Research ArticleApplication of Local Fractional Series Expansion Method toSolve Klein-Gordon Equations on Cantor Sets

Ai-Min Yang12 Yu-Zhu Zhang23 Carlo Cattani4 Gong-Nan Xie5

Mohammad Mehdi Rashidi6 Yi-Jun Zhou7 and Xiao-Jun Yang8

1 College of Science Hebei United University Tangshan 063009 China2 College of Mechanical Engineering Yanshan University Qinhuangdao 066004 China3 College of Metallurgy and Energy Hebei United University Tangshan 063009 China4Department of Mathematics University of Salerno Via Ponte don Melillo Fisciano 84084 Salerno Italy5 School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan Shaanxi 710048 China6Department of Mechanical Engineering Bu-Ali Sina University PO Box 65175-4161 Hamedan Iran7Qinggong College Hebei United University Tangshan 063009 China8Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Jiangsu 221008 China

Correspondence should be addressed to Yu-Zhu Zhang zyzheuueducn

Received 5 January 2014 Revised 8 February 2014 Accepted 8 February 2014 Published 12 March 2014

Academic Editor Ali H Bhrawy

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

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractionalderivatives The analytical solutions within the nondifferential terms are discussed The obtained results show the simplicity andefficiency of the present technique with application to the problems of the liner differential equations on Cantor sets

1 Introduction

The Klein-Gordon equation [1] has been applied to mathe-matical physics such as solid-state physics nonlinear opticsand quantum field theory Some of the analytical methodsfor solving theKlein-Gordon equation include the variationaliteration method [2] the tanh and the sine-cosine methods[3] the decompositionmethod [4] the differential transformmethod [5] and the homotopy-perturbation method [6]

Recently the solutions for the fractional Klein-Gordonequation with the Caputo fractional derivative were con-sidered in [7ndash9] Golmankhaneh et al used the homotopy-perturbation method to obtain solution for the fractionalKlein-Gordon equation [7] Kurulay [8] pointed out thesolution for the fractional Klein-Gordon equation by usingthe homotopy analysis method Gepreel and Mohamed [9]presented the solution for nonlinear space-time fractionalKlein-Gordon equation by the homotopy analysis method

When some domains cannot be described by smoothfunctions both the classical approach and the fractional

approach based on Riemann-Liouville (or Caputo) deriva-tives are unacceptable In such cases the local fractionalcalculus is an efficient technique for modeling these physicalproblems [10ndash23] Using the fractional complex transformmethod [20] one transforms the classical Klein-Gordonequation into the Klein-Gordon equation on Cantor sets inthe following form

1205972120572119906 (119909 119905)

1205971199052120572minus1205972120572119906 (119909 119905)

1205971199092120572= 119865 (119906 (119909)) (1)

subject to the initial value conditions

119906 (119909 0) = 119891 (119909)

120597120572

120597119906120572119906 (119909 0) = 119892 (119909)

(2)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 372741 6 pageshttpdxdoiorg1011552014372741

2 Abstract and Applied Analysis

where the operator is the local fractional derivative operatorwhich is defined by [16ndash23]

119891(120572)(1199090) =119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(3)

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

0)) and 119906(119909 119905)

and 119892(119909) are the local fractional continuous functions and119865(119906(119909)) are themixed terms of nonlinear and liner functions

In view of (1)-(2) the linear Klein-Gordon equation onCantor sets

1205972120572119906 (119909 119905)

1205971199052120572minus1205972120572119906 (119909 119905)

1205971199092120572= 119906 (119909 119905) 119909 gt 0 119905 gt 0 (4)

subject to the initial value conditions

119906 (119909 0) = 119891 (119909)

120597120572

120597119906120572119906 (119909 0) = 119892 (119909)

(5)

is under consideration where 119891(119909) and 119892(119909) are localfractional continuous functions

On the other hand the local fractional series expansionmethodwas applied to solve the wave and diffusion equationson Cantor sets [21] the local fractional Schrodinger equationin the one-dimensional Cantorian system [22] and the localfractional Helmholtz equation [23] In this paper our aimis to investigate a new application of this technology tosolve the linear Klein-Gordon equations on Cantor sets Thepaper is organized as follows In Section 2 the idea of localfractional series expansion method is given In Section 3 thesolutions for linear Klein-Gordon equations on Cantor setsare presented Finally Section 4 is the conclusions

