Research Article Solving Initial-Boundary Value Problems ...downloads.hindawi.com/journals/aaa/2014/912464.pdf · where the operators are described by the local fractional di erential

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Research ArticleSolving Initial-Boundary Value Problems for Local FractionalDifferential Equation by Local Fractional Fourier Series Method

Yu Zhang

College of Mathematics and Information Science North China University of Water Resources and Electric PowerZhengzhou 450000 China

Correspondence should be addressed to Yu Zhang b42old163com

Received 22 June 2014 Accepted 1 July 2014 Published 13 July 2014

Academic Editor Xiao-Jun Yang

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

The initial-boundary value problems for the local fractional differential equation are investigated in this paper The local fractionalFourier series solutions with the nondifferential terms are obtained Two illustrative examples are given to show efficiency andaccuracy of the presented method to process the local fractional differential equations

1 Introduction

In various fields of physics mathematics and engineeringbecause of the different operators there are classical differen-tial equations [1] fractional differential equation [2ndash4] andlocal fractional differential equations [5 6] There are moretechniques to achieve analytical approximations to the solu-tions to differential equations in mathematical physics suchas the decomposition method [7] the variational iterationmethod [8] the homotopy perturbationmethod [9] the heat-balance integral method [10] the Fourier transform [11] theLaplace transform [11] and the references therein

Recently a new Fourier series (local fractional Fourierseries) via local fractional operator was proposed [6] and hadvarious applications in the applied fields such as fractal waveproblems in fractal string [12 13] and the heat-conductionproblems arising in fractal heat transfer [14 15] For a detaileddescription of the local fractional Fourier series method werefer the readers to the recent works [14ndash16] This is themain advantage of local fractional differential equations incomparison with classical integer-order and fractional-ordermodels

In the present paper we consider the local fractionaldifferential equation

120597120572

119906 (119909 119910)

120597119910120572minus1205972120572

119906 (119909 119910)

1205971199092120572= 0 (1)

subject to the initial-boundary value conditions

120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) = 119892 (119909)

(2)

where the operators are described by the local fractionaldifferential operators [5 6 12ndash15] The paper is organized asfollows In Section 2 the basic theory of the local fractionalcalculus and local fractional Fourier series is presented InSection 3 we discuss the initial-boundary problems for localfractional differential equation Finally Section 4 is devotedto the conclusions

2 Analysis of the Method

In this section we present the basic theory of the localfractional calculus and analyze the local fractional Fourierseries method

Definition 1 Let 119865 be a subset of the real line and be a fractalThe mass function 120574120572[119865 119886 119887] can be written as [5]

120574120572

[119865 119886 119887] = limmax0lt119894lt119899minus1

(119909119894+1minus119909119894)rarr0

119899minus1

sum

119894=0

(119909119894+1

minus 119909119894)120572

Γ (1 + 120572) (3)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 912464 5 pageshttpdxdoiorg1011552014912464

2 Abstract and Applied Analysis

The following properties are present as follows [5]

(i) If 119865 cap (119886 119887) = then 120574120572[119865 119886 119887] = 0(ii) If 119886 lt 119887 lt 119888 and 120574

120572

[119865 119886 119887] lt 0 then 120574120572

[119865 119886 119887] +

120574120572

[119865 119887 119888] = 120574120572

[119865 119886 119888]

If 119891 (119865 119889) rarr (Ω1015840

1198891015840

) is a bi-Lipschitz mapping then wehave [5 12]

120588119904

119867119904

(119865) le 119867119904

(119891 (119865)) le 120591119904

119867119904

(119865) (4)

such that

12058812057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572

le1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572

(5)

In view of (5) we have1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572 (6)

such that1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 lt 120576120572

(7)

where 120572 is the fractal dimension of 119865 This result is directlydeduced from fractal geometry and relates to the fractalcoarse-grained mass function 120574120572[119865 119886 119887] which reads [5 13]

120574120572

[119865 119886 119887] =119867120572

(119865 cap (119886 119887))

