Research ArticlePicard Successive Approximation Method for SolvingDifferential Equations Arising in Fractal Heat Transfer withLocal Fractional Derivative

Ai-Min Yang12 Cheng Zhang3 Hossein Jafari4 Carlo Cattani5 and Ying Jiao6

1 College of Science Hebei United University Tangshan China2 College of Mechanical Engineering Yanshan University Qinhuangdao China3 School of Civil Engineering and Architecture Chongqing Jiaotong University Chongqing 400074 China4 Faculty of Basic Sciences Department of Mathematics Ayatollah Amoli Branch Islamic Azad University Amol 4615143358 Iran5 Department of Mathematics University of Salerno Via Ponte don Melillo Fisciano 84084 Salerno Italy6Qinggong College Hebei United University Tangshan 063000 China

Correspondence should be addressed to Ai-Min Yang aimin heut163com

Received 12 December 2013 Accepted 1 January 2014 Published 11 February 2014

Academic Editor Ming Li

Copyright copy 2014 Ai-Min 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

TheFourier law of one-dimensional heat conduction equation in fractalmedia is investigated in this paper An approximate solutionto one-dimensional local fractional Volterra integral equation of the second kind which is derived from the transformation ofFourier flux equation in discontinuous media is considered The Picard successive approximation method is applied to solve thetemperature field based on the givenMittag-Leffler-type Fourier flux distribution in fractal mediaThe nondifferential approximatesolutions are given to show the efficiency of the present method

1 Introduction

Engineering problems can be mathematically described bydifferential equations Many initial and boundary valueproblems associated with differential equations can be trans-formed into problems of solving some approximate integralequations Heat transfer is described by theory of integralequations Integral equation arising in heat transfer withsmooth condition is valid for continuous media [1ndash4] Thecommon methods for solving the equations of heat transferare purelymathematical are among them the finite differencetechniques (FDT) [5] the regression analysis (RA) [6] theAdomian decomposition method (ADM) [7] the combinedLaplace-Adomian method (CLAM) [8] the homotopy anal-ysis method (HAM) [9 10] the differential transformationmethod (DTM) [11] the spline-wavelets techniques (SWT)[12] the boundary element method (BEM) [13] the heat-balance integral method (HBIM) [14 15] the variationaliteration method (VIM) [16] the local fractional variational

iteration method (LFVIM) [17] and the Picard successiveapproximation method (PSAM) [18]

On the other hand the nanoscale heat problem can becharacterized as fractal behaviors As usual the materials arecalled the Cantor materials Heat transfer in fractal mediawith nonsmooth conditions is a hot topic For example theheat transfer equations in a medium with fractal geometry[19] and fractal domains [20] were considered The localfractional transient heat conduction equations based uponthe Fourier law within local fractional derivative arising inheat transfer from discontinuous media were presented in[21ndash24]

Fractional calculus was successfully used to deal with thereal world problems [25ndash30] There is its limit that the oper-ators do not deal with the local fractional continuous func-tions (nondifferential functions) Hence the local fractionalFourier flux [21] is not handled by using some approachesfrom the classical and fractional operatorsThis paper focuseson analytical solution to local fractional Fourier flux in fractal

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 395710 5 pageshttpdxdoiorg1011552014395710

2 Abstract and Applied Analysis

media by using Picardrsquos successive approximation method[18] This paper is organized as follows In Section 2 wegive notations to local fractional derivative and integrals andinvestigate the heat transfer in fractal media Section 3 isdevoted to Picardrsquos successive approximation method basedupon local fractional integrals Analysis solution is shown inSection 4 Conclusions are in Section 5

2 Heat Transfer in Fractal Mediawith Local Fractional Derivative

In order to study the non-differential solution for the heatproblem in fractal media with local fractional derivative wehere begin with the Fourier flux equation in discontinuousmedia

The temperature field reads as [21]

119879 (119909 119910 119911 120591) = 119891 (119909 119910 119911 120591) at 120591 gt 1205910and in Ω (1)

where 119891(119909 119910 119911 120591) is local fractional continuous at fractaldomainΩ

For a given temperature field 119879 a local fractional temper-ature gradient [21] can be written as follows

nabla120572119879 =

120597120572119879

120597119906120572

1

997888119890120572

1+120597120572119879

120597119906120572

2

997888119890120572

2+120597120572119879

120597119906120572

3

997888119890120572

3 (2)

where the local fractional partial derivative is defined by [21ndash24]

119891(120572)

119909(1199090 119910) =

120597120572119891 (119909 119910)

120597119909120572

100381610038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909 119910) minus 119891 (119909

0 119910))

(119909 minus 1199090)120572

(3)

whereΔ120572(119891(119909 119910)minus119891(1199090 119910)) cong Γ(1+120572)Δ(119891(119909 119910)minus119891(119909

0 119910))

Here the local fractional derivative is defined on thefractal set like a Cantor set For example when we considerthe Cantor set we can find the local fractional derivative ofdiscontinuous function 119879 (however 119879 is a local fractionalcontinuous function)

We consider the heat flux per unit fractal area 997888119902 is

proportional to the temperature gradient in fractal mediumFourier law of heat conduction in fractal medium with localfractional derivative is expressed by [21]

997888119902 (119909 119910 119911 119905) = minus119870

2120572nabla120572119879 (119909 119910 119911 119905) (4)

where 1198702120572 denotes the thermal conductivity of the fractal

material and it is related to fractal dimensions of materialsIt is shown that the fractal dimensions of materials are animportant characteristic value Here we consider the fractalFourier flow which is discontinuous however it is found thatit is local fractional continuous Like classical Fourier flow itsthermal conductivity is an approximate value for fractal onewhen 120572 = 1

Fourier lawof one-dimensional heat conduction equationin fractal media reads as [21]

119902 (119909 119905) = minus1198702120572 119889120572119879 (119909 119905)

