Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 462535 5 pageshttpdxdoiorg1011552013462535

Research ArticleAnalytical Solutions of the One-Dimensional HeatEquations Arising in Fractal Transient Conduction withLocal Fractional Derivative

Ai-Ming Yang12 Carlo Cattani3 Hossein Jafari45 and Xiao-Jun Yang6

1 College of Science Hebei United University Tangshan 063009 China2 College of Mechanical Engineering Yanshan University Qinhuangdao 066004 China3Department of Mathematics University of Salerno Via Ponte don Melillo Fisciano 84084 Salerno Italy4Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran Babolsar Iran5 International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical SciencesNorth-West University Mafikeng Campus Private Bag X 2046 Mmabatho 2735 South Africa

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

Correspondence should be addressed to Ai-Ming Yang aimin heut163com

Received 26 September 2013 Accepted 17 October 2013

Academic Editor Abdon Atangana

Copyright copy 2013 Ai-Ming Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The one-dimensional heat equations with the heat generation arising in fractal transient conduction associated with local fractionalderivative operators are investigated Analytical solutions are obtained by using the local fractional Adomian decompositionmethod via local fractional calculus theoryThemethod in general is easy to implement and yields good results Illustrative examplesare included to demonstrate the validity and applicability of the new technique

1 Introduction

The Adomian decomposition method [1ndash3] was applied toprocess linear and nonlinear problems in the fields of scienceand engineering Tatari and Dehghan [4] applied Adomiandecomposition method to process the multipoint boundaryvalue problem Wazwaz [5] used Adomian decompositionmethod to deal with the Bratu-type equations Daftardar-Gejji and Jafari [6] considered Adomian decompositionmethod to analyze the Bagley Torvik equation Larsson [7]presented the solution for Helmholtz equation by usingthe Adomain decomposition method Tatari and coworkers[8] investigated solution for the Fokker-Planck equation byAdomain decomposition method

Fractional calculus [9ndash12] was applied to model the phys-ical and engineering problems for expressions of stress-strainconstitutive relations of different viscoelastic fractional orderproperties of materials diffusion processes with fractionalorder properties fractional order flows analytical mechanicsof fractional order discrete system vibrations [13ndash15] and

so on Recently the application of Adomian decompositionmethod for solving the linear and nonlinear fractional partialdifferential equations in the fields of the physics and engineer-ing had been established in [16 17] Adomian decompositionmethod was applied to handle the time-fractional Navier-Stokes equation [18] fractional space diffusion equation [19]fractional KdV-Burgers equation [20] linear and nonlinearfractional diffusion and wave equations [21] KdV-Burgers-Kuramoto equation [22] fractional Burgersrsquo equation [23]and so on For more details on some methods for solvingfractional differential equations see [24ndash28]

Recently local fractional calculus theory was applied tomodel some nondifferentiable problems for mathematicalphysics (see [29ndash36] and the references therein) The Ado-mian decomposition method as one of efficient tools forsolving the linear and nonlinear differential equations wasextended to find the solutions for local fractional differen-tial equations [37ndash40] and nondifferentiable solutions wereobtained

2 Abstract and Applied Analysis

The partial differential equationfs describing thermalprocess of fractal heat conduction were suggested in [30 38]in the following form

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 0 (1)

The initial and boundary conditions are

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(2)

where the operator is the local fractional differential operator[29 30 34 37 38] which is applied to model the heatconduction problems in fractal media fractal materialsfractal fracture mechanics fractal wave behavior Navier-Stokes equations on Cantor sets Schrodinger equation withlocal fractional derivative and diffusion equations on cantorspace-time

The one-dimensional heat equations with the heat gener-ation arising in fractal transient conduction were consideredin [30] as

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 120601 (119909 119905) (3)

where 120601(119909 119905) is the heat generation termWe use initial and boundary conditions as follows

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(4)

The aim of this paper is to investigate the one-dimensionalheat equations with the heat generation arising in fractaltransient conduction by using the local fractional Adomiandecomposition method

This paper is structured as follows In Section 2 we givethe basic notations and definitions of local fractional oper-ators In Section 3 local fractional Adomian decompositionmethod for heat generation arising in fractal transient con-duction is presented Three examples are shown in Section 4Finally Section 5 presents conclusions

2 Preliminaries

In this section we present some basic definitions and nota-tions of the local fractional operators which are used furtherthrough the paper

Let us denote local fractional continuity of 119891(119909) as

119891 (119909) isin 119862120572(119886 119887) (5)

Definition 1 Local fractional derivative operator of 119891(119909) atthe point 119909

0is given by [29 30 34ndash38]

119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(6)

whereΔ120572(119891(119909)minus119891(1199090)) cong Γ(1+120572)Δ(119891(119909)minus119891(119909

0)) and119891(119909) isin

119862120572(119886 119887)Local fractional derivative of high order and local frac-

tional partial derivative of high order are written in the form[29 30 38]

119891(119896120572)

(119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863(120572)

119909sdot sdot sdot 119863(120572)

119909119891 (119909)

(7)

120597119896120572

120597119909119896120572119891 (119909 119910) =

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

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909 119910)