Γ (1 + 120572)(8)

with

119867120572

(119865 cap (119886 119887)) = (119887 minus 119886)120572

(9)

where119867120572 is 120572 dimensional Hausdorff measure

Definition 2 If there is [5 6 12ndash15]1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 lt 120576120572 (10)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then 119891(119909) is called

local fractional continuous at 119909 = 1199090

If119891(119909) is local fractional continuous on the interval (119886 119887)then we can write it in the form [5 6 12]

119891 (119909) isin 119862120572(119886 119887) (11)

Definition 3 Local fractional derivative of 119891(119909) of order 120572 at119909 = 119909

0is defined as follows [5 6 12ndash15]

119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572|119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(12)

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

0))

From (12) we can rewrite the local fractional derivative as

119891(120572)

(1199090) = lim119909rarr1199090

119891 (119909) minus 119891 (1199090)

120574120572 [119865 1199090 119909]

(13)

where

120574120572

[119865 1199090 119909] =

119867120572

(119865 cap (119886 119887))

Γ (1 + 120572) (14)

Definition 4 The partition of the interval [119886 119887] is (119905119895 119905119895+1

)119895 = 0 119873 minus 1 119905

0= 119886 and 119905

119873= 119887 with Δ119905

119895= 119905119895+1

minus 119905119895and

Δ119905 = maxΔ1199051 Δ1199052 Δ119905119895 Local fractional integral of 119891(119909)

of order 120572 in the interval [119886 119887] is given by [5 6 12ndash15]

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

(15)

Following (14) we have

119886119868(120572)

119887119891 (119909) =

1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) 120574120572

[119865 119905119895 119905119895+1

] (16)

where

120574120572


(119909119894+1minus119909119894)rarr0

119899minus1

sum

119894=0

(119909119894+1

minus 119909119894)120572

Γ (1 + 120572) (17)

If 119865 are Cantor sets we can get the derivative and integral onCantor sets

Some properties of local fractional integrals are listed asfollows

0119868(120572)

119909119864120572(119909120572

) = 119864120572(119909120572

) minus 1

0119868(120572)

119909

119909119899120572

Γ (1 + 119899120572)=

119909(119899+1)120572

Γ (1 + (119899 + 1) 120572)

0119868(120572)

119909sin120572(119886120572

119909120572

) =1

119886120572[cos120572(119886120572

119909120572

) minus 1]

0119868(120572)

119909cos120572(119886120572

119909120572

) =1

119886120572sin120572(119886120572

119909120572

)

0119868(120572)

119909

119909120572

Γ (1 + 120572)sin120572(119886120572

119909120572

)

= minus1

119886120572[

119909120572

Γ (1 + 120572)cos120572(119886120572

119909120572

) minus1

119886120572sin120572(119886120572

119909120572

)]

0119868(120572)

119909

119909120572

Γ (1 + 120572)cos120572(119886120572

119909120572

)

=1

119886120572

119909120572

Γ (1 + 120572)sin120572(119886120572

119909120572

) minus1

119886120572[cos120572(119886120572

119909120572

) minus 1]

0119868(120572)

119909119864120572(119909120572

) sin120572(119886120572

119909120572

)

=119864120572(119909120572

) [sin120572(119886120572

119909120572

) minus 119886120572cos120572(119886120572

119909120572

)] + 119886120572

1 + 1198862120572

0119868(120572)

119909119864120572(119909120572

) cos120572(119886120572

119909120572

)

=119864120572(119909120572

) [cos120572(119886120572

119909120572

) + 119886120572 sin (119886120572119909120572)] minus 1

1 + 1198862120572

(18)

Abstract and Applied Analysis 3

Definition 5 Local fractional trigonometric Fourier series of119891(119905) is given by [6 12ndash16]

119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

1205960

120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

1205960

120572

119905120572

) (19)

for 119909 isin 119877 and 0 lt 120572 le 1The local fractional Fourier coefficients of (19) can be

computed by

1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

1205960

120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

1205960

120572

119905120572

)

(20)

If 1205960= 1 then we get

119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

119905120572

) (21)

where the local fractional Fourier coefficients can be com-puted by

1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

119905120572

)

(22)

3 The Initial-Boundary Problems for theLocal Fractional Differential Equation

In this section we consider (1) with the various initial-boundary conditions

Example 6 The initial-boundary condition (2) becomes

120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) = 119864

120572(119909120572

)