119889119909120572 at 120591 gt 120591

0and in 119860 (5)

where 1198702120572 denotes the thermal conductivity of the fractal

materialsWhen 120591 = 120591

0 from (5) we have

119902 (119909) = minus1198702120572 119889120572119879 (119909)

119889119909120572 (6)

at 120591 gt 1205910and in 119860 where 1198702120572 is the thermal conductivity

of the fractal materials Namely 119879 is a bi-Lipschitz mappingand shows the fractal characteristic behavior [21]

Local fractional heat conduction equation with heatgeneration in fractal media can be written as [21]

1198702120572nabla2120572119879 + 119892 minus 120588

120572119888120572

120597120572119879

120597119905120572= 0 at 120591 gt 120591

0and in Ω (7)

Local fractional heat conduction equation with no heatgeneration in fractal media is suggested as [21 22]

1198702120572nabla2120572119879 minus 120588120572119888120572

120597120572119879

120597119905120572= 0 at 120591 gt 120591

0and in Ω (8)

where nabla2120572 is a local fractional Laplace operator [21]

3 The Method

In this section we discuss the Picard successive approxima-tion method Meanwhile we transfer the Fourier law of one-dimensional heat conduction equation in fractal media intothe local fractional Volterra integral equation of the secondkind

31 Picardrsquos Successive Approximation Method This methodis first proposed in [18] Here wewill give a short introductionto Picardrsquos successive approximation method within the localfractional calculus

In this method we set

1199060(119909) = 119891 (119909) (9)

We give the first approximation 1199061(119909) by

1199061(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 1199060(119909) (119889119905)

120572 (10)

where the local fractional integral of 119891(119909) of order 120572 in theinterval [119886 119887] is defined as follows [21ndash24]

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

(11)

withΔ119905119895= 119905119895+1

minus119905119895Δ119905 = maxΔ119905

0 Δ1199051 Δ1199052 and [119905

119895 119905119895+1

]119895 = 0 119873 minus 1 119905

0= 119886 and 119905

119873= 119887

Here we find that the equality 1199061(119909) is a local fractional

continuous function if 119891(119909) 119870(119909 119905) and 1199060(119909) are local

fractional continuous functions

Abstract and Applied Analysis 3

Continuing in thismanner we have the infinite sequencesof functions

1199060(119909) 119906

1(119909) 119906

2(119909) 119906

119899(119909) (12)

such that the recurrence equations are given by

119906119899(119909) = 119891 (119909) minus

1

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(119899 = 1 2 3 )

(13)

where 1199060(119909) is equivalent to any selected function which is

the local fractional continuous functionHence we have successive approximation as follows

1199061(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119891 (119905) (119889119905)120572

119906119899(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(14)

Thus at the limit the solution 119906(119909) is written as

119906 (119909) = lim119899rarrinfin

119906119899 (119909) (15)

32 An Alternative Method from Local Fractional Derivativeto Local Fractional Volterra Integral Equations We directlyobserve that the local fractional differential equation of 120572order

1198702120572 119889120572119879

119889119909120572= 119902 (119909) (0 le 119909 le 119887) (16)

can be written immediately as the local fractional Volterraintegral equation of thee second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119902 (119905) (119889119905)120572 (17)

where 120582 = 11198702

TheMittag-Leffler type Fourier flux distribution in fractalmedia can be written as follows

119902 (119909) = 119902119905(119905 119909) = 119864

120572(119909 minus 119905)

120572119879 (119905) (0 le 119905 le 119887) (18)

Making use of (18) we can get the local fractional Volterraintegral equation of the second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119879 (119905) (119889119905)

120572 (19)

4 Approximate Solutions for LocalFractional Volterra Integral Equationof the Second Kind

Let us assume that the zeroth approximation is

1199060(119909) = 0 (20)

Then the first approximation can be written as follows

1199061(119909) = 119891 (119909) = 119879 (119886) (21)

Here we obtain the second approximation which reads as

1199062 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199061 (119905) (119889119905)

120572

(22)

Proceeding in this manner we have the third approximationin the following form

1199063 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199062 (119905) (119889119905)

120572 (23)

Hence continuing in this manner we obtain

119906119899 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119906119899minus1 (119905) (119889119905)

120572

= 119891 (119909) +120582120572

Γ (1 + 120572)

times int

119909

0

(1 + sdot sdot sdot +120582(119899minus1)120572

(119909 minus 119905)(119899minus1)120572

Γ (1 + (119899 minus 1) 120572))

times 119864120572(119909 minus 119905)

120572119891 (119905) (119889119905)

120572

(24)

Taking the limit we have

119906 (119909) = lim119899rarrinfin

119906119899 (119909)

= 119879 (119886) +120582120572

Γ (1 + 120572)

times int

119909

0

119864120572[(119909 minus 119905)

120572(1 + 120582

120572)] 119879 (119886) (119889119905)

120572

= 119879 (119886) 2120582120572

1 + 120582120572119864120572[119909120572(1 + 120582

120572)]

= 119879 (119886) 2

1198702120572 + 1119864120572[119909120572(1 +

1

1198702120572)]

(25)

where the term119864120572[119909120572(1+120582120572)] is aMittag-Leffler type Fourier

flux distribution in fractal media which is related to thefractal coarse-grained mass function [21 24] When 119870 = 1we get 119906(119909) = 119879(119886)119864

120572[2119909120572] The nondifferentiable solution

of (25) for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 1

is shown in Figure 1 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 2 isshown in Figure 2 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 3 isshown in Figure 3 the non-differentiable solution of (25) forparameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 4 is shown inFigure 4

5 Conclusions

This work studied the Fourier law of one-dimensional heatconduction equation in fractal media Mittag-Leffler type