(8)

respectivelyAs inverse of local fractional differential operator the

local fractional integral operator of 119891(119909) in the interval [119886 119887]is defined as [29 30 36ndash38]

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

(9)

where a partition of the interval [119886 119887] is denoted as Δ119905119895

=

119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895 and 119895 = 0 119873 minus 1

1199050= 119886 and 119905

119873= 119887

The properties are only presented as follows [29 30 37]

119863(120572)

119909[119891 (119909) 119892 (119909)]

= (119863(120572)

119909119891 (119909)) 119892 (119909) + 119891 (119909) (119863

(120572)

119909119892 (119909))

119886119868(120572)

119909119891 (119909) 119892

(120572)

(119909)

= [119891 (119909) 119892 (119909)]1003816100381610038161003816119909

119886minus119886119868(120572)

119909119891(120572)

(119909) 119892 (119909)

119863(120572)

119909

119909119896120572

Γ (1 + 119896120572)=

119909(119896minus1)120572

Γ [1 + (119896 minus 1) 120572]

0119868(120572)

119887

119909119896120572

Γ (1 + 119896120572)=

119909(119896+1)120572

Γ [1 + (119896 + 1) 120572]

(10)

3 Analysis of the Method

Let us rewrite the heat equations with the heat generationarising in fractal transient conduction in the form

119871(120572)

119905119906 minus 119871(2120572)

119909119909119906 = 120601 (11)

subject to the initial and boundary conditions

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(12)

where 120597120572120597119905120572 and 1205972120572

1205971199092120572 symbolize 119871(120572)

119905and 119871

(2120572)

119909119909 respec-

tively

Abstract and Applied Analysis 3

By defining the twofold local fractional integral operatoras 119871(minus2120572)119909119909

we have

119871(minus2120572)

119909119909[119871(120572)

119905119906 minus 120601] = 119871

(minus2120572)

119909119909119871(2120572)

119909119909119906 (13)

so that

119906 = 119871(minus2120572)

119909119909119871(120572)

119905119906 minus 119871(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (14)

Hence we get

119906 (119909 119905) = 1199060(119909 119905) + 119871

(minus2120572)

119909119909[119871(120572)

119905119906 (119909 119905)] (15)

where

1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (16)

So from (15) we have iterative formula as follows

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] 119899 ge 0 (17)

where 1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 + (119909

120572

Γ(1 + 120572))119892(119905) + 119891(119905)Finally the exact solution can be constructed as follows

119906 (119909 119905) = lim119899rarrinfin

119899

sum

119894=1

119906119894(119909 119905) (18)

4 Illustrative Examples

Example 1 In view of (3) we consider 120601(119909 119905) = 1 119891(119905) =

119905120572

Γ(1 + 120572) and 119892(119905) = 119905120572

Γ(1 + 120572)We have

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (19)

subject to the initial value condition

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(20)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (21)

In view of (21) the first few terms of the decomposition seriesread

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)+

1199092120572

Γ (1 + 2120572)

(22)

From (25) we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (23)

02

46

8

02

46

80

05

1

15

2

25

3

x

t

u(xt)

Figure 1 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Therefore the exact solution of (19) can be written as

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(24)

The value of the fractal-dimension order 120572 = ln 2 ln 3 of thebehavior of the solution is shown in Figure 1

Example 2 When 120601(119909 119905) = 1 119891(119905) = 119905120572

Γ(1 + 120572) and 119892(119905) =

0 we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (25)

We give the initial value condition as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572) (26)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (27)

From (27) we have the first few terms of the decompositionseries as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

(28)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (29)

So the exact solution of (19) reads

119906 (119909 119905) =119905120572

Γ (1 + 120572) (30)

The solution with fractal-dimension order 120572 = ln 2 ln 3 isshown in Figure 2

4 Abstract and Applied Analysis

02

46

8

02

46

80

05

1

15

x

t

u(xt)

Figure 2 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Example 3 When 120601(119909 119905) = 1 119891(119905) = 0 and 119892(119905) = 119905120572

Γ(1 +

120572) we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (31)

The initial value condition is presented as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572) (32)

From (19) the recursive relations follow

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (33)

In view of (27) we get the few terms of the series namely

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)

(34)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (35)

So the exact solution of (19) reads

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

(36)

Figure 3 shows the exact solution when 120572 = ln 2 ln 3

5 Conclusions

In this work analytical solutions for the one-dimensionalheat equationswith the heat generation arising in fractal tran-sient conduction associated with local fractional derivativeoperators were discussed The obtained solutions are nondif-ferentiable functions which are Cantor functions and they

02

46

8

02

46

80

05

1

15

2

25

3

x

t

Figure 3 The surface shows the exact solution 119906(119909 119905) with a fixedvalue 120572 = ln 2 ln 3

discontinuously depend on the local fractional derivative Itis shown that the local fractional Adomian decompositionmethod is an efficient and simple tool for solving localfractional differential equations

Conflict of Interests

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

Acknowledgments

This paperwas supported by theNational Scientific andTech-nological Support Projects (no 2012BAE09B00) theNationalNatural Science Foundation of China (no 11126213 and no61170317) and theNationalNatural Science Foundation of theHebei Province (no E2013209215)

