research article local fractional sumudu transform with application...

Research ArticleLocal Fractional Sumudu Transform withApplication to IVPs on Cantor Sets

H M Srivastava1 Alireza Khalili Golmankhaneh2

Dumitru Baleanu345 and Xiao-Jun Yang6

1 Department of Mathematics and Statistics University of Victoria Victoria BC Canada V8W 3R42Department of Physics Urmia Branch Islamic Azad University PO Box 969 Orumiyeh Iran3Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz UniversityPO Box 80204 Jeddah 21589 Saudi Arabia

4 Institute of Space Sciences Magurele 077125 Bucharest Romania5 Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University 06530 Ankara Turkey6Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Jiangsu 221008 China

Correspondence should be addressed to Dumitru Baleanu dumitrubaleanugmailcom

Received 13 February 2014 Accepted 10 May 2014 Published 26 May 2014

Academic Editor Jordan Hristov

Copyright copy 2014 H M Srivastava et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Local fractional derivatives were investigated intensively during the last few years The coupling method of Sumudu transform andlocal fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper The presented method isapplied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative The obtainedresults are given to show the advantages

1 Introduction

Fractals are sets and their topological dimension exceedsthe fractal dimensions Mathematical techniques on fractalsets are presented (see eg [1ndash4]) Nonlocal fractionalderivative has many applications in fractional dynamicalsystems having memory properties Fractional calculus hasbeen applied to the phenomena with fractal structure [5ndash12] Because of the limit of fractional calculus the fractalcalculus as a framework for themodel of anomalous diffusion[13ndash16] had been constructed The Newtonian mechanicsMaxwellrsquos equations and Hamiltonian mechanics on fractalsets [17ndash19] were generalized The alternative definitions ofcalculus on fractal sets had been suggested in [20 21] andthe systems of Navier-Stokes equations on Cantor sets hadbeen studied in [22] Maxwellrsquos equations on Cantor setswith local fractional vector calculus had been considered[23] The local fractional Fourier analysis had been adapted

to find Heisenberg uncertainty principle [24] A family oflocal fractional Fredholm andVolterra integral equations wasinvestigated in [25] Local fractional variational iteration anddecomposition methods for wave equation on Cantor setswere reported in [26]The local fractional Laplace transformswere developed in [27ndash30]

The Sumudu transforms (ST) had been considered forapplication to solve differential equations and to deal withcontrol engineering [31ndash37] The aims of this paper areto couple the Sumudu transforms and the local fractionalcalculus (LFC) and to give some illustrative examples in orderto show the advantages

The structures of the paper are as follows In Section 2the local fractional derivatives and integrals are presented InSection 3 the notions and properties of local fractionalSumudu transform are proposed In Section 4 some exam-ples for initial value problems are shown Finally the conclu-sions are given in Section 5

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 620529 7 pageshttpdxdoiorg1011552014620529

2 Abstract and Applied Analysis

2 Local Fractional Calculus and PolynomialFunctions on Cantor Sets

In this section we give the concepts of local fractional deriva-tives and integrals and polynomial functions on Cantor sets

Definition 1 (see [20 21 24ndash26]) Let the function 119891(119909) isin119862120572(119886 119887) if there are

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 0 lt 120572 le 1 (1)

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

Definition 2 (see [20 21 24]) Let 119891(119909) isin 119862120572(119886 119887) The local

fractional derivative of 119891(119909) of order 120572 in the interval [119886 119887]is defined as

119889120572119891 (1199090)

119889119909120572=Δ120572 (119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(2)

where

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

0)] (3)

The local fractional partial differential operator of order 120572(0 lt 120572 le 1) was given by [20 21]

120597120572

120597119905120572119906 (1199090 119905) =

Δ120572 (119906 (1199090 119905) minus 119906 (119909

0 1199050))

(119905 minus 1199050)120572

(4)

where

Δ120572 (119906 (1199090 119905) minus 119906 (119909

0 1199050)) cong Γ (1 + 120572) [119906 (119909

0 119905) minus 119906 (119909

0 1199050)] (5)

Definition 3 (see [20 21 24ndash26]) Let 119891(119909) isin 119862120572[119886 119887] The

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

119886119868(120572)119887

119891 (119909) =1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(6)

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

) 119895 = 0 119873 minus 1 1199050

= 119886 and 119905119873

= 119887 withΔ119905119895= 119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

Theorem 4 (local fractional Taylorrsquo theorem (see [20 21]))Suppose that 119891((119896+1)120572)(119909) isin 119862