4 Abstract and Applied Analysis

0 10

5

10

15

20

25

30

35

40

02 04 06 08

Figure 1 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 1

0

10

20

30

40

50

60

70

80

0 102 04 06 08

Figure 2 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 2

0

20

40

60

80

100

120

0 102 04 06 08

Figure 3 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 3

0

50

100

150

0 102 04 06 08

Figure 4 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 4

Fourier flux distribution in fractal media with temperaturefield effect was considered An approximation solution forthe local fractional Volterra integral equation of the secondkind derived from Fourier law of one-dimensional heatconduction equation for heat conduction in discontinuousmediawas studied by using Picardrsquos successive approximationmethod The non-differential approximate solutions weregiven to show the efficiency of the present method

Conflict of Interests

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

Acknowledgments

This work was supported by National Scientific and Techno-logical Support Projects (no 2012BAE09B00) the NationalNatural Science Foundation of China (no 51274270) and theNational Natural Science Foundation of Hebei Province (noE2013209215)

References

[1] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 pp 50ndash67 1947

[2] T M Shih ldquoA literature survey on numerical heat transferrdquoNumerical Heat Transfer vol 5 no 4 pp 369ndash420 1982

[3] B C Choi and S W Churchill ldquoA technique for obtainingapproximate solutions for a class of integral equations arising inradiative heat transferrdquo International Journal of Heat and FluidFlow vol 6 no 1 pp 42ndash48 1985

[4] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2004

Abstract and Applied Analysis 5

[5] M Dehghan ldquoThe one-dimensional heat equation subject to aboundary integral specificationrdquoChaos Solitons amp Fractals vol32 no 2 pp 661ndash675 2007

[6] Y Ioannou M M Fyrillas and C Doumanidis ldquoApproximatesolution to Fredholm integral equations using linear regressionand applications to heat andmass transferrdquoEngineeringAnalysiswith Boundary Elements vol 36 no 8 pp 1278ndash1283 2012

[7] S Nadeem and N S Akbar ldquoEffects of heat transfer on theperistaltic transport ofMHDNewtonian fluid with variable vis-cosity application of adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 11 pp 3844ndash3855 2009

[8] A-M Wazwaz and M S Mehanna ldquoThe combined Laplace-Adomian method for handling singular integral equation ofheat transferrdquo International Journal of Nonlinear Science vol 10no 2 pp 248ndash252 2010

[9] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

[10] B Raftari and K Vajravelu ldquoHomotopy analysis method forMHD viscoelastic fluid flow and heat transfer in a channel witha stretching wallrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4149ndash4162 2012

[11] A A Joneidi D D Ganji andM Babaelahi ldquoDifferential trans-formation method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[12] C Cattani and E Laserra ldquoSpline-wavelets techniques for heatpropagationrdquo Journal of Information amp Optimization Sciencesvol 24 no 3 pp 485ndash496 2003

[13] N Simoes A Tadeu J Antonio and W Mansur ldquoTransientheat conduction under nonzero initial conditions a solutionusing the boundary element method in the frequency domainrdquoEngineering Analysis with Boundary Elements vol 36 no 4 pp562ndash567 2012

[14] J Hristov ldquoApproximate solutions to fractional sub-diffusionequations the heat-balance integral methodrdquo The EuropeanPhysical Journal vol 193 no 1 pp 229ndash243 2011

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

[16] D D Ganji and H Sajjadi ldquoNew analytical solution for naturalconvection of Darcian fluid in porous media prescribed surfaceheat fluxrdquoThermal Science vol 15 no 2 pp 221ndash227 2011

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

[18] X-J Yang ldquoPicardrsquos approximation method for solving a classof local fractional Volterra integral equationsrdquo Advances inIntelligent Transportation Systems vol 1 no 3 pp 67ndash70 2012

[19] R R Nigmatullin ldquoThe realization of the generalized transferequation in a medium with fractal geometryrdquo Physica StatusSolidi B vol 133 no 1 pp 425ndash430 1986

[20] K Davey and R Prosser ldquoAnalytical solutions for heat transferon fractal and pre-fractal domainsrdquo Applied MathematicalModelling vol 37 no 1-2 pp 554ndash569 2013

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

[22] Y Z Zhang A M Yang and X-J Yang ldquo1-D heat conductionin a fractal medium a solution by the local fractional FourierseriesmethodrdquoThermal Science vol 17 no 3 pp 953ndash956 2013

[23] M-S Hu D Baleanu and X-J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

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

[25] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[26] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[27] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[28] M H Heydari M R Hooshmandasl F M Maalek Ghaini andM Li ldquoChebyshev wavelets method for solution of nonlinearfractional integro-differential equations in a large intervalrdquoAdvances in Mathematical Physics vol 2013 Article ID 48208312 pages 2013

[29] 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

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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

media by using Picardrsquos successive approximation method[18] This paper is organized as follows In Section 2 wegive notations to local fractional derivative and integrals andinvestigate the heat transfer in fractal media Section 3 isdevoted to Picardrsquos successive approximation method basedupon local fractional integrals Analysis solution is shown inSection 4 Conclusions are in Section 5

2 Heat Transfer in Fractal Mediawith Local Fractional Derivative

In order to study the non-differential solution for the heatproblem in fractal media with local fractional derivative wehere begin with the Fourier flux equation in discontinuousmedia

The temperature field reads as [21]

119879 (119909 119910 119911 120591) = 119891 (119909 119910 119911 120591) at 120591 gt 1205910and in Ω (1)

where 119891(119909 119910 119911 120591) is local fractional continuous at fractaldomainΩ

For a given temperature field 119879 a local fractional temper-ature gradient [21] can be written as follows

nabla120572119879 =

120597120572119879

120597119906120572

1

997888119890120572

1+120597120572119879

120597119906120572

2

997888119890120572

2+120597120572119879

120597119906120572

3

997888119890120572

3 (2)

where the local fractional partial derivative is defined by [21ndash24]

119891(120572)

119909(1199090 119910) =

120597120572119891 (119909 119910)