References

[1] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press Orlando Fla USA 1986

[2] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994

[3] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[4] M Tatari andM Dehghan ldquoThe use of the Adomian decompo-sitionmethod for solvingmultipoint boundary value problemsrdquoPhysica Scripta vol 73 no 6 pp 672ndash676 2006

[5] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[6] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

Abstract and Applied Analysis 5

[7] E Larsson ldquoA domain decomposition method for theHelmholtz equation in a multilayer domainrdquo SIAM Journal onScientific Computing vol 20 no 5 pp 1713ndash1731 1999

[8] M Tatari M Dehghan and M Razzaghi ldquoApplication ofthe Adomian decomposition method for the Fokker-Planckequationrdquo Mathematical and Computer Modelling vol 45 no5-6 pp 639ndash650 2007

[9] J A T Machado D Baleanu and A C LuoNonlinear Dynam-ics of Complex Systems Applications in Physical Biological andFinancial Systems Springer New York NY USA 2011

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

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

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

[13] K S Hedrih ldquoAnalytical mechanics of fractional order discretesystem vibrationsrdquo Advances in Nonlinear Sciences vol 3 pp101ndash148 2011

[14] KHedrih ldquoModes of the homogeneous chain dynamicsrdquo SignalProcessing vol 86 no 10 pp 2678ndash2702 2006

[15] K Hedrih ldquoDynamics of coupled systemsrdquo Nonlinear AnalysisHybrid Systems vol 2 no 2 pp 310ndash334 2008

[16] C Li and Y Wang ldquoNumerical algorithm based on Adomiandecomposition for fractional differential equationsrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1672ndash16812009

[17] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

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

[19] S S Ray and R K Bera ldquoAnalytical solution of a fractional dif-fusion equation by Adomian decomposition methodrdquo AppliedMathematics and Computation vol 174 no 1 pp 329ndash3362006

[20] Q Wang ldquoNumerical solutions for fractional KdV-Burgersequation by Adomian decomposition methodrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1048ndash1055 2006

[21] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[22] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[23] M El-Shahed ldquoAdomian decomposition method for solvingBurgers equation with fractional derivativerdquo Journal of Frac-tional Calculus vol 24 pp 23ndash28 2003

[24] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

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

[26] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[27] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[28] A Atangana and A K Kılıcman ldquoAnalytical solutions ofboundary values problem of 2D and 3D poisson andbiharmonic equations by homotopy decomposition methodrdquoAbstract and Applied Analysis vol 2013 Article ID 380484 9pages 2013

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

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

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

[32] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo Zeitschrift fur AngewandteMathematik und Mechanik vol 90 no 3 pp 203ndash210 2010

[33] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[34] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] X-J Yang D Baleanu and J A Tenreiro Machado ldquoSystemsof Navier-Stokes equations on Cantor setsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 769724 8 pages 2013

[36] X-J Yang D Baleanu and J A Tenreiro Machado ldquoMathe-matical aspects ofHeisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013no 1 pp 131ndash146 2013

[37] X-J YangD Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations on cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[38] X-J Yang D Baleanu M P Lazarevic andM S Cajic ldquoFractalboundary value problems for integral and differential equationswith local fractional operatorsrdquoThermal Science 2013

[39] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

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

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

The partial differential equationfs describing thermalprocess of fractal heat conduction were suggested in [30 38]in the following form

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 0 (1)

The initial and boundary conditions are

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(2)

where the operator is the local fractional differential operator[29 30 34 37 38] which is applied to model the heatconduction problems in fractal media fractal materialsfractal fracture mechanics fractal wave behavior Navier-Stokes equations on Cantor sets Schrodinger equation withlocal fractional derivative and diffusion equations on cantorspace-time

The one-dimensional heat equations with the heat gener-ation arising in fractal transient conduction were consideredin [30] as

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 120601 (119909 119905) (3)

where 120601(119909 119905) is the heat generation termWe use initial and boundary conditions as follows

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(4)

The aim of this paper is to investigate the one-dimensionalheat equations with the heat generation arising in fractaltransient conduction by using the local fractional Adomiandecomposition method

This paper is structured as follows In Section 2 we givethe basic notations and definitions of local fractional oper-ators In Section 3 local fractional Adomian decompositionmethod for heat generation arising in fractal transient con-duction is presented Three examples are shown in Section 4Finally Section 5 presents conclusions

2 Preliminaries

In this section we present some basic definitions and nota-tions of the local fractional operators which are used furtherthrough the paper

Let us denote local fractional continuity of 119891(119909) as

119891 (119909) isin 119862120572(119886 119887) (5)

Definition 1 Local fractional derivative operator of 119891(119909) atthe point 119909

0is given by [29 30 34ndash38]

119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(6)

whereΔ120572(119891(119909)minus119891(1199090)) cong Γ(1+120572)Δ(119891(119909)minus119891(119909

0)) and119891(119909) isin

119862120572(119886 119887)Local fractional derivative of high order and local frac-

tional partial derivative of high order are written in the form[29 30 38]

119891(119896120572)

(119909) =

119896 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863(120572)

119909sdot sdot sdot 119863(120572)

119909119891 (119909)

(7)