(23)

Let 119906 = 119883119884 in (1) Separation of the variables yields

119883119884(120572)

= 119884119883(2120572)

(24)

Setting

119884(120572)

119884=119883(2120572)

119883= minus1205822120572

(25)

we obtain

119883(2120572)

+ 1205822120572

119883 = 0

119884(120572)

+ 1205822120572

119884 = 0

(26)

Hence we have their solutions which read

119883 = 119886 cos120572(120582120572

119909120572

) + 119887 sin120572(120582120572

119909120572

)

119884 = 119888119864120572(minus1205822120572

119910120572

)

(27)

Therefore a solution is written in the form

119906 (119909 119910) = 119883119884

= 119864120572(minus1205822120572

119910120572

) (119860 cos120572(120582120572

119909120572

) + 119861 sin120572(120582120572

119909120572

))

(28)

where 119860 = 119886119888 119861 = 119887119888For the given condition

120597120572

119906 (0 119905)

120597119909120572= 0 (29)

there is 119861 = 0 so that

119906 (119909 119910) = 119860119864120572(minus1205822120572

119910120572

) cos120572(120582120572

119909120572

) (30)

For the given condition

120597120572

119906 (119871 119905)

120597119909120572= 0 (31)

we obtain

sin120572(120582120572

119909120572

) = 0 (32)

120582120572

= (119898120587

119871)

120572

119898 isin 119885+

cup 0 (33)

Thus from (33) we deduce that

119906 (119909 119910) = 119860119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

119898 isin 119885+

cup 0

(34)

To satisfy the condition (23) (34) is written in the form

119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(35)


Then we derive

1198600=Γ (1 + 120572)

119871120572(119864120572(119871120572

) minus 1)

119860119898

= (2

119871)

120572

Γ (1 + 120572) 119864120572(119909120572

)

times[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572

119906 (119909 119910)

=Γ (1 + 120572)

2119871120572(119864120572(119871120572

) minus 1)

+

infin

sum

119898=1

[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572

times 119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

times 119864120572(119909120572

) (2

119871)

120572

Γ (1 + 120572)

(36)

Example 7 Let us consider (1) with the initial-boundary valuecondition which becomes

120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) =

119909120572

Γ (1 + 120572)

(37)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(38)

where

1198600=

1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

119860119898=

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]

(39)

Hence we get

119906 (119909 119910)

=1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

+

infin

sum

119898=1

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(40)

4 Conclusions

In this work the initial-boundary value problems for the localfractional differential equation are discussed by using thelocal fractional Fourier series method Analytical solutionsfor the local fractional differential equation with the nondif-ferentiable conditions are obtained

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A M Wazwaz Partial Differential Equations Methods andApplications Balkema Rotterdam The Netherlands 2002

[2] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

[3] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[4] J Sabatier O P Agrawal and J A Tenreiro Machado Advancesin Fractional Calculus Theoretical Developments and Applica-tions in Physics and Engineering Springer New York NY USA2007

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

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

[7] G Adomian Solving Frontier Problems of PhysicsTheDecompo-sition Method Kluwer Academic Dordrecht The Netherlands1994

[8] M A Noor and S T Mohyud-Din ldquoVariational iterationmethod for solving higher-order nonlinear boundary valueproblems using Hersquos polynomialsrdquo International Journal ofNonlinear Sciences and Numerical Simulation vol 9 no 2 pp141ndash156 2008

[9] H Jafari and S Momani ldquoSolving fractional diffusion and waveequations bymodified homotopy perturbationmethodrdquoPhysicsLetters A vol 370 no 5-6 pp 388ndash396 2007

[10] J Hristov ldquoApproximate solutions to fractional subdiffusionequationsrdquoEuropean Physical Journal Special Topics vol 193 no1 pp 229ndash243 2011


[11] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific BostonMass USA 2012

[12] M Hu R P Agarwal and X J Yang ldquoLocal fractional Fourierseries with application to wave equation in fractal vibratingstringrdquo Abstract and Applied Analysis vol 2012 Article ID567401 15 pages 2012