120597119909120572

100381610038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909 119910) minus 119891 (119909

0 119910))

(119909 minus 1199090)120572

(3)

whereΔ120572(119891(119909 119910)minus119891(1199090 119910)) cong Γ(1+120572)Δ(119891(119909 119910)minus119891(119909

0 119910))

Here the local fractional derivative is defined on thefractal set like a Cantor set For example when we considerthe Cantor set we can find the local fractional derivative ofdiscontinuous function 119879 (however 119879 is a local fractionalcontinuous function)

We consider the heat flux per unit fractal area 997888119902 is

proportional to the temperature gradient in fractal mediumFourier law of heat conduction in fractal medium with localfractional derivative is expressed by [21]

997888119902 (119909 119910 119911 119905) = minus119870

2120572nabla120572119879 (119909 119910 119911 119905) (4)

where 1198702120572 denotes the thermal conductivity of the fractal

material and it is related to fractal dimensions of materialsIt is shown that the fractal dimensions of materials are animportant characteristic value Here we consider the fractalFourier flow which is discontinuous however it is found thatit is local fractional continuous Like classical Fourier flow itsthermal conductivity is an approximate value for fractal onewhen 120572 = 1

Fourier lawof one-dimensional heat conduction equationin fractal media reads as [21]

119902 (119909 119905) = minus1198702120572 119889120572119879 (119909 119905)

119889119909120572 at 120591 gt 120591

0and in 119860 (5)

where 1198702120572 denotes the thermal conductivity of the fractal

materialsWhen 120591 = 120591

0 from (5) we have

119902 (119909) = minus1198702120572 119889120572119879 (119909)

119889119909120572 (6)

at 120591 gt 1205910and in 119860 where 1198702120572 is the thermal conductivity

of the fractal materials Namely 119879 is a bi-Lipschitz mappingand shows the fractal characteristic behavior [21]

Local fractional heat conduction equation with heatgeneration in fractal media can be written as [21]

1198702120572nabla2120572119879 + 119892 minus 120588

120572119888120572

120597120572119879

120597119905120572= 0 at 120591 gt 120591

0and in Ω (7)

Local fractional heat conduction equation with no heatgeneration in fractal media is suggested as [21 22]

1198702120572nabla2120572119879 minus 120588120572119888120572

120597120572119879

120597119905120572= 0 at 120591 gt 120591

0and in Ω (8)

where nabla2120572 is a local fractional Laplace operator [21]

3 The Method

In this section we discuss the Picard successive approxima-tion method Meanwhile we transfer the Fourier law of one-dimensional heat conduction equation in fractal media intothe local fractional Volterra integral equation of the secondkind

31 Picardrsquos Successive Approximation Method This methodis first proposed in [18] Here wewill give a short introductionto Picardrsquos successive approximation method within the localfractional calculus

In this method we set

1199060(119909) = 119891 (119909) (9)

We give the first approximation 1199061(119909) by

1199061(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 1199060(119909) (119889119905)

120572 (10)

where the local fractional integral of 119891(119909) of order 120572 in theinterval [119886 119887] is defined as follows [21ndash24]

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

(11)

withΔ119905119895= 119905119895+1

minus119905119895Δ119905 = maxΔ119905

0 Δ1199051 Δ1199052 and [119905

119895 119905119895+1

]119895 = 0 119873 minus 1 119905

0= 119886 and 119905

119873= 119887

Here we find that the equality 1199061(119909) is a local fractional

continuous function if 119891(119909) 119870(119909 119905) and 1199060(119909) are local

fractional continuous functions

Abstract and Applied Analysis 3

Continuing in thismanner we have the infinite sequencesof functions

1199060(119909) 119906

1(119909) 119906

2(119909) 119906

119899(119909) (12)

such that the recurrence equations are given by

119906119899(119909) = 119891 (119909) minus

1

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(119899 = 1 2 3 )

(13)

where 1199060(119909) is equivalent to any selected function which is

the local fractional continuous functionHence we have successive approximation as follows

1199061(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119891 (119905) (119889119905)120572

119906119899(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(14)

Thus at the limit the solution 119906(119909) is written as

119906 (119909) = lim119899rarrinfin

119906119899 (119909) (15)

32 An Alternative Method from Local Fractional Derivativeto Local Fractional Volterra Integral Equations We directlyobserve that the local fractional differential equation of 120572order

1198702120572 119889120572119879

119889119909120572= 119902 (119909) (0 le 119909 le 119887) (16)

can be written immediately as the local fractional Volterraintegral equation of thee second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119902 (119905) (119889119905)120572 (17)

where 120582 = 11198702

TheMittag-Leffler type Fourier flux distribution in fractalmedia can be written as follows

119902 (119909) = 119902119905(119905 119909) = 119864

120572(119909 minus 119905)

120572119879 (119905) (0 le 119905 le 119887) (18)

Making use of (18) we can get the local fractional Volterraintegral equation of the second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119879 (119905) (119889119905)

120572 (19)

4 Approximate Solutions for LocalFractional Volterra Integral Equationof the Second Kind

Let us assume that the zeroth approximation is

1199060(119909) = 0 (20)

Then the first approximation can be written as follows

1199061(119909) = 119891 (119909) = 119879 (119886) (21)

Here we obtain the second approximation which reads as

1199062 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199061 (119905) (119889119905)

120572

(22)

Proceeding in this manner we have the third approximationin the following form

1199063 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199062 (119905) (119889119905)

120572 (23)

Hence continuing in this manner we obtain

119906119899 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119906119899minus1 (119905) (119889119905)

120572

= 119891 (119909) +120582120572

Γ (1 + 120572)

times int

119909

0

(1 + sdot sdot sdot +120582(119899minus1)120572

(119909 minus 119905)(119899minus1)120572

Γ (1 + (119899 minus 1) 120572))

times 119864120572(119909 minus 119905)