2 The Local Fractional SeriesExpansion Method

In order to illustrate the idea of the local fractional seriesexpansion method [21ndash23] we consider the local fractionaldifferential operator equation in the following form

1199062120572

2119905= 119871120572119906 (6)

where 119871120572is the linear local fractional operator and 120601 is a local

fractional continuous functionFrom (6) the multiterm separated functions with respect

to 119909 119905 read as

119906 (119909 119905) =

infin

sum

119894=0

120601119894(119905) 120595119894(119909) (7)

where 120601119894(119905) and 120595

119894(119909) are the local fractional continuous

functionsFrom (7) we have

120601119894(119905) =

119905119894120572

Γ (1 + 119894120572) (8)

so that

119906 (119909 119905) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909) (9)

In view of (9) we obtain

1199062120572

119905=

infin

sum

119894=0

1

Γ (1 + 119894120572)119905119894120572120595119894+2(119909)

119871120572119906 = 119871

120572[

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909)] =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572120595119894) (119909)

(10)

Making use of (10) we have

infin

sum

119894=0

1

Γ (1 + 119894120572)119905119894120572120595119894+2(119909) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572120595119894) (119909) (11)

so that

120595119894+2(119909) = (119871

120572120595119894) (119909) (12)

Hence from (12) we get

119906 (119909 119905) =

infin

sum

119894=0

119906119894(119909 119905) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909) (13)

We now rewrite (4) in the local fractional operator form asfollows

1199062120572

2119905= 119871120572119906 (14)

subject to the initial value conditions

119906 (119909 0) = 119891 (119909)

120597120572

120597119906120572119906 (119909 0) = 119892 (119909)

(15)

where the linear local fractional operator is defined as follows

119871120572=1205972120572

1205971199092120572+ 119868 (16)

Hence (16) is a special case of (6) and it is usedwith the linearKlein-Gordon equations on Cantor sets in next section

3 Analytical Solutions for LinearKlein-Gordon Equations on Cantor Sets

In this section we present the nondifferentiable solutions forlinear Klein-Gordon equations on Cantor sets

Example 1 Let us consider the Klein-Gordon equations onCantor sets in the following form

1199062120572

2119905= 119871120572119906 (17)

Abstract and Applied Analysis 3

subject to the initial value conditions

119906 (119909 0) = 0

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(18)

From (12) and (18) we can structure the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) = 0

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(19)

Hence we can calculate

1205950(119909) = 0

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 0

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 0

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(20)

and so onTherefore we have

119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum

119894=1

119905(2119894minus1)120572

Γ (1 + (2119894 minus 1) 120572)

+

infin

sum

119894=1

119905(1+2119894)120572

Γ (1 + (1 + 2119894) 120572)

(21)

and the corresponding graph is illustrated in Figure 1

Example 2 We consider the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (22)

subject to the initial value conditions

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) = 0

(23)

002

0406

081

0

10

05

05

1

15

2

25

tx

u(xt)

Figure 1 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

From (12) and (23) we get the following iterative formulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) = 0

(24)

Hence we get

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) = 0

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) = 0

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) = 0

(25)

and so onHereby we obtain the solution of (22)

119906 (119909 119905) =119909120572

Γ (1 + 120572)

infin

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572) (26)

and the corresponding graph is depicted in Figure 2

Example 3 We present the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (27)

4 Abstract and Applied Analysis

002

0406

081

0

10

05

05

1

15

2

3

25

tx

u(xt)

Figure 2 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

subject to the initial value conditions

119906 (119909 0) =1199092120572

Γ (1 + 2120572)

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(28)

From (12) and (27)-(28) we get the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

1199092120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(29)

From (29) we obtain

1205950(119909) =

1199092120572

Γ (1 + 2120572)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(30)

and so on

Therefore we obtain the exact solution of (27)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) + 119864120572(119905120572) minus119905120572

Γ (1 + 120572)minus 1

(31)

and its graph is shown in Figure 3

Example 4 The Klein-Gordon equation on Cantor sets ispresented as

1199062120572

2119905= 119871120572119906 (32)

and the initial value conditions are written as

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) =

119909120572

Γ (1 + 120572)

(33)