120597119896120572

120597119909119896120572119891 (119909 119910) =

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

120597119909120572sdot sdot sdot

120597120572

120597119909120572119891 (119909 119910)

(8)

respectivelyAs inverse of local fractional differential operator the

local fractional integral operator of 119891(119909) in the interval [119886 119887]is defined as [29 30 36ndash38]

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

(9)

where a partition of the interval [119886 119887] is denoted as Δ119905119895

=

119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895 and 119895 = 0 119873 minus 1

1199050= 119886 and 119905

119873= 119887

The properties are only presented as follows [29 30 37]

119863(120572)

119909[119891 (119909) 119892 (119909)]

= (119863(120572)

119909119891 (119909)) 119892 (119909) + 119891 (119909) (119863

(120572)

119909119892 (119909))

119886119868(120572)

119909119891 (119909) 119892

(120572)

(119909)

= [119891 (119909) 119892 (119909)]1003816100381610038161003816119909

119886minus119886119868(120572)

119909119891(120572)

(119909) 119892 (119909)

119863(120572)

119909

119909119896120572

Γ (1 + 119896120572)=

119909(119896minus1)120572

Γ [1 + (119896 minus 1) 120572]

0119868(120572)

119887

119909119896120572

Γ (1 + 119896120572)=

119909(119896+1)120572

Γ [1 + (119896 + 1) 120572]

(10)

3 Analysis of the Method

Let us rewrite the heat equations with the heat generationarising in fractal transient conduction in the form

119871(120572)

119905119906 minus 119871(2120572)

119909119909119906 = 120601 (11)

subject to the initial and boundary conditions

119906 (0 119905) = 119891 (119905)

120597120572

119906 (0 119905)

120597119909120572= 119892 (119905)

(12)

where 120597120572120597119905120572 and 1205972120572

1205971199092120572 symbolize 119871(120572)

119905and 119871

(2120572)

119909119909 respec-

tively

Abstract and Applied Analysis 3

By defining the twofold local fractional integral operatoras 119871(minus2120572)119909119909

we have

119871(minus2120572)

119909119909[119871(120572)

119905119906 minus 120601] = 119871

(minus2120572)

119909119909119871(2120572)

119909119909119906 (13)

so that

119906 = 119871(minus2120572)

119909119909119871(120572)

119905119906 minus 119871(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (14)

Hence we get

119906 (119909 119905) = 1199060(119909 119905) + 119871

(minus2120572)

119909119909[119871(120572)

119905119906 (119909 119905)] (15)

where

1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (16)

So from (15) we have iterative formula as follows

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] 119899 ge 0 (17)

where 1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 + (119909

120572

Γ(1 + 120572))119892(119905) + 119891(119905)Finally the exact solution can be constructed as follows

119906 (119909 119905) = lim119899rarrinfin

119899

sum

119894=1

119906119894(119909 119905) (18)

4 Illustrative Examples

Example 1 In view of (3) we consider 120601(119909 119905) = 1 119891(119905) =

119905120572

Γ(1 + 120572) and 119892(119905) = 119905120572

Γ(1 + 120572)We have

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (19)

subject to the initial value condition

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(20)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (21)

In view of (21) the first few terms of the decomposition seriesread

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)+

1199092120572

Γ (1 + 2120572)

(22)

From (25) we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (23)

02

46

8

02

46

80

05

1

15

2

25

3

x

t

u(xt)

Figure 1 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Therefore the exact solution of (19) can be written as

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(24)

The value of the fractal-dimension order 120572 = ln 2 ln 3 of thebehavior of the solution is shown in Figure 1

Example 2 When 120601(119909 119905) = 1 119891(119905) = 119905120572

Γ(1 + 120572) and 119892(119905) =

0 we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (25)

We give the initial value condition as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572) (26)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (27)

From (27) we have the first few terms of the decompositionseries as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

(28)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (29)

So the exact solution of (19) reads

119906 (119909 119905) =119905120572

Γ (1 + 120572) (30)

The solution with fractal-dimension order 120572 = ln 2 ln 3 isshown in Figure 2

4 Abstract and Applied Analysis

02

46

8

02

46

80

05

1

15

x

t

u(xt)

Figure 2 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Example 3 When 120601(119909 119905) = 1 119891(119905) = 0 and 119892(119905) = 119905120572

Γ(1 +

120572) we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (31)

The initial value condition is presented as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572) (32)

From (19) the recursive relations follow

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (33)

In view of (27) we get the few terms of the series namely

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)

(34)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (35)

So the exact solution of (19) reads

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

(36)

Figure 3 shows the exact solution when 120572 = ln 2 ln 3

5 Conclusions

In this work analytical solutions for the one-dimensionalheat equationswith the heat generation arising in fractal tran-sient conduction associated with local fractional derivativeoperators were discussed The obtained solutions are nondif-ferentiable functions which are Cantor functions and they

02

46

8

02

46

80

05

1

15

2

25

3

x

t

Figure 3 The surface shows the exact solution 119906(119909 119905) with a fixedvalue 120572 = ln 2 ln 3

discontinuously depend on the local fractional derivative Itis shown that the local fractional Adomian decompositionmethod is an efficient and simple tool for solving localfractional differential equations