120572(119886 119887) for 119896 = 0 1 119899 and

0 lt 120572 le 1 Then one has

119891 (119909) =119899

sum119896=0

119891(119896120572) (1199090)

Γ (1 + 119896120572)(119909 minus 119909

0)119896120572

+119891((119899+1)120572) (120585)

Γ (1 + (119899 + 1) 120572)(119909 minus 119909

0)(119899+1)120572

(7)

with 119886 lt 1199090lt 120585 lt 119909 lt 119887 forall119909 isin (119886 119887) where

119891((119896+1)120572) (119909) =

119896+1 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863119909

(120572) 119863119909

(120572) 119891 (119909) (8)

Proof (see [20 21]) Local fractional Mc-Laurinrsquos series of theMittag-Leffler functions on Cantor sets is given by [20 21]

119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572) 119909 isin 119877 0 lt 120572 le 1 (9)

and local fractional Mc-Laurinrsquos series of the Mittag-Lefflerfunctions on Cantor sets with the parameter 120577 reads asfollows

119864120572(120577120572119909120572) =

infin

sum119896=0

120577119896120572119909120572119896

Γ (1 + 119896120572) 119909 isin 119877 0 lt 120572 le 1 (10)

As generalizations of (9) and (10) we have

119891 (119909) =infin

sum119896=0

119886119896119909120572119896 (11)

where 119886119896(119896 = 0 1 2 119899) are coefficients of the generalized

polynomial function on Cantor setsMaking use of (10) we get

119864120572(119894120572119909120572) =

infin

sum119896=0

119894119896120572119909120572119896

Γ (1 + 119896120572) (12)

where 119894120572 is the imaginary unit with 119864120572(119894120572(2120587)120572) = 1

Let us consider the polynomial function on Cantor sets inthe form

119891 (119909) =infin

sum119896=0

119894120572119896119909120572119896 (13)

where |119909| lt 1Hence we have the closed form of (13) as follows

119891 (119909) =1

1 minus 119894120572119909120572 (14)

Definition 5 The local fractional Laplace transform of 119891(119909)of order 120572 is defined as [27ndash30]

119871120572119891 (119909) = 119891119871120572

119904(119904)

=1

Γ (1 + 120572)intinfin

0

119864120572(minus119904120572119909120572) 119891 (119909) (119889119909)

120572(15)

If 119865120572119891(119909) equiv 119891119865120572

120596(120596) the inverse formula of (42) is defined

as [27ndash30]

119891 (119909) = 119871minus1120572119891119871120572119904

(119904)

=1

(2120587)120572int120573+119894infin

120573minus119894infin

119864120572(119904120572119909120572) 119891119871120572

119904(119904) (119889119904)

120572(16)

where 119891(119909) is local fractional continuous 119904120572 = 120573120572 + 119894120572infin120572and Re(119904) = 120573 gt 0

Abstract and Applied Analysis 3

Theorem 6 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

119871120572119891(120572) (119909) = 119904120572119871

120572119891 (119909) minus 119891 (0) (17)

Proof See [21]

Theorem 7 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

1198711205720119868(120572)

119909119891 (119909) =

1

119904120572119871120572119891 (119909) (18)

Proof See [21]

Theorem 8 (see [21]) If 1198711205721198911(119909) = 119891119871120572

1199041(119904) and

1198711205721198912(119909) = 119891119871120572

1199042(119904) then one has

1198711205721198911(119909) lowast 119891

2(119909) = 119891119871120572

1199041(119904) 119891119871120572

1199042(119904) (19)

where

1198911(119909) lowast 119891

2(119909) =

1


0

1198911(119905) 1198912(119909 minus 119905) (119889119905)

120572 (20)

Proof See [21]

3 Local Fractional Sumudu Transform

In this section we derive the local fractional Sumudu trans-form (LFST) and some properties are discussed

If there is a new transform operator LFS120572 119891(119909) rarr 119865(119906)

namely

LFS120572119891 (119909) = LFS

120572infin

sum119896=0

119886119896119909120572119896 =

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

(21)

As typical examples we have

LFS120572119864120572(119894120572119909120572) =

infin

sum119896=0

119894120572119896119911120572119896

LFS120572

119909120572

Γ (1 + 120572) = 119911120572

(22)

As the generalized result we give the following definition

Definition 9 The local fractional Sumudu transform of 119891(119909)of order 120572 is defined as

LFS120572119891 (119909)

= 119865120572(119911) =

1

Γ (1 + 120572)

times intinfin

0

119864120572(minus119911minus120572119909120572)