From (12) and (27)-(28) the following iterative formulas areas follows

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

119909120572

Γ (1 + 120572)

(34)

From (29) we give

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) =

119909120572

Γ (1 + 120572)

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) =

119909120572

Γ (1 + 120572)

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) =

119909120572

Γ (1 + 120572)

(35)

and so onTherefore we give the exact solution of (32)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) (36)

and its graph is shown in Figure 4

Abstract and Applied Analysis 5

0 0204

0608

1

0

10

05

2

8

4

6

tx

u(xt)

Figure 3 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

002

0406

081

0

10

05

1

2

3

4

5

6

tx

u(xt)

Figure 4 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

4 Conclusions

In this work the Klein-Gordon equations on Cantor setswithin the local fractional differential operator had beenanalyzed using the local fractional series expansion methodThe nondifferentiable solutions for local fractional Klein-Gordon equations were obtained The present method is apowerful mathematical tool for solving the local fractionallinear differential equations

Conflict of Interests

The authors declare that they have 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] A-M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons amp Fractals vol 28 no 4 pp 1005ndash1013 2006

[2] E Yusufoglu ldquoThe variational iterationmethod for studying theKlein-Gordon equationrdquo Applied Mathematics Letters vol 21no 7 pp 669ndash674 2008

[3] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[4] S M El-Sayed ldquoThe decomposition method for studying theKlein-Gordon equationrdquo Chaos Solitons amp Fractals vol 18 no5 pp 1025ndash1030 2003

[5] A S V Ravi Kanth and K Aruna ldquoDifferential transformmethod for solving the linear and nonlinear Klein-GordonequationrdquoComputer Physics Communications vol 180 no 5 pp708ndash711 2009

[6] M S H Chowdhury and I Hashim ldquoApplication of homotopy-perturbation method to Klein-Gordon and sine-Gordon equa-tionsrdquo Chaos Solitons amp Fractals vol 39 no 4 pp 1928ndash19352009

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

[8] M Kurulay ldquoSolving the fractional nonlinear Klein-Gordonequation bymeans of the homotopy analysismethodrdquoAdvancesin Difference Equations vol 2012 no 1 article 187 pp 1ndash8 2012

[9] K A Gepreel and M S Mohamed ldquoAnalytical approximatesolution for nonlinear space-time fractional Klein-Gordonequationrdquo Chinese Physics B vol 22 no 1 Article ID 0102012013

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

[11] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[12] A K Golmankhaneh and D Baleanu ldquoOn a new measure onfractalsrdquo Journal of Inequalities and Applications vol 2013 no1 article 522 2013

[13] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoLagrangian and Hamiltonian mechanics on fractals subset ofreal-linerdquo International Journal ofTheoretical Physics vol 52 no11 pp 4210ndash4217 2013

[14] K G Alireza V Fazlollahi and D Baleanu ldquoNewtonianmechanics on fractals subset of real-linerdquo Romania Reports inPhysics vol 65 pp 84ndash93 2013

[15] A S Balankin ldquoStresses and strains in a deformable fractalmedium and in its fractal continuum modelrdquo Physics Letters Avol 377 no 38 pp 2535ndash2541 2013

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

[17] 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

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

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

where the operator is the local fractional derivative operatorwhich is defined by [16ndash23]

119891(120572)(1199090) =119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090

= lim119909rarr1199090

Δ120572(119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(3)

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

0)) and 119906(119909 119905)

and 119892(119909) are the local fractional continuous functions and119865(119906(119909)) are themixed terms of nonlinear and liner functions

In view of (1)-(2) the linear Klein-Gordon equation onCantor sets

1205972120572119906 (119909 119905)

1205971199052120572minus1205972120572119906 (119909 119905)

1205971199092120572= 119906 (119909 119905) 119909 gt 0 119905 gt 0 (4)

subject to the initial value conditions

119906 (119909 0) = 119891 (119909)

120597120572

120597119906120572119906 (119909 0) = 119892 (119909)

(5)

is under consideration where 119891(119909) and 119892(119909) are localfractional continuous functions