Conflict of Interests

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

Acknowledgments

This paperwas supported by theNational Scientific andTech-nological Support Projects (no 2012BAE09B00) theNationalNatural Science Foundation of China (no 11126213 and no61170317) and theNationalNatural Science Foundation of theHebei Province (no E2013209215)

References

[1] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press Orlando Fla USA 1986

[2] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994

[3] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[4] M Tatari andM Dehghan ldquoThe use of the Adomian decompo-sitionmethod for solvingmultipoint boundary value problemsrdquoPhysica Scripta vol 73 no 6 pp 672ndash676 2006

[5] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[6] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

Abstract and Applied Analysis 5

[7] E Larsson ldquoA domain decomposition method for theHelmholtz equation in a multilayer domainrdquo SIAM Journal onScientific Computing vol 20 no 5 pp 1713ndash1731 1999

[8] M Tatari M Dehghan and M Razzaghi ldquoApplication ofthe Adomian decomposition method for the Fokker-Planckequationrdquo Mathematical and Computer Modelling vol 45 no5-6 pp 639ndash650 2007

[9] J A T Machado D Baleanu and A C LuoNonlinear Dynam-ics of Complex Systems Applications in Physical Biological andFinancial Systems Springer New York NY USA 2011

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

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

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

[13] K S Hedrih ldquoAnalytical mechanics of fractional order discretesystem vibrationsrdquo Advances in Nonlinear Sciences vol 3 pp101ndash148 2011

[14] KHedrih ldquoModes of the homogeneous chain dynamicsrdquo SignalProcessing vol 86 no 10 pp 2678ndash2702 2006

[15] K Hedrih ldquoDynamics of coupled systemsrdquo Nonlinear AnalysisHybrid Systems vol 2 no 2 pp 310ndash334 2008

[16] C Li and Y Wang ldquoNumerical algorithm based on Adomiandecomposition for fractional differential equationsrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1672ndash16812009

[17] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

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

[19] S S Ray and R K Bera ldquoAnalytical solution of a fractional dif-fusion equation by Adomian decomposition methodrdquo AppliedMathematics and Computation vol 174 no 1 pp 329ndash3362006

[20] Q Wang ldquoNumerical solutions for fractional KdV-Burgersequation by Adomian decomposition methodrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1048ndash1055 2006

[21] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[22] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[23] M El-Shahed ldquoAdomian decomposition method for solvingBurgers equation with fractional derivativerdquo Journal of Frac-tional Calculus vol 24 pp 23ndash28 2003

[24] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

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

[26] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[27] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[28] A Atangana and A K Kılıcman ldquoAnalytical solutions ofboundary values problem of 2D and 3D poisson andbiharmonic equations by homotopy decomposition methodrdquoAbstract and Applied Analysis vol 2013 Article ID 380484 9pages 2013

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

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

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

[32] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo Zeitschrift fur AngewandteMathematik und Mechanik vol 90 no 3 pp 203ndash210 2010

[33] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[34] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] X-J Yang D Baleanu and J A Tenreiro Machado ldquoSystemsof Navier-Stokes equations on Cantor setsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 769724 8 pages 2013

[36] X-J Yang D Baleanu and J A Tenreiro Machado ldquoMathe-matical aspects ofHeisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013no 1 pp 131ndash146 2013

[37] X-J YangD Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations on cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[38] X-J Yang D Baleanu M P Lazarevic andM S Cajic ldquoFractalboundary value problems for integral and differential equationswith local fractional operatorsrdquoThermal Science 2013

[39] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

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

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

By defining the twofold local fractional integral operatoras 119871(minus2120572)119909119909

we have

119871(minus2120572)

119909119909[119871(120572)

119905119906 minus 120601] = 119871

(minus2120572)

119909119909119871(2120572)

119909119909119906 (13)

so that

119906 = 119871(minus2120572)

119909119909119871(120572)

119905119906 minus 119871(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (14)

Hence we get

119906 (119909 119905) = 1199060(119909 119905) + 119871

(minus2120572)

119909119909[119871(120572)

119905119906 (119909 119905)] (15)

where

1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 +

119909120572

Γ (1 + 120572)119892 (119905) + 119891 (119905) (16)

So from (15) we have iterative formula as follows

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] 119899 ge 0 (17)

where 1199060(119909 119905) = minus119871

(minus2120572)

119909119909120601 + (119909

120572

Γ(1 + 120572))119892(119905) + 119891(119905)Finally the exact solution can be constructed as follows

119906 (119909 119905) = lim119899rarrinfin

119899

sum

119894=1

119906119894(119909 119905) (18)

4 Illustrative Examples

Example 1 In view of (3) we consider 120601(119909 119905) = 1 119891(119905) =

119905120572

Γ(1 + 120572) and 119892(119905) = 119905120572

Γ(1 + 120572)We have

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (19)

subject to the initial value condition

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(20)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (21)

In view of (21) the first few terms of the decomposition seriesread

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)+

1199092120572

Γ (1 + 2120572)

(22)

From (25) we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (23)

02

46

8

02

46

80

05

1

15

2

25

3

x

t

u(xt)

Figure 1 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Therefore the exact solution of (19) can be written as