119891 (119909)

119911120572(119889119909)120572 0 lt 120572 le 1

(23)

Following (23) its inverse formula is defined as

LFSminus1120572119865120572(119911) = 119891 (119909) 0 lt 120572 le 1 (24)

Theorem 10 (linearity) If LFS120572119891(119909) = 119865

120572(119911) and

LFS120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) + 119892 (119909) = 119865

120572(119911) + 119866

120572(119911) (25)

Proof As a direct result of the definition of local fractionalSumudu transform we get the following result

Theorem 11 (local fractional Laplace-Sumudu duality) If119871120572119891(119909) = 119891119871120572

119904(119904) and LFS

120572119891(119909) = 119865

120572(119911) then one has

LFS120572119891 (119909) =

1

119911120572119871120572119891(

1

119909) (26)

119871120572119891 (119909) =

LFS120572[119891 (1119904)]

119904120572 (27)

Proof Definitions of the local fractional Sumudu and Laplacetransforms directly give the results

Theorem 12 (local fractional Sumudu transform of localfractional derivative) If LFS

120572119891(119909) = 119865

120572(119911) then one has

LFS120572119889120572119891 (119909)

119889119909120572 =

119865120572(119911) minus 119891 (0)

119911120572 (28)

Proof From (17) and (26) the local fractional Sumudutransform of the local fractional derivative of 119891(119909) read as

LFS120572119867 (119909) =

119871120572119867 (1119909)

119911120572

=119871120572119891 (1119909) 119911120572 minus 119891 (0)

119911120572=119865120572(119911) minus 119891 (0)

119911120572

(29)

where

119867(119909) =119889120572119891 (119909)

119889119909120572 (30)

This completes the proof

As the direct result of (28) we have the following resultsIf LFS

120572119891(119909) = 119865

120572(119911) then we have

LFS120572119889119899120572119891 (119909)

119889119909119899120572 =

1

119911119899120572[119865120572(119911) minus119899minus1

sum119896=0

119911119896120572119891(119896120572) (0)] (31)

When 119899 = 2 from (31) we get

LFS1205721198892120572119891 (119909)

1198891199092120572 =

1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)]

(32)


120572119891(119909) = 119865

120572(119911) then one has

LFS1205720119868(120572)119909

119891 (119909) = 119911120572119865120572(119911) (33)


Table 1 Local fractional Sumudu transform of special functions

Mathematical operation in the 119905-domain Corresponding operation in the 119911-domain Remarks

119886 119886 119886 is a constant119909120572

Γ (1 + 120572)119911120572

infin

sum119896=0

119886119896119909120572119896

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

119864120572(119886119909120572)

1

1 minus 119886119911120572119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572)

sin120572(119886119909120572)

119886119911120572

1 + 11988621199112120572sin120572119909120572 =

infin

sum119896=0

(minus1)119896

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cos120572(119886119909120572)

1

1 + 11988621199112120572cos120572119909120572 =

infin

sum119896=0

(minus1)119896

1199092120572119896

Γ (1 + 2120572119896)

sinh120572(119886119909120572)

119886119911120572

1 minus 11988621199112120572sinh120572119909120572 =

infin

sum119896=0

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cosh120572(119886119909120572)

1

1 minus 11988621199112120572cosh120572119909120572 =

infin

sum119896=0

1199092120572119896

Γ (1 + 2120572119896)

Proof From (18) and (26) we have

1198711205720119868(120572)119909

119891 (119909) =1

119904120572119871120572119891 (119909) (34)

so that

LFS120572119861 (119909) =

119871120572119861 (1119909)

119911120572= 119871120572119891(

1

119909) = 119911120572119865

120572(119911)

(35)where

119861 (119909) =0119868(120572)119909

119891 (119909) (36)This completes the proof

Theorem 14 (local fractional convolution) If LFS120572119891(119909) =

119865120572(119911) and LFS

120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) lowast 119892 (119909) = 119911120572119865

120572(119911) 119866120572(119911) (37)

where

119891 (119909) lowast 119892 (119909) =1


0

119891 (119905) 119892 (119909 minus 119905) (119889119905)120572 (38)


LFS120572119891 (119909) lowast 119892 (119909) =

119871120572119891 (119909) lowast 119892 (119909)

119911120572

=119871120572119891 (1119909) 119871

120572119892 (1119909)

119911120572

= 119911120572119865120572(119911) 119866120572(119911)

(39)

where

119865120572(119911) =

119871120572119891 (1119909)