On the other hand the local fractional series expansionmethodwas applied to solve the wave and diffusion equationson Cantor sets [21] the local fractional Schrodinger equationin the one-dimensional Cantorian system [22] and the localfractional Helmholtz equation [23] In this paper our aimis to investigate a new application of this technology tosolve the linear Klein-Gordon equations on Cantor sets Thepaper is organized as follows In Section 2 the idea of localfractional series expansion method is given In Section 3 thesolutions for linear Klein-Gordon equations on Cantor setsare presented Finally Section 4 is the conclusions

2 The Local Fractional SeriesExpansion Method

In order to illustrate the idea of the local fractional seriesexpansion method [21ndash23] we consider the local fractionaldifferential operator equation in the following form

1199062120572

2119905= 119871120572119906 (6)

where 119871120572is the linear local fractional operator and 120601 is a local

fractional continuous functionFrom (6) the multiterm separated functions with respect

to 119909 119905 read as

119906 (119909 119905) =

infin

sum

119894=0

120601119894(119905) 120595119894(119909) (7)

where 120601119894(119905) and 120595

119894(119909) are the local fractional continuous

functionsFrom (7) we have

120601119894(119905) =

119905119894120572

Γ (1 + 119894120572) (8)

so that

119906 (119909 119905) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909) (9)

In view of (9) we obtain

1199062120572

119905=

infin

sum

119894=0

1

Γ (1 + 119894120572)119905119894120572120595119894+2(119909)

119871120572119906 = 119871

120572[

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909)] =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572120595119894) (119909)

(10)

Making use of (10) we have

infin

sum

119894=0

1

Γ (1 + 119894120572)119905119894120572120595119894+2(119909) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)(119871120572120595119894) (119909) (11)

so that

120595119894+2(119909) = (119871

120572120595119894) (119909) (12)

Hence from (12) we get

119906 (119909 119905) =

infin

sum

119894=0

119906119894(119909 119905) =

infin

sum

119894=0

119905119894120572

Γ (1 + 119894120572)120595119894(119909) (13)

We now rewrite (4) in the local fractional operator form asfollows

1199062120572

2119905= 119871120572119906 (14)

subject to the initial value conditions

119906 (119909 0) = 119891 (119909)

120597120572

120597119906120572119906 (119909 0) = 119892 (119909)

(15)

where the linear local fractional operator is defined as follows

119871120572=1205972120572

1205971199092120572+ 119868 (16)

Hence (16) is a special case of (6) and it is usedwith the linearKlein-Gordon equations on Cantor sets in next section

3 Analytical Solutions for LinearKlein-Gordon Equations on Cantor Sets

In this section we present the nondifferentiable solutions forlinear Klein-Gordon equations on Cantor sets

Example 1 Let us consider the Klein-Gordon equations onCantor sets in the following form

1199062120572

2119905= 119871120572119906 (17)

Abstract and Applied Analysis 3

subject to the initial value conditions

119906 (119909 0) = 0

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(18)

From (12) and (18) we can structure the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) = 0

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(19)

Hence we can calculate

1205950(119909) = 0

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 0

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 0

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(20)

and so onTherefore we have

119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum

119894=1

119905(2119894minus1)120572

Γ (1 + (2119894 minus 1) 120572)

+

infin

sum

119894=1

119905(1+2119894)120572

Γ (1 + (1 + 2119894) 120572)

(21)

and the corresponding graph is illustrated in Figure 1

Example 2 We consider the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (22)

subject to the initial value conditions

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) = 0

(23)

002

0406

081

0

10

05

05

1

15

2

25

tx

u(xt)

Figure 1 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

From (12) and (23) we get the following iterative formulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) = 0

(24)

Hence we get

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) = 0

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) = 0

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) = 0

(25)

and so onHereby we obtain the solution of (22)

119906 (119909 119905) =119909120572

Γ (1 + 120572)

infin

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572) (26)

and the corresponding graph is depicted in Figure 2

Example 3 We present the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (27)

4 Abstract and Applied Analysis

002

0406

081

0

10

05

05

1

15

2

3

25

tx

u(xt)

Figure 2 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

subject to the initial value conditions

119906 (119909 0) =1199092120572

Γ (1 + 2120572)

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(28)

From (12) and (27)-(28) we get the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

1199092120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(29)

From (29) we obtain

1205950(119909) =

1199092120572

Γ (1 + 2120572)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(30)

and so on

Therefore we obtain the exact solution of (27)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) + 119864120572(119905120572) minus119905120572