120572119891 (119905) (119889119905)

120572

(24)

Taking the limit we have

119906 (119909) = lim119899rarrinfin

119906119899 (119909)

= 119879 (119886) +120582120572

Γ (1 + 120572)

times int

119909

0

119864120572[(119909 minus 119905)

120572(1 + 120582

120572)] 119879 (119886) (119889119905)

120572

= 119879 (119886) 2120582120572

1 + 120582120572119864120572[119909120572(1 + 120582

120572)]

= 119879 (119886) 2

1198702120572 + 1119864120572[119909120572(1 +

1

1198702120572)]

(25)

where the term119864120572[119909120572(1+120582120572)] is aMittag-Leffler type Fourier

flux distribution in fractal media which is related to thefractal coarse-grained mass function [21 24] When 119870 = 1we get 119906(119909) = 119879(119886)119864

120572[2119909120572] The nondifferentiable solution

of (25) for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 1

is shown in Figure 1 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 2 isshown in Figure 2 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 3 isshown in Figure 3 the non-differentiable solution of (25) forparameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 4 is shown inFigure 4

5 Conclusions

This work studied the Fourier law of one-dimensional heatconduction equation in fractal media Mittag-Leffler type

4 Abstract and Applied Analysis

0 10

5

10

15

20

25

30

35

40

02 04 06 08

Figure 1 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 1

0

10

20

30

40

50

60

70

80

0 102 04 06 08

Figure 2 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 2

0

20

40

60

80

100

120

0 102 04 06 08

Figure 3 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 3

0

50

100

150

0 102 04 06 08

Figure 4 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 4

Fourier flux distribution in fractal media with temperaturefield effect was considered An approximation solution forthe local fractional Volterra integral equation of the secondkind derived from Fourier law of one-dimensional heatconduction equation for heat conduction in discontinuousmediawas studied by using Picardrsquos successive approximationmethod The non-differential approximate solutions weregiven to show the efficiency of the present method

Conflict of Interests

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

Acknowledgments

This work was supported by National Scientific and Techno-logical Support Projects (no 2012BAE09B00) the NationalNatural Science Foundation of China (no 51274270) and theNational Natural Science Foundation of Hebei Province (noE2013209215)

References

[1] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 pp 50ndash67 1947

[2] T M Shih ldquoA literature survey on numerical heat transferrdquoNumerical Heat Transfer vol 5 no 4 pp 369ndash420 1982

[3] B C Choi and S W Churchill ldquoA technique for obtainingapproximate solutions for a class of integral equations arising inradiative heat transferrdquo International Journal of Heat and FluidFlow vol 6 no 1 pp 42ndash48 1985

[4] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2004

Abstract and Applied Analysis 5

[5] M Dehghan ldquoThe one-dimensional heat equation subject to aboundary integral specificationrdquoChaos Solitons amp Fractals vol32 no 2 pp 661ndash675 2007

[6] Y Ioannou M M Fyrillas and C Doumanidis ldquoApproximatesolution to Fredholm integral equations using linear regressionand applications to heat andmass transferrdquoEngineeringAnalysiswith Boundary Elements vol 36 no 8 pp 1278ndash1283 2012

[7] S Nadeem and N S Akbar ldquoEffects of heat transfer on theperistaltic transport ofMHDNewtonian fluid with variable vis-cosity application of adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 11 pp 3844ndash3855 2009

[8] A-M Wazwaz and M S Mehanna ldquoThe combined Laplace-Adomian method for handling singular integral equation ofheat transferrdquo International Journal of Nonlinear Science vol 10no 2 pp 248ndash252 2010

[9] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

[10] B Raftari and K Vajravelu ldquoHomotopy analysis method forMHD viscoelastic fluid flow and heat transfer in a channel witha stretching wallrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4149ndash4162 2012

[11] A A Joneidi D D Ganji andM Babaelahi ldquoDifferential trans-formation method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[12] C Cattani and E Laserra ldquoSpline-wavelets techniques for heatpropagationrdquo Journal of Information amp Optimization Sciencesvol 24 no 3 pp 485ndash496 2003

[13] N Simoes A Tadeu J Antonio and W Mansur ldquoTransientheat conduction under nonzero initial conditions a solutionusing the boundary element method in the frequency domainrdquoEngineering Analysis with Boundary Elements vol 36 no 4 pp562ndash567 2012

[14] J Hristov ldquoApproximate solutions to fractional sub-diffusionequations the heat-balance integral methodrdquo The EuropeanPhysical Journal vol 193 no 1 pp 229ndash243 2011

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

[16] D D Ganji and H Sajjadi ldquoNew analytical solution for naturalconvection of Darcian fluid in porous media prescribed surfaceheat fluxrdquoThermal Science vol 15 no 2 pp 221ndash227 2011

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

[18] X-J Yang ldquoPicardrsquos approximation method for solving a classof local fractional Volterra integral equationsrdquo Advances inIntelligent Transportation Systems vol 1 no 3 pp 67ndash70 2012

[19] R R Nigmatullin ldquoThe realization of the generalized transferequation in a medium with fractal geometryrdquo Physica StatusSolidi B vol 133 no 1 pp 425ndash430 1986

[20] K Davey and R Prosser ldquoAnalytical solutions for heat transferon fractal and pre-fractal domainsrdquo Applied MathematicalModelling vol 37 no 1-2 pp 554ndash569 2013

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

[22] Y Z Zhang A M Yang and X-J Yang ldquo1-D heat conductionin a fractal medium a solution by the local fractional FourierseriesmethodrdquoThermal Science vol 17 no 3 pp 953ndash956 2013

[23] M-S Hu D Baleanu and X-J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

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

[25] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[26] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[27] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[28] M H Heydari M R Hooshmandasl F M Maalek Ghaini andM Li ldquoChebyshev wavelets method for solution of nonlinearfractional integro-differential equations in a large intervalrdquoAdvances in Mathematical Physics vol 2013 Article ID 48208312 pages 2013