119911120572

119866120572(119911) =

119871120572119892 (1119909)

119911120572

(40)


In the following we present some of the basic formulaswhich are in Table 1

The above results are easily obtained by using localfractional Mc-Laurinrsquos series of special functions

4 Illustrative Examples

In this sectionwe give applications of the LFST to initial valueproblems

Example 1 Let us consider the following initial value prob-lems

119889120572119891 (119909)

119889119909120572= 119891 (119909) (41)

subject to the initial value condition

119891 (0) = 5 (42)

Taking the local fractional Sumudu transform gives

119865120572(119911) minus 119891 (0)

119911120572= 119865120572(119911) (43)

where

LFS120572119891 (119909) = 119865

120572(119911) (44)

Making use of (43) we obtain

119865120572(119911) =

5

1 minus 119911120572 (45)

Hence from (45) we get

119891 (119909) = 5119864120572(119909120572) (46)

and we draw its graphs as shown in Figure 1


0 02 04 06 08 1

4

6

8

10

12

14

16

18

20

22

x

f(x)

Figure 1 The plot of nondifferentiable solution of (41) with theparameter 120572 = ln 2 ln 3

Example 2 We consider the following initial value problems

119889120572119891 (119909)

119889119909120572+ 119891 (119909) =

119909120572

Γ (1 + 120572)(47)

and the initial boundary value reads as

119891 (0) = minus1 (48)

Taking the local fractional Sumudu transform from (47) and(48) we have

119865120572(119911) minus 119891 (0)

119911120572+ 119865120572(119911) = 119911120572 (49)

so that

119865120572(119911) = 119911120572 minus 1 (50)

Therefore the nondifferentiable solution of (47) is

119891 (119909) =119909120572

Γ (1 + 120572)minus 1 (51)


Example 3 We give the following initial value problems

1198892120572119891 (119909)

1198891199092120572= 119891 (119909) (52)

together with the initial value conditions

119891(120572) (0) = 0

119891 (0) = 2(53)

Taking the local fractional Sumudu transform from (52) weobtain

1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)] = 119865

120572(119911) (54)

0 02 04 06 08 1

minus1

minus08

minus04

minus02

0

02

04

x

f(x)

minus06


0 02 04 06 08 1

2

25

3

35

4

45

5

x

f(x)


which leads to

119865120572(119911) =

119891 (0) + 119911120572119891(120572) (0)

1 minus 1199112120572=

2

1 minus 1199112120572 (55)

Therefore form (55) we give the nondifferentiable solutionof (52)

119891 (119909) = 2cosh120572(119909120572) (56)


5 Conclusions

In this work we proposed the local fractional Sumudutransformbased on the local fractional calculus and its resultswere discussed Applications to initial value problems werepresented and the nondifferentiable solutions are obtained It


is shown that it is an alternative method of local fractionalLaplace transform to solve a class of local fractional differen-tiable equations

Conflict of Interests

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

References

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[2] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons New York NY USA 1990

[3] K FalconerTechniques in Fractal Geometry JohnWileyamp SonsChichester UK 1997

[4] G A Edgar Integral Probability and Fractal MeasuresSpringer New York NY USA 1998

[5] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[6] K S Miller and B Ross An Introduction to the FractionalIntegrals and Derivatives Theory and Applications John Wileyand Sons New York NY USA 1993

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[8] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[9] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[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 Hacken-sack NJ USA 2012

[12] V V Uchaikin Fractional Derivatives for Physicists and Engi-neers Background and Theory Nonlinear Physical ScienceSpringer Berlin Germany 2013

[13] B J West M Bologna and P Grigolini Physics of FractalOperators Institute for Nonlinear Science Springer New YorkNY USA 2003

[14] W Chen H Sun X Zhang and D Korosak ldquoAnomalous diffu-sion modeling by fractal and fractional derivativesrdquo Computersamp Mathematics with Applications vol 59 no 5 pp 1754ndash17582010

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

[16] A Parvate and A D Gangal ldquoCalculus on fractal subsets of realline I Formulationrdquo Fractals Complex Geometry Patterns andScaling in Nature and Society vol 17 no 1 pp 53ndash81 2009

[17] A Parvate S Satin and A D Gangal ldquoCalculus on fractalcurves inR119899rdquo Fractals Complex Geometry Patterns and Scalingin Nature and Society vol 19 no 1 pp 15ndash27 2011

[18] A K Golmankhaneh A K Golmankhaneh and D BaleanuldquoAbout Maxwellrsquos equations on fractal subsets of R3rdquo CentralEuropean Journal of Physics vol 11 no 6 pp 863ndash867 2013