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)+

119905120572

Γ (1 + 120572)

(24)

The value of the fractal-dimension order 120572 = ln 2 ln 3 of thebehavior of the solution is shown in Figure 1

Example 2 When 120601(119909 119905) = 1 119891(119905) = 119905120572

Γ(1 + 120572) and 119892(119905) =

0 we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (25)

We give the initial value condition as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572) (26)

From (19) we have the following recursive relations

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (27)

From (27) we have the first few terms of the decompositionseries as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199092120572

Γ (1 + 2120572)

(28)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (29)

So the exact solution of (19) reads

119906 (119909 119905) =119905120572

Γ (1 + 120572) (30)

The solution with fractal-dimension order 120572 = ln 2 ln 3 isshown in Figure 2

4 Abstract and Applied Analysis

02

46

8

02

46

80

05

1

15

x

t

u(xt)

Figure 2 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Example 3 When 120601(119909 119905) = 1 119891(119905) = 0 and 119892(119905) = 119905120572

Γ(1 +

120572) we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (31)

The initial value condition is presented as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572) (32)

From (19) the recursive relations follow

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (33)

In view of (27) we get the few terms of the series namely

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)

(34)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (35)

So the exact solution of (19) reads

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

(36)

Figure 3 shows the exact solution when 120572 = ln 2 ln 3

5 Conclusions

In this work analytical solutions for the one-dimensionalheat equationswith the heat generation arising in fractal tran-sient conduction associated with local fractional derivativeoperators were discussed The obtained solutions are nondif-ferentiable functions which are Cantor functions and they

02

46

8

02

46

80

05

1

15

2

25

3

x

t

Figure 3 The surface shows the exact solution 119906(119909 119905) with a fixedvalue 120572 = ln 2 ln 3

discontinuously depend on the local fractional derivative Itis shown that the local fractional Adomian decompositionmethod is an efficient and simple tool for solving localfractional differential equations

Conflict of Interests

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

Acknowledgments

This paperwas supported by theNational Scientific andTech-nological Support Projects (no 2012BAE09B00) theNationalNatural Science Foundation of China (no 11126213 and no61170317) and theNationalNatural Science Foundation of theHebei Province (no E2013209215)

References

[1] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press Orlando Fla USA 1986

[2] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994

[3] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[4] M Tatari andM Dehghan ldquoThe use of the Adomian decompo-sitionmethod for solvingmultipoint boundary value problemsrdquoPhysica Scripta vol 73 no 6 pp 672ndash676 2006

[5] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[6] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

Abstract and Applied Analysis 5

[7] E Larsson ldquoA domain decomposition method for theHelmholtz equation in a multilayer domainrdquo SIAM Journal onScientific Computing vol 20 no 5 pp 1713ndash1731 1999

[8] M Tatari M Dehghan and M Razzaghi ldquoApplication ofthe Adomian decomposition method for the Fokker-Planckequationrdquo Mathematical and Computer Modelling vol 45 no5-6 pp 639ndash650 2007

[9] J A T Machado D Baleanu and A C LuoNonlinear Dynam-ics of Complex Systems Applications in Physical Biological andFinancial Systems Springer New York NY USA 2011

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

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

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

[13] K S Hedrih ldquoAnalytical mechanics of fractional order discretesystem vibrationsrdquo Advances in Nonlinear Sciences vol 3 pp101ndash148 2011

[14] KHedrih ldquoModes of the homogeneous chain dynamicsrdquo SignalProcessing vol 86 no 10 pp 2678ndash2702 2006

[15] K Hedrih ldquoDynamics of coupled systemsrdquo Nonlinear AnalysisHybrid Systems vol 2 no 2 pp 310ndash334 2008

[16] C Li and Y Wang ldquoNumerical algorithm based on Adomiandecomposition for fractional differential equationsrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1672ndash16812009

[17] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

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

[19] S S Ray and R K Bera ldquoAnalytical solution of a fractional dif-fusion equation by Adomian decomposition methodrdquo AppliedMathematics and Computation vol 174 no 1 pp 329ndash3362006

[20] Q Wang ldquoNumerical solutions for fractional KdV-Burgersequation by Adomian decomposition methodrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1048ndash1055 2006

[21] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[22] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[23] M El-Shahed ldquoAdomian decomposition method for solvingBurgers equation with fractional derivativerdquo Journal of Frac-tional Calculus vol 24 pp 23ndash28 2003

[24] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

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

[26] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[27] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[28] A Atangana and A K Kılıcman ldquoAnalytical solutions ofboundary values problem of 2D and 3D poisson andbiharmonic equations by homotopy decomposition methodrdquoAbstract and Applied Analysis vol 2013 Article ID 380484 9pages 2013

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

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

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

[32] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo Zeitschrift fur AngewandteMathematik und Mechanik vol 90 no 3 pp 203ndash210 2010

[33] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[34] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] X-J Yang D Baleanu and J A Tenreiro Machado ldquoSystemsof Navier-Stokes equations on Cantor setsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 769724 8 pages 2013

[36] X-J Yang D Baleanu and J A Tenreiro Machado ldquoMathe-matical aspects ofHeisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013no 1 pp 131ndash146 2013