Γ (1 + 120572)minus 1

(31)

and its graph is shown in Figure 3

Example 4 The Klein-Gordon equation on Cantor sets ispresented as

1199062120572

2119905= 119871120572119906 (32)

and the initial value conditions are written as

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) =

119909120572

Γ (1 + 120572)

(33)

From (12) and (27)-(28) the following iterative formulas areas follows

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

119909120572

Γ (1 + 120572)

(34)

From (29) we give

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) =

119909120572

Γ (1 + 120572)

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) =

119909120572

Γ (1 + 120572)

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) =

119909120572

Γ (1 + 120572)

(35)

and so onTherefore we give the exact solution of (32)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) (36)

and its graph is shown in Figure 4

Abstract and Applied Analysis 5

0 0204

0608

1

0

10

05

2

8

4

6

tx

u(xt)

Figure 3 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

002

0406

081

0

10

05

1

2

3

4

5

6

tx

u(xt)

Figure 4 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

4 Conclusions

In this work the Klein-Gordon equations on Cantor setswithin the local fractional differential operator had beenanalyzed using the local fractional series expansion methodThe nondifferentiable solutions for local fractional Klein-Gordon equations were obtained The present method is apowerful mathematical tool for solving the local fractionallinear differential equations

Conflict of Interests

The authors declare that they have 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] A-M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons amp Fractals vol 28 no 4 pp 1005ndash1013 2006

[2] E Yusufoglu ldquoThe variational iterationmethod for studying theKlein-Gordon equationrdquo Applied Mathematics Letters vol 21no 7 pp 669ndash674 2008

[3] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[4] S M El-Sayed ldquoThe decomposition method for studying theKlein-Gordon equationrdquo Chaos Solitons amp Fractals vol 18 no5 pp 1025ndash1030 2003

[5] A S V Ravi Kanth and K Aruna ldquoDifferential transformmethod for solving the linear and nonlinear Klein-GordonequationrdquoComputer Physics Communications vol 180 no 5 pp708ndash711 2009

[6] M S H Chowdhury and I Hashim ldquoApplication of homotopy-perturbation method to Klein-Gordon and sine-Gordon equa-tionsrdquo Chaos Solitons amp Fractals vol 39 no 4 pp 1928ndash19352009

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

[8] M Kurulay ldquoSolving the fractional nonlinear Klein-Gordonequation bymeans of the homotopy analysismethodrdquoAdvancesin Difference Equations vol 2012 no 1 article 187 pp 1ndash8 2012

[9] K A Gepreel and M S Mohamed ldquoAnalytical approximatesolution for nonlinear space-time fractional Klein-Gordonequationrdquo Chinese Physics B vol 22 no 1 Article ID 0102012013

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

[11] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[12] A K Golmankhaneh and D Baleanu ldquoOn a new measure onfractalsrdquo Journal of Inequalities and Applications vol 2013 no1 article 522 2013

[13] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoLagrangian and Hamiltonian mechanics on fractals subset ofreal-linerdquo International Journal ofTheoretical Physics vol 52 no11 pp 4210ndash4217 2013

[14] K G Alireza V Fazlollahi and D Baleanu ldquoNewtonianmechanics on fractals subset of real-linerdquo Romania Reports inPhysics vol 65 pp 84ndash93 2013

[15] A S Balankin ldquoStresses and strains in a deformable fractalmedium and in its fractal continuum modelrdquo Physics Letters Avol 377 no 38 pp 2535ndash2541 2013

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

[17] 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

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

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

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

subject to the initial value conditions

119906 (119909 0) = 0

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(18)

From (12) and (18) we can structure the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) = 0

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(19)

Hence we can calculate

1205950(119909) = 0

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 0

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 0

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(20)

and so onTherefore we have

119906 (119909 119905) =1199092120572

Γ (1 + 2120572)

infin

sum

119894=1

119905(2119894minus1)120572

Γ (1 + (2119894 minus 1) 120572)

+

infin

sum

119894=1

119905(1+2119894)120572

Γ (1 + (1 + 2119894) 120572)

(21)

and the corresponding graph is illustrated in Figure 1

Example 2 We consider the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (22)

subject to the initial value conditions

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) = 0

(23)

002

0406

081

0

10

05

05

1

15

2

25

tx

u(xt)

Figure 1 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

From (12) and (23) we get the following iterative formulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) = 0