[19] A K Golmankhaneh V Fazlollahi and D Baleanu ldquoNew-tonian mechanics on fractals subset of real-linerdquo RomanianReports in Physics vol 65 no 1 pp 84ndash93 2013

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

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

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

[23] Y Zhao D Baleanu C Cattani D-F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor sets a local fractionalapproachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[24] X-J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of the Heisenberg uncertainty principle within localfractional Fourier analysisrdquoBoundary Value Problems vol 2013131 2013

[25] X-J Ma H M Srivastava D Baleanu and X-J Yang ldquoA newNeumann series method for solving a family of local fractionalFredholm and Volterra integral equationsrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 325121 6 pages 2013

[26] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X-J Yang ldquoLocal fractional variational iteration anddecomposition methods for wave equation on Cantor setswithin local fractional operatorsrdquoAbstract andAppliedAnalysisvol 2014 Article ID 535048 6 pages 2014

[27] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

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

[29] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractionalderivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014

[30] 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

[31] G K Watugala ldquoSumudu transform a new integral trans-form to solve differential equations and control engineeringproblemsrdquo International Journal of Mathematical Education inScience and Technology vol 24 no 1 pp 35ndash43 1993

[32] S Weerakoon ldquoApplication of Sumudu transform to partialdifferential equationsrdquo International Journal of MathematicalEducation in Science and Technology vol 25 no 2 pp 277ndash2831994

[33] M A B Deakin ldquoThe Sumudu transform and the Laplacetransformrdquo International Journal of Mathematical Education inScience and Technology vol 28 no 1 pp 159ndash160 1997

[34] M A Asiru ldquoSumudu transform and the solution of integralequations of convolution typerdquo International Journal of Mathe-matical Education in Science and Technology vol 32 no 6 pp906ndash910 2001


[35] M A Asiru ldquoFurther properties of the Sumudu transformand its applicationsrdquo International Journal of MathematicalEducation in Science and Technology vol 33 no 3 pp 441ndash4492002

[36] H Eltayeb and A Kılıcman ldquoA note on the Sumudu transformsand differential equationsrdquo Applied Mathematical Sciences vol4 no 21ndash24 pp 1089ndash1098 2010

[37] F B M Belgacem A A Karaballi and S L Kalla ldquoAnalyt-ical investigations of the Sumudu transform and applicationsto integral production equationsrdquo Mathematical Problems inEngineering no 3-4 pp 103ndash118 2003

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Local Fractional Calculus and PolynomialFunctions on Cantor Sets

In this section we give the concepts of local fractional deriva-tives and integrals and polynomial functions on Cantor sets

Definition 1 (see [20 21 24ndash26]) Let the function 119891(119909) isin119862120572(119886 119887) if there are

1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572 0 lt 120572 le 1 (1)

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

Definition 2 (see [20 21 24]) Let 119891(119909) isin 119862120572(119886 119887) The local

fractional derivative of 119891(119909) of order 120572 in the interval [119886 119887]is defined as

119889120572119891 (1199090)

119889119909120572=Δ120572 (119891 (119909) minus 119891 (119909

0))

(119909 minus 1199090)120572

(2)

where

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

0)] (3)

The local fractional partial differential operator of order 120572(0 lt 120572 le 1) was given by [20 21]

120597120572

120597119905120572119906 (1199090 119905) =

Δ120572 (119906 (1199090 119905) minus 119906 (119909

0 1199050))

(119905 minus 1199050)120572

(4)

where

Δ120572 (119906 (1199090 119905) minus 119906 (119909

0 1199050)) cong Γ (1 + 120572) [119906 (119909

0 119905) minus 119906 (119909

0 1199050)] (5)

Definition 3 (see [20 21 24ndash26]) Let 119891(119909) isin 119862120572[119886 119887] The

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

119886119868(120572)119887

119891 (119909) =1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(6)

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

) 119895 = 0 119873 minus 1 1199050

= 119886 and 119905119873

= 119887 withΔ119905119895= 119905119895+1

minus 119905119895and Δ119905 = maxΔ119905

0 Δ1199051 Δ119905119895

Theorem 4 (local fractional Taylorrsquo theorem (see [20 21]))Suppose that 119891((119896+1)120572)(119909) isin 119862

120572(119886 119887) for 119896 = 0 1 119899 and

0 lt 120572 le 1 Then one has

119891 (119909) =119899

sum119896=0

119891(119896120572) (1199090)