[29] 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

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

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 3

Continuing in thismanner we have the infinite sequencesof functions

1199060(119909) 119906

1(119909) 119906

2(119909) 119906

119899(119909) (12)

such that the recurrence equations are given by

119906119899(119909) = 119891 (119909) minus

1

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(119899 = 1 2 3 )

(13)

where 1199060(119909) is equivalent to any selected function which is

the local fractional continuous functionHence we have successive approximation as follows

1199061(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119891 (119905) (119889119905)120572

119906119899(119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119870 (119909 119905) 119906119899minus1

(119909) (119889119905)120572

(14)

Thus at the limit the solution 119906(119909) is written as

119906 (119909) = lim119899rarrinfin

119906119899 (119909) (15)

32 An Alternative Method from Local Fractional Derivativeto Local Fractional Volterra Integral Equations We directlyobserve that the local fractional differential equation of 120572order

1198702120572 119889120572119879

119889119909120572= 119902 (119909) (0 le 119909 le 119887) (16)

can be written immediately as the local fractional Volterraintegral equation of thee second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119902 (119905) (119889119905)120572 (17)

where 120582 = 11198702

TheMittag-Leffler type Fourier flux distribution in fractalmedia can be written as follows

119902 (119909) = 119902119905(119905 119909) = 119864

120572(119909 minus 119905)

120572119879 (119905) (0 le 119905 le 119887) (18)

Making use of (18) we can get the local fractional Volterraintegral equation of the second kind in the form

119879 (119909) = 119879 (119886) +120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119879 (119905) (119889119905)

120572 (19)

4 Approximate Solutions for LocalFractional Volterra Integral Equationof the Second Kind

Let us assume that the zeroth approximation is

1199060(119909) = 0 (20)

Then the first approximation can be written as follows

1199061(119909) = 119891 (119909) = 119879 (119886) (21)

Here we obtain the second approximation which reads as

1199062 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199061 (119905) (119889119905)

120572

(22)

Proceeding in this manner we have the third approximationin the following form

1199063 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

1205721199062 (119905) (119889119905)

120572 (23)

Hence continuing in this manner we obtain

119906119899 (119909) = 119891 (119909) +

120582120572

Γ (1 + 120572)int

119909

0

119864120572(119909 minus 119905)

120572119906119899minus1 (119905) (119889119905)

120572

= 119891 (119909) +120582120572

Γ (1 + 120572)

times int

119909

0

(1 + sdot sdot sdot +120582(119899minus1)120572

(119909 minus 119905)(119899minus1)120572

Γ (1 + (119899 minus 1) 120572))

times 119864120572(119909 minus 119905)

120572119891 (119905) (119889119905)

120572

(24)

Taking the limit we have

119906 (119909) = lim119899rarrinfin

119906119899 (119909)

= 119879 (119886) +120582120572

Γ (1 + 120572)

times int

119909

0

119864120572[(119909 minus 119905)

120572(1 + 120582

120572)] 119879 (119886) (119889119905)

120572

= 119879 (119886) 2120582120572

1 + 120582120572119864120572[119909120572(1 + 120582

120572)]

= 119879 (119886) 2

1198702120572 + 1119864120572[119909120572(1 +

1

1198702120572)]

(25)

where the term119864120572[119909120572(1+120582120572)] is aMittag-Leffler type Fourier

flux distribution in fractal media which is related to thefractal coarse-grained mass function [21 24] When 119870 = 1we get 119906(119909) = 119879(119886)119864

120572[2119909120572] The nondifferentiable solution

of (25) for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 1

is shown in Figure 1 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 2 isshown in Figure 2 the non-differentiable solution of (25)for parameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 3 isshown in Figure 3 the non-differentiable solution of (25) forparameters 119870 = 1 120572 = ln 2 ln 3 and 119879(119886) = 4 is shown inFigure 4

5 Conclusions

This work studied the Fourier law of one-dimensional heatconduction equation in fractal media Mittag-Leffler type

4 Abstract and Applied Analysis

0 10

5

10

15

20

25

30

35

40

02 04 06 08

Figure 1 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 1

0

10

20

30

40

50

60

70

80

0 102 04 06 08

Figure 2 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 2

0

20

40

60

80

100

120

0 102 04 06 08

Figure 3 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 3

0

50

100

150

0 102 04 06 08

Figure 4 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 4

Fourier flux distribution in fractal media with temperaturefield effect was considered An approximation solution forthe local fractional Volterra integral equation of the secondkind derived from Fourier law of one-dimensional heatconduction equation for heat conduction in discontinuousmediawas studied by using Picardrsquos successive approximationmethod The non-differential approximate solutions weregiven to show the efficiency of the present method

Conflict of Interests

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

Acknowledgments

This work was supported by National Scientific and Techno-logical Support Projects (no 2012BAE09B00) the NationalNatural Science Foundation of China (no 51274270) and theNational Natural Science Foundation of Hebei Province (noE2013209215)

References

[1] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 pp 50ndash67 1947

[2] T M Shih ldquoA literature survey on numerical heat transferrdquoNumerical Heat Transfer vol 5 no 4 pp 369ndash420 1982

[3] B C Choi and S W Churchill ldquoA technique for obtainingapproximate solutions for a class of integral equations arising inradiative heat transferrdquo International Journal of Heat and FluidFlow vol 6 no 1 pp 42ndash48 1985

[4] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2004

Abstract and Applied Analysis 5

[5] M Dehghan ldquoThe one-dimensional heat equation subject to aboundary integral specificationrdquoChaos Solitons amp Fractals vol32 no 2 pp 661ndash675 2007

[6] Y Ioannou M M Fyrillas and C Doumanidis ldquoApproximatesolution to Fredholm integral equations using linear regressionand applications to heat andmass transferrdquoEngineeringAnalysiswith Boundary Elements vol 36 no 8 pp 1278ndash1283 2012