[37] X-J YangD Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations on cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[38] X-J Yang D Baleanu M P Lazarevic andM S Cajic ldquoFractalboundary value problems for integral and differential equationswith local fractional operatorsrdquoThermal Science 2013

[39] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

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

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

02

46

8

02

46

80

05

1

15

x

t

u(xt)

Figure 2 Solution for the one-dimensional heat equations with afixed value 120572 = ln 2 ln 3

Example 3 When 120601(119909 119905) = 1 119891(119905) = 0 and 119892(119905) = 119905120572

Γ(1 +

120572) we get

120597120572

119906 (119909 119905)

120597119905120572minus

1205972120572

119906 (119909 119905)

1205971199092120572= 1 (31)

The initial value condition is presented as follows

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572) (32)

From (19) the recursive relations follow

119906119899+1

(119909 119905) = 119871(minus2120572)

119909119909[119871(120572)

119905119906119899(119909 119905)] (33)

In view of (27) we get the few terms of the series namely

1199060(119909 119905) = minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

1199061(119909 119905) =

1199093120572

Γ (1 + 3120572)

(34)

Hence we get

1199062(119909 119905) = 119906

3(119909 119905) = sdot sdot sdot = 119906

119899(119909 119905) = 0 (35)

So the exact solution of (19) reads

119906 (119909 119905) =1199093120572

Γ (1 + 3120572)minus

1199092120572

Γ (1 + 2120572)+

119909120572

Γ (1 + 120572)

119905120572

Γ (1 + 120572)

(36)

Figure 3 shows the exact solution when 120572 = ln 2 ln 3

5 Conclusions

In this work analytical solutions for the one-dimensionalheat equationswith the heat generation arising in fractal tran-sient conduction associated with local fractional derivativeoperators were discussed The obtained solutions are nondif-ferentiable functions which are Cantor functions and they

02

46

8

02

46

80

05

1

15

2

25

3

x

t

Figure 3 The surface shows the exact solution 119906(119909 119905) with a fixedvalue 120572 = ln 2 ln 3

discontinuously depend on the local fractional derivative Itis shown that the local fractional Adomian decompositionmethod is an efficient and simple tool for solving localfractional differential equations

Conflict of Interests

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

Acknowledgments

This paperwas supported by theNational Scientific andTech-nological Support Projects (no 2012BAE09B00) theNationalNatural Science Foundation of China (no 11126213 and no61170317) and theNationalNatural Science Foundation of theHebei Province (no E2013209215)

References

[1] G Adomian Nonlinear Stochastic Operator Equations Aca-demic Press Orlando Fla USA 1986

[2] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994

[3] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[4] M Tatari andM Dehghan ldquoThe use of the Adomian decompo-sitionmethod for solvingmultipoint boundary value problemsrdquoPhysica Scripta vol 73 no 6 pp 672ndash676 2006

[5] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[6] V Daftardar-Gejji and H Jafari ldquoAdomian decomposition atool for solving a system of fractional differential equationsrdquoJournal of Mathematical Analysis and Applications vol 301 no2 pp 508ndash518 2005

Abstract and Applied Analysis 5

[7] E Larsson ldquoA domain decomposition method for theHelmholtz equation in a multilayer domainrdquo SIAM Journal onScientific Computing vol 20 no 5 pp 1713ndash1731 1999

[8] M Tatari M Dehghan and M Razzaghi ldquoApplication ofthe Adomian decomposition method for the Fokker-Planckequationrdquo Mathematical and Computer Modelling vol 45 no5-6 pp 639ndash650 2007

[9] J A T Machado D Baleanu and A C LuoNonlinear Dynam-ics of Complex Systems Applications in Physical Biological andFinancial Systems Springer New York NY USA 2011

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

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

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

[13] K S Hedrih ldquoAnalytical mechanics of fractional order discretesystem vibrationsrdquo Advances in Nonlinear Sciences vol 3 pp101ndash148 2011

[14] KHedrih ldquoModes of the homogeneous chain dynamicsrdquo SignalProcessing vol 86 no 10 pp 2678ndash2702 2006

[15] K Hedrih ldquoDynamics of coupled systemsrdquo Nonlinear AnalysisHybrid Systems vol 2 no 2 pp 310ndash334 2008

[16] C Li and Y Wang ldquoNumerical algorithm based on Adomiandecomposition for fractional differential equationsrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1672ndash16812009

[17] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

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

[19] S S Ray and R K Bera ldquoAnalytical solution of a fractional dif-fusion equation by Adomian decomposition methodrdquo AppliedMathematics and Computation vol 174 no 1 pp 329ndash3362006

[20] Q Wang ldquoNumerical solutions for fractional KdV-Burgersequation by Adomian decomposition methodrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1048ndash1055 2006

[21] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[22] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[23] M El-Shahed ldquoAdomian decomposition method for solvingBurgers equation with fractional derivativerdquo Journal of Frac-tional Calculus vol 24 pp 23ndash28 2003

[24] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

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

[26] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[27] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[28] A Atangana and A K Kılıcman ldquoAnalytical solutions ofboundary values problem of 2D and 3D poisson andbiharmonic equations by homotopy decomposition methodrdquoAbstract and Applied Analysis vol 2013 Article ID 380484 9pages 2013