Γ (1 + 119896120572)(119909 minus 119909

0)119896120572

+119891((119899+1)120572) (120585)

Γ (1 + (119899 + 1) 120572)(119909 minus 119909

0)(119899+1)120572

(7)

with 119886 lt 1199090lt 120585 lt 119909 lt 119887 forall119909 isin (119886 119887) where

119891((119896+1)120572) (119909) =

119896+1 times⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119863119909

(120572) 119863119909

(120572) 119891 (119909) (8)

Proof (see [20 21]) Local fractional Mc-Laurinrsquos series of theMittag-Leffler functions on Cantor sets is given by [20 21]

119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572) 119909 isin 119877 0 lt 120572 le 1 (9)

and local fractional Mc-Laurinrsquos series of the Mittag-Lefflerfunctions on Cantor sets with the parameter 120577 reads asfollows

119864120572(120577120572119909120572) =

infin

sum119896=0

120577119896120572119909120572119896

Γ (1 + 119896120572) 119909 isin 119877 0 lt 120572 le 1 (10)

As generalizations of (9) and (10) we have

119891 (119909) =infin

sum119896=0

119886119896119909120572119896 (11)

where 119886119896(119896 = 0 1 2 119899) are coefficients of the generalized

polynomial function on Cantor setsMaking use of (10) we get

119864120572(119894120572119909120572) =

infin

sum119896=0

119894119896120572119909120572119896

Γ (1 + 119896120572) (12)

where 119894120572 is the imaginary unit with 119864120572(119894120572(2120587)120572) = 1

Let us consider the polynomial function on Cantor sets inthe form

119891 (119909) =infin

sum119896=0

119894120572119896119909120572119896 (13)

where |119909| lt 1Hence we have the closed form of (13) as follows

119891 (119909) =1

1 minus 119894120572119909120572 (14)

Definition 5 The local fractional Laplace transform of 119891(119909)of order 120572 is defined as [27ndash30]

119871120572119891 (119909) = 119891119871120572

119904(119904)

=1


0

119864120572(minus119904120572119909120572) 119891 (119909) (119889119909)

120572(15)

If 119865120572119891(119909) equiv 119891119865120572

120596(120596) the inverse formula of (42) is defined

as [27ndash30]

119891 (119909) = 119871minus1120572119891119871120572119904

(119904)

=1

(2120587)120572int120573+119894infin

120573minus119894infin

119864120572(119904120572119909120572) 119891119871120572

119904(119904) (119889119904)

120572(16)

where 119891(119909) is local fractional continuous 119904120572 = 120573120572 + 119894120572infin120572and Re(119904) = 120573 gt 0


Theorem 6 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

119871120572119891(120572) (119909) = 119904120572119871

120572119891 (119909) minus 119891 (0) (17)

Proof See [21]

Theorem 7 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

1198711205720119868(120572)

119909119891 (119909) =

1

119904120572119871120572119891 (119909) (18)

Proof See [21]

Theorem 8 (see [21]) If 1198711205721198911(119909) = 119891119871120572

1199041(119904) and

1198711205721198912(119909) = 119891119871120572

1199042(119904) then one has

1198711205721198911(119909) lowast 119891

2(119909) = 119891119871120572

1199041(119904) 119891119871120572

1199042(119904) (19)

where

1198911(119909) lowast 119891

2(119909) =

1


0

1198911(119905) 1198912(119909 minus 119905) (119889119905)

120572 (20)

Proof See [21]




namely

LFS120572119891 (119909) = LFS

120572infin

sum119896=0

119886119896119909120572119896 =

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

(21)


LFS120572119864120572(119894120572119909120572) =

infin

sum119896=0

119894120572119896119911120572119896

LFS120572

119909120572

Γ (1 + 120572) = 119911120572

(22)



LFS120572119891 (119909)

= 119865120572(119911) =

1

Γ (1 + 120572)

times intinfin

0

119864120572(minus119911minus120572119909120572)

119891 (119909)

119911120572(119889119909)120572 0 lt 120572 le 1

(23)


LFSminus1120572119865120572(119911) = 119891 (119909) 0 lt 120572 le 1 (24)


120572(119911) and

LFS120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) + 119892 (119909) = 119865

120572(119911) + 119866

120572(119911) (25)



119904(119904) and LFS

120572119891(119909) = 119865

120572(119911) then one has

LFS120572119891 (119909) =

1

119911120572119871120572119891(

1

119909) (26)