[13] Y Yang D Baleanu and X Yang ldquoAnalysis of fractal waveequations by local fractional Fourier series methodrdquo Advancesin Mathematical Physics vol 2013 Article ID 632309 6 pages2013

[14] 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

[15] AM Yang C Cattani C Zhang et al ldquoLocal fractional Fourierseries solutions for non-homogeneous heat equations arising infractal heat flow with local fractional derivativerdquo 5 vol 2014Article ID 514639 5 pages 2014

[16] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on cantor sets a local fractionalfourier series approachrdquo Advances in Mathematical Physics vol2014 Article ID 561434 7 pages 2014

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Stochastic AnalysisInternational Journal of


The following properties are present as follows [5]

(i) If 119865 cap (119886 119887) = then 120574120572[119865 119886 119887] = 0(ii) If 119886 lt 119887 lt 119888 and 120574

120572

[119865 119886 119887] lt 0 then 120574120572

[119865 119886 119887] +

120574120572

[119865 119887 119888] = 120574120572

[119865 119886 119888]

If 119891 (119865 119889) rarr (Ω1015840

1198891015840

) is a bi-Lipschitz mapping then wehave [5 12]

120588119904

119867119904

(119865) le 119867119904

(119891 (119865)) le 120591119904

119867119904

(119865) (4)

such that

12058812057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572

le1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572

(5)

In view of (5) we have1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 le 12059112057210038161003816100381610038161199091 minus 119909

2

1003816100381610038161003816120572 (6)

such that1003816100381610038161003816119891 (1199091) minus 119891 (119909

2)1003816100381610038161003816 lt 120576120572

(7)

where 120572 is the fractal dimension of 119865 This result is directlydeduced from fractal geometry and relates to the fractalcoarse-grained mass function 120574120572[119865 119886 119887] which reads [5 13]

120574120572

[119865 119886 119887] =119867120572

(119865 cap (119886 119887))

Γ (1 + 120572)(8)

with

119867120572

(119865 cap (119886 119887)) = (119887 minus 119886)120572

(9)

where119867120572 is 120572 dimensional Hausdorff measure

Definition 2 If there is [5 6 12ndash15]1003816100381610038161003816119891 (119909) minus 119891 (119909

0)1003816100381610038161003816 lt 120576120572 (10)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then 119891(119909) is called

local fractional continuous at 119909 = 1199090

If119891(119909) is local fractional continuous on the interval (119886 119887)then we can write it in the form [5 6 12]

119891 (119909) isin 119862120572(119886 119887) (11)

Definition 3 Local fractional derivative of 119891(119909) of order 120572 at119909 = 119909

0is defined as follows [5 6 12ndash15]

119891(120572)

(1199090) =

119889120572

119891 (119909)

119889119909120572|119909=1199090

= lim119909rarr1199090

Δ120572

(119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(12)

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

0))

From (12) we can rewrite the local fractional derivative as

119891(120572)

(1199090) = lim119909rarr1199090

119891 (119909) minus 119891 (1199090)

120574120572 [119865 1199090 119909]

(13)

where

120574120572

[119865 1199090 119909] =

119867120572

(119865 cap (119886 119887))

Γ (1 + 120572) (14)

Definition 4 The partition of the interval [119886 119887] is (119905119895 119905119895+1

)119895 = 0 119873 minus 1 119905

0= 119886 and 119905

119873= 119887 with Δ119905

119895= 119905119895+1

minus 119905119895and

Δ119905 = maxΔ1199051 Δ1199052 Δ119905119895 Local fractional integral of 119891(119909)

of order 120572 in the interval [119886 119887] is given by [5 6 12ndash15]

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

(15)


119886119868(120572)

119887119891 (119909) =

1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum

119895=0

119891 (119905119895) 120574120572

[119865 119905119895 119905119895+1

] (16)

where

120574120572


(119909119894+1minus119909119894)rarr0

119899minus1

sum

119894=0

(119909119894+1

minus 119909119894)120572

Γ (1 + 120572) (17)

If 119865 are Cantor sets we can get the derivative and integral onCantor sets

Some properties of local fractional integrals are listed asfollows

0119868(120572)