(24)

Hence we get

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) = 0

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) = 0

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) = 0

(25)

and so onHereby we obtain the solution of (22)

119906 (119909 119905) =119909120572

Γ (1 + 120572)

infin

sum

119894=0

1199052119894120572

Γ (1 + 2119894120572) (26)

and the corresponding graph is depicted in Figure 2

Example 3 We present the following Klein-Gordon equa-tions on Cantor sets

1199062120572

2119905= 119871120572119906 (27)

4 Abstract and Applied Analysis

002

0406

081

0

10

05

05

1

15

2

3

25

tx

u(xt)

Figure 2 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

subject to the initial value conditions

119906 (119909 0) =1199092120572

Γ (1 + 2120572)

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(28)

From (12) and (27)-(28) we get the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

1199092120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(29)

From (29) we obtain

1205950(119909) =

1199092120572

Γ (1 + 2120572)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(30)

and so on

Therefore we obtain the exact solution of (27)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) + 119864120572(119905120572) minus119905120572

Γ (1 + 120572)minus 1

(31)

and its graph is shown in Figure 3

Example 4 The Klein-Gordon equation on Cantor sets ispresented as

1199062120572

2119905= 119871120572119906 (32)

and the initial value conditions are written as

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) =

119909120572

Γ (1 + 120572)

(33)

From (12) and (27)-(28) the following iterative formulas areas follows

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

119909120572

Γ (1 + 120572)

(34)

From (29) we give

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) =

119909120572

Γ (1 + 120572)

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) =

119909120572

Γ (1 + 120572)

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) =

119909120572

Γ (1 + 120572)

(35)

and so onTherefore we give the exact solution of (32)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) (36)

and its graph is shown in Figure 4

Abstract and Applied Analysis 5

0 0204

0608

1

0

10

05

2

8

4

6

tx

u(xt)

Figure 3 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

002

0406

081

0

10

05

1

2

3

4

5

6

tx

u(xt)

Figure 4 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

4 Conclusions

In this work the Klein-Gordon equations on Cantor setswithin the local fractional differential operator had beenanalyzed using the local fractional series expansion methodThe nondifferentiable solutions for local fractional Klein-Gordon equations were obtained The present method is apowerful mathematical tool for solving the local fractionallinear differential equations

Conflict of Interests

The authors declare that they have 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] A-M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons amp Fractals vol 28 no 4 pp 1005ndash1013 2006

[2] E Yusufoglu ldquoThe variational iterationmethod for studying theKlein-Gordon equationrdquo Applied Mathematics Letters vol 21no 7 pp 669ndash674 2008

[3] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[4] S M El-Sayed ldquoThe decomposition method for studying theKlein-Gordon equationrdquo Chaos Solitons amp Fractals vol 18 no5 pp 1025ndash1030 2003

[5] A S V Ravi Kanth and K Aruna ldquoDifferential transformmethod for solving the linear and nonlinear Klein-GordonequationrdquoComputer Physics Communications vol 180 no 5 pp708ndash711 2009

[6] M S H Chowdhury and I Hashim ldquoApplication of homotopy-perturbation method to Klein-Gordon and sine-Gordon equa-tionsrdquo Chaos Solitons amp Fractals vol 39 no 4 pp 1928ndash19352009

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

[8] M Kurulay ldquoSolving the fractional nonlinear Klein-Gordonequation bymeans of the homotopy analysismethodrdquoAdvancesin Difference Equations vol 2012 no 1 article 187 pp 1ndash8 2012

[9] K A Gepreel and M S Mohamed ldquoAnalytical approximatesolution for nonlinear space-time fractional Klein-Gordonequationrdquo Chinese Physics B vol 22 no 1 Article ID 0102012013

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

[11] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[12] A K Golmankhaneh and D Baleanu ldquoOn a new measure onfractalsrdquo Journal of Inequalities and Applications vol 2013 no1 article 522 2013

[13] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoLagrangian and Hamiltonian mechanics on fractals subset ofreal-linerdquo International Journal ofTheoretical Physics vol 52 no11 pp 4210ndash4217 2013

[14] K G Alireza V Fazlollahi and D Baleanu ldquoNewtonianmechanics on fractals subset of real-linerdquo Romania Reports inPhysics vol 65 pp 84ndash93 2013