119871120572119891 (119909) =

LFS120572[119891 (1119904)]

119904120572 (27)



120572119891(119909) = 119865

120572(119911) then one has

LFS120572119889120572119891 (119909)

119889119909120572 =

119865120572(119911) minus 119891 (0)

119911120572 (28)


LFS120572119867 (119909) =

119871120572119867 (1119909)

119911120572

=119871120572119891 (1119909) 119911120572 minus 119891 (0)

119911120572=119865120572(119911) minus 119891 (0)

119911120572

(29)

where

119867(119909) =119889120572119891 (119909)

119889119909120572 (30)



120572119891(119909) = 119865

120572(119911) then we have

LFS120572119889119899120572119891 (119909)

119889119909119899120572 =

1

119911119899120572[119865120572(119911) minus119899minus1

sum119896=0

119911119896120572119891(119896120572) (0)] (31)


LFS1205721198892120572119891 (119909)

1198891199092120572 =

1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)]

(32)


120572119891(119909) = 119865

120572(119911) then one has

LFS1205720119868(120572)119909

119891 (119909) = 119911120572119865120572(119911) (33)




119886 119886 119886 is a constant119909120572

Γ (1 + 120572)119911120572

infin

sum119896=0

119886119896119909120572119896

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

119864120572(119886119909120572)

1

1 minus 119886119911120572119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572)

sin120572(119886119909120572)

119886119911120572

1 + 11988621199112120572sin120572119909120572 =

infin

sum119896=0

(minus1)119896

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cos120572(119886119909120572)

1

1 + 11988621199112120572cos120572119909120572 =

infin

sum119896=0

(minus1)119896

1199092120572119896

Γ (1 + 2120572119896)

sinh120572(119886119909120572)

119886119911120572

1 minus 11988621199112120572sinh120572119909120572 =

infin

sum119896=0

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cosh120572(119886119909120572)

1

1 minus 11988621199112120572cosh120572119909120572 =

infin

sum119896=0

1199092120572119896

Γ (1 + 2120572119896)


1198711205720119868(120572)119909

119891 (119909) =1

119904120572119871120572119891 (119909) (34)

so that

LFS120572119861 (119909) =

119871120572119861 (1119909)

119911120572= 119871120572119891(

1

119909) = 119911120572119865

120572(119911)

(35)where

119861 (119909) =0119868(120572)119909



119865120572(119911) and LFS

120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) lowast 119892 (119909) = 119911120572119865

120572(119911) 119866120572(119911) (37)

where

119891 (119909) lowast 119892 (119909) =1


0

119891 (119905) 119892 (119909 minus 119905) (119889119905)120572 (38)


LFS120572119891 (119909) lowast 119892 (119909) =

119871120572119891 (119909) lowast 119892 (119909)

119911120572

=119871120572119891 (1119909) 119871

120572119892 (1119909)

119911120572

= 119911120572119865120572(119911) 119866120572(119911)

(39)

where

119865120572(119911) =

119871120572119891 (1119909)

119911120572

119866120572(119911) =

119871120572119892 (1119909)

119911120572

(40)







119889120572119891 (119909)

119889119909120572= 119891 (119909) (41)


119891 (0) = 5 (42)


119865120572(119911) minus 119891 (0)

119911120572= 119865120572(119911) (43)

where

LFS120572119891 (119909) = 119865

120572(119911) (44)


119865120572(119911) =

5

1 minus 119911120572 (45)


119891 (119909) = 5119864120572(119909120572) (46)



0 02 04 06 08 1

4

6

8

10

12

14

16

18

20

22

x

f(x)



119889120572119891 (119909)

119889119909120572+ 119891 (119909) =

119909120572

Γ (1 + 120572)(47)


119891 (0) = minus1 (48)


119865120572(119911) minus 119891 (0)

119911120572+ 119865120572(119911) = 119911120572 (49)

so that

119865120572(119911) = 119911120572 minus 1 (50)


119891 (119909) =119909120572

Γ (1 + 120572)minus 1 (51)



1198892120572119891 (119909)

1198891199092120572= 119891 (119909) (52)


119891(120572) (0) = 0

119891 (0) = 2(53)


1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)] = 119865

120572(119911) (54)

0 02 04 06 08 1

minus1

minus08

minus04

minus02

0

02

04

x

f(x)

minus06


0 02 04 06 08 1

2

25

3

35

4

45

5

x

f(x)


which leads to

119865120572(119911) =

119891 (0) + 119911120572119891(120572) (0)

1 minus 1199112120572=

2

1 minus 1199112120572 (55)