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

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

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

[32] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo Zeitschrift fur AngewandteMathematik und Mechanik vol 90 no 3 pp 203ndash210 2010

[33] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[34] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] X-J Yang D Baleanu and J A Tenreiro Machado ldquoSystemsof Navier-Stokes equations on Cantor setsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 769724 8 pages 2013

[36] X-J Yang D Baleanu and J A Tenreiro Machado ldquoMathe-matical aspects ofHeisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013no 1 pp 131ndash146 2013

[37] X-J YangD Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations on cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[38] X-J Yang D Baleanu M P Lazarevic andM S Cajic ldquoFractalboundary value problems for integral and differential equationswith local fractional operatorsrdquoThermal Science 2013

[39] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

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

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

[7] E Larsson ldquoA domain decomposition method for theHelmholtz equation in a multilayer domainrdquo SIAM Journal onScientific Computing vol 20 no 5 pp 1713ndash1731 1999

[8] M Tatari M Dehghan and M Razzaghi ldquoApplication ofthe Adomian decomposition method for the Fokker-Planckequationrdquo Mathematical and Computer Modelling vol 45 no5-6 pp 639ndash650 2007

[9] J A T Machado D Baleanu and A C LuoNonlinear Dynam-ics of Complex Systems Applications in Physical Biological andFinancial Systems Springer New York NY USA 2011

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

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

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

[13] K S Hedrih ldquoAnalytical mechanics of fractional order discretesystem vibrationsrdquo Advances in Nonlinear Sciences vol 3 pp101ndash148 2011

[14] KHedrih ldquoModes of the homogeneous chain dynamicsrdquo SignalProcessing vol 86 no 10 pp 2678ndash2702 2006

[15] K Hedrih ldquoDynamics of coupled systemsrdquo Nonlinear AnalysisHybrid Systems vol 2 no 2 pp 310ndash334 2008

[16] C Li and Y Wang ldquoNumerical algorithm based on Adomiandecomposition for fractional differential equationsrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1672ndash16812009

[17] H Jafari and V Daftardar-Gejji ldquoSolving a system of nonlinearfractional differential equations using Adomian decomposi-tionrdquo Journal of Computational and Applied Mathematics vol196 no 2 pp 644ndash651 2006

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

[19] S S Ray and R K Bera ldquoAnalytical solution of a fractional dif-fusion equation by Adomian decomposition methodrdquo AppliedMathematics and Computation vol 174 no 1 pp 329ndash3362006

[20] Q Wang ldquoNumerical solutions for fractional KdV-Burgersequation by Adomian decomposition methodrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1048ndash1055 2006

[21] H Jafari and V Daftardar-Gejji ldquoSolving linear and nonlinearfractional diffusion and wave equations by Adomian decompo-sitionrdquo Applied Mathematics and Computation vol 180 no 2pp 488ndash497 2006

[22] M Safari D D Ganji and M Moslemi ldquoApplication of Hersquosvariational iteration method and Adomianrsquos decompositionmethod to the fractional KdV-Burgers-Kuramoto equationrdquoComputers amp Mathematics with Applications vol 58 no 11-12pp 2091ndash2097 2009

[23] M El-Shahed ldquoAdomian decomposition method for solvingBurgers equation with fractional derivativerdquo Journal of Frac-tional Calculus vol 24 pp 23ndash28 2003

[24] J Hristov ldquoTransient flow of a generalized second grade fluiddue to a constant surface shear stress an approximate integral-balance solutionrdquo International Review of Chemical Engineeringvol 3 no 6 pp 802ndash809 2011

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

[26] H Jafari H Tajadodi N Kadkhoda and D Baleanu ldquoFrac-tional subequation method for Cahn-Hilliard and Klein-Gordon equationsrdquo Abstract and Applied Analysis vol 2013Article ID 587179 5 pages 2013

[27] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[28] A Atangana and A K Kılıcman ldquoAnalytical solutions ofboundary values problem of 2D and 3D poisson andbiharmonic equations by homotopy decomposition methodrdquoAbstract and Applied Analysis vol 2013 Article ID 380484 9pages 2013

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

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

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

[32] A Carpinteri and A Sapora ldquoDiffusion problems in fractalmedia defined on Cantor setsrdquo Zeitschrift fur AngewandteMathematik und Mechanik vol 90 no 3 pp 203ndash210 2010

[33] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNewto-nianmechanics on fractals subset of real-linerdquoRomania Reportsin Physics vol 65 pp 84ndash93 2013

[34] X-J Yang H M Srivastava J-H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] X-J Yang D Baleanu and J A Tenreiro Machado ldquoSystemsof Navier-Stokes equations on Cantor setsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 769724 8 pages 2013

[36] X-J Yang D Baleanu and J A Tenreiro Machado ldquoMathe-matical aspects ofHeisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013no 1 pp 131ndash146 2013

[37] X-J YangD Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations on cantor space-timerdquo Proceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[38] X-J Yang D Baleanu M P Lazarevic andM S Cajic ldquoFractalboundary value problems for integral and differential equationswith local fractional operatorsrdquoThermal Science 2013

[39] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of