[15] A S Balankin ldquoStresses and strains in a deformable fractalmedium and in its fractal continuum modelrdquo Physics Letters Avol 377 no 38 pp 2535ndash2541 2013

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

[17] 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

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

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

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

002

0406

081

0

10

05

05

1

15

2

3

25

tx

u(xt)

Figure 2 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

subject to the initial value conditions

119906 (119909 0) =1199092120572

Γ (1 + 2120572)

120597120572

120597119906120572119906 (119909 0) =

1199092120572

Γ (1 + 2120572)

(28)

From (12) and (27)-(28) we get the following iterativeformulas

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

1199092120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

(29)

From (29) we obtain

1205950(119909) =

1199092120572

Γ (1 + 2120572)

1205951(119909) =

1199092120572

Γ (1 + 2120572)

1205952(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205953(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205954(119909) = 1 +

1199092120572

Γ (1 + 2120572)

1205955(119909) = 1 +

1199092120572

Γ (1 + 2120572)

(30)

and so on

Therefore we obtain the exact solution of (27)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) + 119864120572(119905120572) minus119905120572

Γ (1 + 120572)minus 1

(31)

and its graph is shown in Figure 3

Example 4 The Klein-Gordon equation on Cantor sets ispresented as

1199062120572

2119905= 119871120572119906 (32)

and the initial value conditions are written as

119906 (119909 0) =119909120572

Γ (1 + 120572)

120597120572

120597119906120572119906 (119909 0) =

119909120572

Γ (1 + 120572)

(33)

From (12) and (27)-(28) the following iterative formulas areas follows

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205950(119909) =

119909120572

Γ (1 + 120572)

120595119894+2(119909) = (1205972120572120595119894

1205971199092120572+ 120595119894) (119909)

1205951(119909) =

119909120572

Γ (1 + 120572)

(34)

From (29) we give

1205950(119909) =

119909120572

Γ (1 + 120572)

1205951(119909) =

119909120572

Γ (1 + 120572)

1205952(119909) =

119909120572

Γ (1 + 120572)

1205953(119909) =

119909120572

Γ (1 + 120572)

1205954(119909) =

119909120572

Γ (1 + 120572)

1205955(119909) =

119909120572

Γ (1 + 120572)

(35)

and so onTherefore we give the exact solution of (32)

119906 (119909 119905) =119909120572

Γ (1 + 120572)119864120572(119905120572) (36)

and its graph is shown in Figure 4

Abstract and Applied Analysis 5

0 0204

0608

1

0

10

05

2

8

4

6

tx

u(xt)

Figure 3 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

002

0406

081

0

10

05

1

2

3

4

5

6

tx

u(xt)

Figure 4 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

4 Conclusions

In this work the Klein-Gordon equations on Cantor setswithin the local fractional differential operator had beenanalyzed using the local fractional series expansion methodThe nondifferentiable solutions for local fractional Klein-Gordon equations were obtained The present method is apowerful mathematical tool for solving the local fractionallinear differential equations

Conflict of Interests

The authors declare that they have 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] A-M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons amp Fractals vol 28 no 4 pp 1005ndash1013 2006

[2] E Yusufoglu ldquoThe variational iterationmethod for studying theKlein-Gordon equationrdquo Applied Mathematics Letters vol 21no 7 pp 669ndash674 2008

[3] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[4] S M El-Sayed ldquoThe decomposition method for studying theKlein-Gordon equationrdquo Chaos Solitons amp Fractals vol 18 no5 pp 1025ndash1030 2003

[5] A S V Ravi Kanth and K Aruna ldquoDifferential transformmethod for solving the linear and nonlinear Klein-GordonequationrdquoComputer Physics Communications vol 180 no 5 pp708ndash711 2009

[6] M S H Chowdhury and I Hashim ldquoApplication of homotopy-perturbation method to Klein-Gordon and sine-Gordon equa-tionsrdquo Chaos Solitons amp Fractals vol 39 no 4 pp 1928ndash19352009

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

[8] M Kurulay ldquoSolving the fractional nonlinear Klein-Gordonequation bymeans of the homotopy analysismethodrdquoAdvancesin Difference Equations vol 2012 no 1 article 187 pp 1ndash8 2012