119909119864120572(119909120572

) = 119864120572(119909120572

) minus 1

0119868(120572)

119909

119909119899120572

Γ (1 + 119899120572)=

119909(119899+1)120572

Γ (1 + (119899 + 1) 120572)

0119868(120572)

119909sin120572(119886120572

119909120572

) =1

119886120572[cos120572(119886120572

119909120572

) minus 1]

0119868(120572)

119909cos120572(119886120572

119909120572

) =1

119886120572sin120572(119886120572

119909120572

)

0119868(120572)

119909

119909120572

Γ (1 + 120572)sin120572(119886120572

119909120572

)

= minus1

119886120572[

119909120572

Γ (1 + 120572)cos120572(119886120572

119909120572

) minus1

119886120572sin120572(119886120572

119909120572

)]

0119868(120572)

119909

119909120572

Γ (1 + 120572)cos120572(119886120572

119909120572

)

=1

119886120572

119909120572

Γ (1 + 120572)sin120572(119886120572

119909120572

) minus1

119886120572[cos120572(119886120572

119909120572

) minus 1]

0119868(120572)

119909119864120572(119909120572

) sin120572(119886120572

119909120572

)

=119864120572(119909120572

) [sin120572(119886120572

119909120572

) minus 119886120572cos120572(119886120572

119909120572

)] + 119886120572

1 + 1198862120572

0119868(120572)

119909119864120572(119909120572

) cos120572(119886120572

119909120572

)

=119864120572(119909120572

) [cos120572(119886120572

119909120572

) + 119886120572 sin (119886120572119909120572)] minus 1

1 + 1198862120572

(18)



119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

1205960

120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

1205960

120572

119905120572

) (19)


computed by

1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

1205960

120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

1205960

120572

119905120572

)

(20)


119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

119905120572

) (21)


1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

119905120572

)

(22)




120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) = 119864

120572(119909120572

)

(23)


119883119884(120572)

= 119884119883(2120572)

(24)

Setting

119884(120572)

119884=119883(2120572)

119883= minus1205822120572

(25)

we obtain

119883(2120572)

+ 1205822120572

119883 = 0

119884(120572)

+ 1205822120572

119884 = 0

(26)


119883 = 119886 cos120572(120582120572

119909120572

) + 119887 sin120572(120582120572

119909120572

)

119884 = 119888119864120572(minus1205822120572

119910120572

)

(27)


119906 (119909 119910) = 119883119884

= 119864120572(minus1205822120572

119910120572

) (119860 cos120572(120582120572

119909120572

) + 119861 sin120572(120582120572

119909120572

))

(28)


120597120572

119906 (0 119905)

120597119909120572= 0 (29)


119906 (119909 119910) = 119860119864120572(minus1205822120572

119910120572

) cos120572(120582120572

119909120572

) (30)


120597120572

119906 (119871 119905)

120597119909120572= 0 (31)

we obtain

sin120572(120582120572

119909120572

) = 0 (32)

120582120572

= (119898120587

119871)

120572

119898 isin 119885+

cup 0 (33)


119906 (119909 119910) = 119860119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

119898 isin 119885+

cup 0

(34)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(35)


Then we derive

1198600=Γ (1 + 120572)

119871120572(119864120572(119871120572

) minus 1)

119860119898

= (2

119871)

120572

Γ (1 + 120572) 119864120572(119909120572

)

times[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572

119906 (119909 119910)

=Γ (1 + 120572)

2119871120572(119864120572(119871120572

) minus 1)

+

infin

sum

119898=1

[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

times 119864120572(119909120572

) (2

119871)

120572

Γ (1 + 120572)

(36)


120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) =

119909120572

Γ (1 + 120572)

(37)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(38)

where

1198600=

1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

119860119898=

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]

(39)

Hence we get

119906 (119909 119910)

=1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

+

infin

sum

119898=1

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(40)

4 Conclusions




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

1205960

120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

1205960

120572

119905120572

) (19)


computed by

1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

1205960

120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

1205960

120572

119905120572

)

(20)


119891 (119905) = 1198860+

infin

sum

119894=1

119886119896sin120572(119896120572

119905120572

) +

infin

sum

119894=1

119887119896cos120572(119896120572

119905120572

) (21)


