+1)120572

Γ (1 + (2119894 + 1) 120572)

(40)

and so onThus the nondifferentiable solution of (36) has the form

119906 (119909 119910) = lim119899rarrinfin

119906119899(119909 119910) = sin

120572(119910120572) + 119864120572(minus119910120572) sin120572(119909120572)

(41)

and its graph is shown in Figure 3

5 Conclusions

The local fractional operators started to be deeply investigatedduring the last few years One of the major problems is

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

08

06

04

02

1

1

0

05

1

002

0406

08

x

y

u(xy)

Figure 3 Plot of the nondifferentiable solution of (36) with theparameter 120572 = ln 2 ln 3

to find new methods and techniques to solve some givenimportant local fractional partial differential equations onCantor set In this line of thought we consider that threelocal fractional Poisson equations with differential initial andboundary values were solved by using the local fractionalvariational iteration method The graphs of the nondifferen-tiable solutions were also obtained

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundationof Henan Province China

References

[1] L C Evans Partial Differential Equations vol 19 of GraduateStudies in Mathematics American Mathematical Society Prov-idence RI USA 1998

[2] H C Elman D J Silvester and A J Wathen Finite Elementsand Fast Iterative Solvers With Applications in IncompressibleFluid Dynamics Oxford University Press New York NY USA2005

[3] D J Griffiths and R College Introduction to ElectrodynamicsPrentice Hall Upper Saddle River NJ USA 1999

[4] R B Kellogg ldquoOn the Poisson equation with intersectinginterfacesrdquo Applicable Analysis vol 4 no 2 pp 101ndash129 1974

[5] R W Hockney ldquoA fast direct solution of Poissonrsquos equationusing Fourier analysisrdquo Journal of the Association for ComputingMachinery vol 12 no 1 pp 95ndash113 1965

[6] P Neittaanmaki and J Saranen ldquoOn finite element approx-imation of the gradient for solution of Poisson equationrdquoNumerische Mathematik vol 37 no 3 pp 333ndash337 1981

[7] F Civan and C M Sliepcevich ldquoSolution of the Poissonequation by differential quadraturerdquo International Journal for

Numerical Methods in Engineering vol 19 no 5 pp 711ndash7241983

[8] W Dorfler ldquoA convergent adaptive algorithm for Poissonrsquosequationrdquo SIAM Journal on Numerical Analysis vol 33 no 3pp 1106ndash1124 1996

[9] Z Cai and S Kim ldquoA finite element method using singularfunctions for the Poisson equation corner singularitiesrdquo SIAMJournal on Numerical Analysis vol 39 no 1 pp 286ndash299 2002

[10] Y Derriennic and M Lin ldquoFractional Poisson equations andergodic theorems for fractional coboundariesrdquo Israel Journal ofMathematics vol 123 no 1 pp 93ndash130 2001

[11] M Sanz-Sole and I Torrecilla ldquoA fractional poisson equa-tion existence regularity and approximations of the solutionrdquoStochastics and Dynamics vol 9 no 4 pp 519ndash548 2009

[12] M H Heydari M R Hooshmandasl F M Maalek Ghainiand F Fereidouni ldquoTwo-dimensional Legendre wavelets forsolving fractional Poisson equation with Dirichlet boundaryconditionsrdquo Engineering Analysis with Boundary Elements vol37 no 11 pp 1331ndash1338 2013

[13] V D Beibalaev and R P Meilanov ldquoThe Dirihlet problemfor the fractional Poissonrsquos equation with Caputo derivativesa finite difference approximation and a numerical solutionrdquoThermal Science vol 16 no 2 pp 385ndash394 2012

[14] B Abdollah and V Sohrab ldquoFractional finite difference methodfor solving the fractional Poisson equation based on the shiftedGrunwald estimaterdquoWalailak Journal of Science andTechnologyvol 15 no 2 pp 427ndash435 2013

[15] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

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

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

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

[19] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[20] S Abbasbandy ldquoNumerical solution of non-linear Klein-Gordon equations by variational iteration methodrdquo Interna-tional Journal for Numerical Methods in Engineering vol 70 no7 pp 876ndash881 2007

[21] S T Mohyud-Din M A Noor and K I Noor ldquoTraveling wavesolutions of seventh-order generalized KdV equations usingHersquos polynomialsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 10 no 2 pp 227ndash234 2009

[22] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008

[23] A-M Wazwaz ldquoA study on linear and nonlinear Schrodingerequations by the variational iteration methodrdquo Chaos Solitonsamp Fractals vol 37 no 4 pp 1136ndash1142 2008

[24] M Dehghan and F Shakeri ldquoApplication of Hersquos variationaliteration method for solving the Cauchy reaction-diffusionproblemrdquo Journal of Computational and Applied Mathematicsvol 214 no 2 pp 435ndash446 2008

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

[25] S Das ldquoApproximate solution of fractional diffusion equation-revisitedrdquo International Review of Chemical Engineering vol 4no 5 pp 501ndash504 2012

[26] J Hristov ldquoAn exercise with the Hersquos variation iteration methodto a fractional Bernoulli equation arising in a transient conduc-tionwith a non-linear boundary heat fluxrdquo International Reviewof Chemical Engineering vol 4 no 5 pp 489ndash497 2012

[27] G-C Wu and E W M Lee ldquoFractional variational iterationmethod and its applicationrdquo Physics Letters A General Atomicand Solid State Physics vol 374 no 25 pp 2506ndash2509 2010

[28] J-H He ldquoA short remark on fractional variational iterationmethodrdquo Physics Letters A General Atomic and Solid StatePhysics vol 375 no 38 pp 3362ndash3364 2011

[29] N Faraz Y Khan H Jafari A Yildirim and M MadanildquoFractional variational iterationmethod viamodifiedRiemann-Liouville derivativerdquo Journal of King Saud University Sciencevol 23 no 4 pp 413ndash417 2011

[30] S Guo and L Mei ldquoThe fractional variational iteration methodusing Hersquos polynomialsrdquo Physics Letters A General Atomic andSolid State Physics vol 375 no 3 pp 309ndash313 2011

[31] E A Hussain and Z M Alwan ldquoComparison of variationaland fractional variational iteration methods in solving time-fractional differential equationsrdquo Journal of Advance in Math-ematics vol 6 no 1 pp 849ndash858 2014

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

[33] 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

[34] 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 andApplications vol 2013 no 1 article 86 pp 1ndash11 2013

[35] A M Yang Z S Chen H M Srivastava and X J YangldquoApplication of the local fractional series expansion methodand the variational iteration method to the Helmholtz equationinvolving local fractional derivative operatorsrdquo Abstract andApplied Analysis vol 2013 Article ID 259125 6 pages 2013

[36] X J Yang andD Baleanu ldquoLocal fractional variational iterationmethod for Fokker-Planck equation on a Cantor setrdquo ActaUniversitaria vol 23 no 2 pp 3ndash8 2013

[37] D Baleanu J A T Machado C Cattani M C Baleanu andX-J Yang ldquoLocal fractional variational iteration and decompo-sition methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014

[38] J H He and F J Liu ldquoLocal fractional variational iterationmethod for fractal heat transfer in silk cocoon hierarchyrdquoNonlinear Science Letters A vol 4 no 1 pp 15ndash20 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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