[9] K A Gepreel and M S Mohamed ldquoAnalytical approximatesolution for nonlinear space-time fractional Klein-Gordonequationrdquo Chinese Physics B vol 22 no 1 Article ID 0102012013

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

[11] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[12] A K Golmankhaneh and D Baleanu ldquoOn a new measure onfractalsrdquo Journal of Inequalities and Applications vol 2013 no1 article 522 2013

[13] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoLagrangian and Hamiltonian mechanics on fractals subset ofreal-linerdquo International Journal ofTheoretical Physics vol 52 no11 pp 4210ndash4217 2013

[14] K G Alireza V Fazlollahi and D Baleanu ldquoNewtonianmechanics on fractals subset of real-linerdquo Romania Reports inPhysics vol 65 pp 84ndash93 2013

[15] A S Balankin ldquoStresses and strains in a deformable fractalmedium and in its fractal continuum modelrdquo Physics Letters Avol 377 no 38 pp 2535ndash2541 2013

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

[17] 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

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

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

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

0 0204

0608

1

0

10

05

2

8

4

6

tx

u(xt)

Figure 3 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

002

0406

081

0

10

05

1

2

3

4

5

6

tx

u(xt)

Figure 4 The plot of 119906(119909 119905) for the parameter 120572 = ln 2 ln 3

4 Conclusions

In this work the Klein-Gordon equations on Cantor setswithin the local fractional differential operator had beenanalyzed using the local fractional series expansion methodThe nondifferentiable solutions for local fractional Klein-Gordon equations were obtained The present method is apowerful mathematical tool for solving the local fractionallinear differential equations

Conflict of Interests

The authors declare that they have 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] A-M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons amp Fractals vol 28 no 4 pp 1005ndash1013 2006

[2] E Yusufoglu ldquoThe variational iterationmethod for studying theKlein-Gordon equationrdquo Applied Mathematics Letters vol 21no 7 pp 669ndash674 2008

[3] A-M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[4] S M El-Sayed ldquoThe decomposition method for studying theKlein-Gordon equationrdquo Chaos Solitons amp Fractals vol 18 no5 pp 1025ndash1030 2003

[5] A S V Ravi Kanth and K Aruna ldquoDifferential transformmethod for solving the linear and nonlinear Klein-GordonequationrdquoComputer Physics Communications vol 180 no 5 pp708ndash711 2009

[6] M S H Chowdhury and I Hashim ldquoApplication of homotopy-perturbation method to Klein-Gordon and sine-Gordon equa-tionsrdquo Chaos Solitons amp Fractals vol 39 no 4 pp 1928ndash19352009

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

[8] M Kurulay ldquoSolving the fractional nonlinear Klein-Gordonequation bymeans of the homotopy analysismethodrdquoAdvancesin Difference Equations vol 2012 no 1 article 187 pp 1ndash8 2012

[9] K A Gepreel and M S Mohamed ldquoAnalytical approximatesolution for nonlinear space-time fractional Klein-Gordonequationrdquo Chinese Physics B vol 22 no 1 Article ID 0102012013

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

[11] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[12] A K Golmankhaneh and D Baleanu ldquoOn a new measure onfractalsrdquo Journal of Inequalities and Applications vol 2013 no1 article 522 2013

[13] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoLagrangian and Hamiltonian mechanics on fractals subset ofreal-linerdquo International Journal ofTheoretical Physics vol 52 no11 pp 4210ndash4217 2013

[14] K G Alireza V Fazlollahi and D Baleanu ldquoNewtonianmechanics on fractals subset of real-linerdquo Romania Reports inPhysics vol 65 pp 84ndash93 2013

[15] A S Balankin ldquoStresses and strains in a deformable fractalmedium and in its fractal continuum modelrdquo Physics Letters Avol 377 no 38 pp 2535ndash2541 2013

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

[17] 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

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

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

[19] J-H He ldquoExp-function method for fractional differentialequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 14 no 6 pp 363ndash366 2013

[20] X-J Yang D Baleanu and J-H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[21] A-M Yang X-J Yang and Z-B Li ldquoLocal fractional seriesexpansion method for solving wave and diffusion equations onCantor setsrdquoAbstract and Applied Analysis vol 2013 Article ID351057 5 pages 2013

[22] Y Zhao D-F Cheng and X-J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

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

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