1198860=

1

119879120572Γ (1 + 120572)

0119868119879119891 (119905)

119886119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) sin

120572(119896120572

119905120572

)

119887119896= (

2

119879)

120572

Γ (1 + 120572)0119868119879119891 (119905) cos

120572(119896120572

119905120572

)

(22)




120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) = 119864

120572(119909120572

)

(23)


119883119884(120572)

= 119884119883(2120572)

(24)

Setting

119884(120572)

119884=119883(2120572)

119883= minus1205822120572

(25)

we obtain

119883(2120572)

+ 1205822120572

119883 = 0

119884(120572)

+ 1205822120572

119884 = 0

(26)


119883 = 119886 cos120572(120582120572

119909120572

) + 119887 sin120572(120582120572

119909120572

)

119884 = 119888119864120572(minus1205822120572

119910120572

)

(27)


119906 (119909 119910) = 119883119884

= 119864120572(minus1205822120572

119910120572

) (119860 cos120572(120582120572

119909120572

) + 119861 sin120572(120582120572

119909120572

))

(28)


120597120572

119906 (0 119905)

120597119909120572= 0 (29)


119906 (119909 119910) = 119860119864120572(minus1205822120572

119910120572

) cos120572(120582120572

119909120572

) (30)


120597120572

119906 (119871 119905)

120597119909120572= 0 (31)

we obtain

sin120572(120582120572

119909120572

) = 0 (32)

120582120572

= (119898120587

119871)

120572

119898 isin 119885+

cup 0 (33)


119906 (119909 119910) = 119860119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

119898 isin 119885+

cup 0

(34)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(35)


Then we derive

1198600=Γ (1 + 120572)

119871120572(119864120572(119871120572

) minus 1)

119860119898

= (2

119871)

120572

Γ (1 + 120572) 119864120572(119909120572

)

times[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572

119906 (119909 119910)

=Γ (1 + 120572)

2119871120572(119864120572(119871120572

) minus 1)

+

infin

sum

119898=1

[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

times 119864120572(119909120572

) (2

119871)

120572

Γ (1 + 120572)

(36)


120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) =

119909120572

Γ (1 + 120572)

(37)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(38)

where

1198600=

1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

119860119898=

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]

(39)

Hence we get

119906 (119909 119910)

=1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

+

infin

sum

119898=1

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(40)

4 Conclusions




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then we derive

1198600=Γ (1 + 120572)

119871120572(119864120572(119871120572

) minus 1)

119860119898

= (2

119871)

120572

Γ (1 + 120572) 119864120572(119909120572

)

times[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572

119906 (119909 119910)

=Γ (1 + 120572)

2119871120572(119864120572(119871120572

) minus 1)

+

infin

sum

119898=1

[cos120572((119898120587119871)

120572

119909120572

) + (119898120587119871)120572 sin ((119898120587119871)120572119909120572)] minus 1

1 + (119898120587119871)2120572


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

times 119864120572(119909120572

) (2

119871)

120572

Γ (1 + 120572)

(36)


120597120572

119906 (0 119905)

120597119909120572= 0

120597120572

119906 (119871 119905)

120597119909120572= 0 119906 (119909 0) =

119909120572

Γ (1 + 120572)

(37)


119906 (119909 119910)

= 1198600+

infin

sum

119898=1

119860119898119864120572(minus(

119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(38)

where

1198600=

1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

119860119898=

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]

(39)

Hence we get

119906 (119909 119910)

=1

119871120572

Γ (1 + 120572)

Γ (1 + 2120572)1199092120572

+

infin

sum

119898=1

1

(2119898120587)120572

119909120572

Γ (1 + 120572)sin120572((

119898120587119909

119871)

120572

)

minus(119871

119898120587)

120572

[cos120572((

119898120587119909

119871)

120572

) minus 1]


119898120587

119871)

2120572

119910120572

) cos120572((

119898120587119909

119871)

120572

)

(40)

4 Conclusions




References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Solving Initial-Boundary Value Problems ...downloads.hindawi.com/journals/aaa/2014/912464.pdf · where the operators are described by the local fractional di erential