[7] S Nadeem and N S Akbar ldquoEffects of heat transfer on theperistaltic transport ofMHDNewtonian fluid with variable vis-cosity application of adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 11 pp 3844ndash3855 2009

[8] A-M Wazwaz and M S Mehanna ldquoThe combined Laplace-Adomian method for handling singular integral equation ofheat transferrdquo International Journal of Nonlinear Science vol 10no 2 pp 248ndash252 2010

[9] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

[10] B Raftari and K Vajravelu ldquoHomotopy analysis method forMHD viscoelastic fluid flow and heat transfer in a channel witha stretching wallrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4149ndash4162 2012

[11] A A Joneidi D D Ganji andM Babaelahi ldquoDifferential trans-formation method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[12] C Cattani and E Laserra ldquoSpline-wavelets techniques for heatpropagationrdquo Journal of Information amp Optimization Sciencesvol 24 no 3 pp 485ndash496 2003

[13] N Simoes A Tadeu J Antonio and W Mansur ldquoTransientheat conduction under nonzero initial conditions a solutionusing the boundary element method in the frequency domainrdquoEngineering Analysis with Boundary Elements vol 36 no 4 pp562ndash567 2012

[14] J Hristov ldquoApproximate solutions to fractional sub-diffusionequations the heat-balance integral methodrdquo The EuropeanPhysical Journal vol 193 no 1 pp 229ndash243 2011

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

[16] D D Ganji and H Sajjadi ldquoNew analytical solution for naturalconvection of Darcian fluid in porous media prescribed surfaceheat fluxrdquoThermal Science vol 15 no 2 pp 221ndash227 2011

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

[18] X-J Yang ldquoPicardrsquos approximation method for solving a classof local fractional Volterra integral equationsrdquo Advances inIntelligent Transportation Systems vol 1 no 3 pp 67ndash70 2012

[19] R R Nigmatullin ldquoThe realization of the generalized transferequation in a medium with fractal geometryrdquo Physica StatusSolidi B vol 133 no 1 pp 425ndash430 1986

[20] K Davey and R Prosser ldquoAnalytical solutions for heat transferon fractal and pre-fractal domainsrdquo Applied MathematicalModelling vol 37 no 1-2 pp 554ndash569 2013

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

[22] Y Z Zhang A M Yang and X-J Yang ldquo1-D heat conductionin a fractal medium a solution by the local fractional FourierseriesmethodrdquoThermal Science vol 17 no 3 pp 953ndash956 2013

[23] M-S Hu D Baleanu and X-J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

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

[25] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[26] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[27] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[28] M H Heydari M R Hooshmandasl F M Maalek Ghaini andM Li ldquoChebyshev wavelets method for solution of nonlinearfractional integro-differential equations in a large intervalrdquoAdvances in Mathematical Physics vol 2013 Article ID 48208312 pages 2013

[29] 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

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

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

0 10

5

10

15

20

25

30

35

40

02 04 06 08

Figure 1 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 1

0

10

20

30

40

50

60

70

80

0 102 04 06 08

Figure 2 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 2

0

20

40

60

80

100

120

0 102 04 06 08

Figure 3 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 3

0

50

100

150

0 102 04 06 08

Figure 4 The nondifferentiable solution for Mittag-Leffler typeFourier flux distribution for parameters 119870 = 1 120572 = ln 2 ln 3 and119879(119886) = 4

Fourier flux distribution in fractal media with temperaturefield effect was considered An approximation solution forthe local fractional Volterra integral equation of the secondkind derived from Fourier law of one-dimensional heatconduction equation for heat conduction in discontinuousmediawas studied by using Picardrsquos successive approximationmethod The non-differential approximate solutions weregiven to show the efficiency of the present method

Conflict of Interests

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

Acknowledgments

This work was supported by National Scientific and Techno-logical Support Projects (no 2012BAE09B00) the NationalNatural Science Foundation of China (no 51274270) and theNational Natural Science Foundation of Hebei Province (noE2013209215)

References

[1] J Crank and P Nicolson ldquoA practical method for numericalevaluation of solutions of partial differential equations of theheat-conduction typerdquo Mathematical Proceedings of the Cam-bridge Philosophical Society vol 43 pp 50ndash67 1947

[2] T M Shih ldquoA literature survey on numerical heat transferrdquoNumerical Heat Transfer vol 5 no 4 pp 369ndash420 1982

[3] B C Choi and S W Churchill ldquoA technique for obtainingapproximate solutions for a class of integral equations arising inradiative heat transferrdquo International Journal of Heat and FluidFlow vol 6 no 1 pp 42ndash48 1985

[4] Y Z Povstenko ldquoFractional heat conduction equation andassociated thermal stressrdquo Journal of Thermal Stresses vol 28no 1 pp 83ndash102 2004

Abstract and Applied Analysis 5

[5] M Dehghan ldquoThe one-dimensional heat equation subject to aboundary integral specificationrdquoChaos Solitons amp Fractals vol32 no 2 pp 661ndash675 2007

[6] Y Ioannou M M Fyrillas and C Doumanidis ldquoApproximatesolution to Fredholm integral equations using linear regressionand applications to heat andmass transferrdquoEngineeringAnalysiswith Boundary Elements vol 36 no 8 pp 1278ndash1283 2012

[7] S Nadeem and N S Akbar ldquoEffects of heat transfer on theperistaltic transport ofMHDNewtonian fluid with variable vis-cosity application of adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 11 pp 3844ndash3855 2009

[8] A-M Wazwaz and M S Mehanna ldquoThe combined Laplace-Adomian method for handling singular integral equation ofheat transferrdquo International Journal of Nonlinear Science vol 10no 2 pp 248ndash252 2010

[9] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

[10] B Raftari and K Vajravelu ldquoHomotopy analysis method forMHD viscoelastic fluid flow and heat transfer in a channel witha stretching wallrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4149ndash4162 2012

[11] A A Joneidi D D Ganji andM Babaelahi ldquoDifferential trans-formation method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[12] C Cattani and E Laserra ldquoSpline-wavelets techniques for heatpropagationrdquo Journal of Information amp Optimization Sciencesvol 24 no 3 pp 485ndash496 2003

[13] N Simoes A Tadeu J Antonio and W Mansur ldquoTransientheat conduction under nonzero initial conditions a solutionusing the boundary element method in the frequency domainrdquoEngineering Analysis with Boundary Elements vol 36 no 4 pp562ndash567 2012

[14] J Hristov ldquoApproximate solutions to fractional sub-diffusionequations the heat-balance integral methodrdquo The EuropeanPhysical Journal vol 193 no 1 pp 229ndash243 2011

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

[16] D D Ganji and H Sajjadi ldquoNew analytical solution for naturalconvection of Darcian fluid in porous media prescribed surfaceheat fluxrdquoThermal Science vol 15 no 2 pp 221ndash227 2011

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

[18] X-J Yang ldquoPicardrsquos approximation method for solving a classof local fractional Volterra integral equationsrdquo Advances inIntelligent Transportation Systems vol 1 no 3 pp 67ndash70 2012

[19] R R Nigmatullin ldquoThe realization of the generalized transferequation in a medium with fractal geometryrdquo Physica StatusSolidi B vol 133 no 1 pp 425ndash430 1986

[20] K Davey and R Prosser ldquoAnalytical solutions for heat transferon fractal and pre-fractal domainsrdquo Applied MathematicalModelling vol 37 no 1-2 pp 554ndash569 2013

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

[22] Y Z Zhang A M Yang and X-J Yang ldquo1-D heat conductionin a fractal medium a solution by the local fractional FourierseriesmethodrdquoThermal Science vol 17 no 3 pp 953ndash956 2013

[23] M-S Hu D Baleanu and X-J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

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

[25] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[26] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[27] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[28] M H Heydari M R Hooshmandasl F M Maalek Ghaini andM Li ldquoChebyshev wavelets method for solution of nonlinearfractional integro-differential equations in a large intervalrdquoAdvances in Mathematical Physics vol 2013 Article ID 48208312 pages 2013

[29] 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

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

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] M Dehghan ldquoThe one-dimensional heat equation subject to aboundary integral specificationrdquoChaos Solitons amp Fractals vol32 no 2 pp 661ndash675 2007

[6] Y Ioannou M M Fyrillas and C Doumanidis ldquoApproximatesolution to Fredholm integral equations using linear regressionand applications to heat andmass transferrdquoEngineeringAnalysiswith Boundary Elements vol 36 no 8 pp 1278ndash1283 2012

[7] S Nadeem and N S Akbar ldquoEffects of heat transfer on theperistaltic transport ofMHDNewtonian fluid with variable vis-cosity application of adomian decomposition methodrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 11 pp 3844ndash3855 2009

[8] A-M Wazwaz and M S Mehanna ldquoThe combined Laplace-Adomian method for handling singular integral equation ofheat transferrdquo International Journal of Nonlinear Science vol 10no 2 pp 248ndash252 2010

[9] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

[10] B Raftari and K Vajravelu ldquoHomotopy analysis method forMHD viscoelastic fluid flow and heat transfer in a channel witha stretching wallrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4149ndash4162 2012

[11] A A Joneidi D D Ganji andM Babaelahi ldquoDifferential trans-formation method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[12] C Cattani and E Laserra ldquoSpline-wavelets techniques for heatpropagationrdquo Journal of Information amp Optimization Sciencesvol 24 no 3 pp 485ndash496 2003

[13] N Simoes A Tadeu J Antonio and W Mansur ldquoTransientheat conduction under nonzero initial conditions a solutionusing the boundary element method in the frequency domainrdquoEngineering Analysis with Boundary Elements vol 36 no 4 pp562ndash567 2012

[14] J Hristov ldquoApproximate solutions to fractional sub-diffusionequations the heat-balance integral methodrdquo The EuropeanPhysical Journal vol 193 no 1 pp 229ndash243 2011

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

[16] D D Ganji and H Sajjadi ldquoNew analytical solution for naturalconvection of Darcian fluid in porous media prescribed surfaceheat fluxrdquoThermal Science vol 15 no 2 pp 221ndash227 2011

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

[18] X-J Yang ldquoPicardrsquos approximation method for solving a classof local fractional Volterra integral equationsrdquo Advances inIntelligent Transportation Systems vol 1 no 3 pp 67ndash70 2012

[19] R R Nigmatullin ldquoThe realization of the generalized transferequation in a medium with fractal geometryrdquo Physica StatusSolidi B vol 133 no 1 pp 425ndash430 1986

[20] K Davey and R Prosser ldquoAnalytical solutions for heat transferon fractal and pre-fractal domainsrdquo Applied MathematicalModelling vol 37 no 1-2 pp 554ndash569 2013

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

[22] Y Z Zhang A M Yang and X-J Yang ldquo1-D heat conductionin a fractal medium a solution by the local fractional FourierseriesmethodrdquoThermal Science vol 17 no 3 pp 953ndash956 2013

[23] M-S Hu D Baleanu and X-J Yang ldquoOne-phase problemsfor discontinuous heat transfer in fractal mediardquoMathematicalProblems in Engineering vol 2013 Article ID 358473 3 pages2013

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

[25] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[26] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[27] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

[28] M H Heydari M R Hooshmandasl F M Maalek Ghaini andM Li ldquoChebyshev wavelets method for solution of nonlinearfractional integro-differential equations in a large intervalrdquoAdvances in Mathematical Physics vol 2013 Article ID 48208312 pages 2013

[29] 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

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

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