119891 (119909) = 2cosh120572(119909120572) (56)


5 Conclusions






References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Theorem 6 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

119871120572119891(120572) (119909) = 119904120572119871

120572119891 (119909) minus 119891 (0) (17)

Proof See [21]

Theorem 7 (see [21]) If 119871120572119891(119909) = 119891119871120572

119904(119904) then one has

1198711205720119868(120572)

119909119891 (119909) =

1

119904120572119871120572119891 (119909) (18)

Proof See [21]

Theorem 8 (see [21]) If 1198711205721198911(119909) = 119891119871120572

1199041(119904) and

1198711205721198912(119909) = 119891119871120572

1199042(119904) then one has

1198711205721198911(119909) lowast 119891

2(119909) = 119891119871120572

1199041(119904) 119891119871120572

1199042(119904) (19)

where

1198911(119909) lowast 119891

2(119909) =

1


0

1198911(119905) 1198912(119909 minus 119905) (119889119905)

120572 (20)

Proof See [21]




namely

LFS120572119891 (119909) = LFS

120572infin

sum119896=0

119886119896119909120572119896 =

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

(21)


LFS120572119864120572(119894120572119909120572) =

infin

sum119896=0

119894120572119896119911120572119896

LFS120572

119909120572

Γ (1 + 120572) = 119911120572

(22)



LFS120572119891 (119909)

= 119865120572(119911) =

1

Γ (1 + 120572)

times intinfin

0

119864120572(minus119911minus120572119909120572)

119891 (119909)

119911120572(119889119909)120572 0 lt 120572 le 1

(23)


LFSminus1120572119865120572(119911) = 119891 (119909) 0 lt 120572 le 1 (24)


120572(119911) and

LFS120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) + 119892 (119909) = 119865

120572(119911) + 119866

120572(119911) (25)



119904(119904) and LFS

120572119891(119909) = 119865

120572(119911) then one has

LFS120572119891 (119909) =

1

119911120572119871120572119891(

1

119909) (26)

119871120572119891 (119909) =

LFS120572[119891 (1119904)]

119904120572 (27)



120572119891(119909) = 119865

120572(119911) then one has

LFS120572119889120572119891 (119909)

119889119909120572 =

119865120572(119911) minus 119891 (0)

119911120572 (28)


LFS120572119867 (119909) =

119871120572119867 (1119909)

119911120572

=119871120572119891 (1119909) 119911120572 minus 119891 (0)

119911120572=119865120572(119911) minus 119891 (0)

119911120572

(29)

where

119867(119909) =119889120572119891 (119909)

119889119909120572 (30)



120572119891(119909) = 119865

120572(119911) then we have

LFS120572119889119899120572119891 (119909)

119889119909119899120572 =

1

119911119899120572[119865120572(119911) minus119899minus1

sum119896=0

119911119896120572119891(119896120572) (0)] (31)


LFS1205721198892120572119891 (119909)

1198891199092120572 =

1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)]

(32)


120572119891(119909) = 119865

120572(119911) then one has

LFS1205720119868(120572)119909

119891 (119909) = 119911120572119865120572(119911) (33)




119886 119886 119886 is a constant119909120572

Γ (1 + 120572)119911120572

infin

sum119896=0

119886119896119909120572119896

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

119864120572(119886119909120572)

1

1 minus 119886119911120572119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572)

sin120572(119886119909120572)

119886119911120572

1 + 11988621199112120572sin120572119909120572 =

infin

sum119896=0

(minus1)119896

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cos120572(119886119909120572)

1

1 + 11988621199112120572cos120572119909120572 =

infin

sum119896=0

(minus1)119896

1199092120572119896

Γ (1 + 2120572119896)

sinh120572(119886119909120572)

119886119911120572

1 minus 11988621199112120572sinh120572119909120572 =

infin

sum119896=0

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cosh120572(119886119909120572)

1

1 minus 11988621199112120572cosh120572119909120572 =

infin

sum119896=0

1199092120572119896

Γ (1 + 2120572119896)


1198711205720119868(120572)119909

119891 (119909) =1

119904120572119871120572119891 (119909) (34)

so that

LFS120572119861 (119909) =

119871120572119861 (1119909)

119911120572= 119871120572119891(

1

119909) = 119911120572119865

120572(119911)

(35)where

119861 (119909) =0119868(120572)119909



119865120572(119911) and LFS

120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) lowast 119892 (119909) = 119911120572119865

120572(119911) 119866120572(119911) (37)

where

119891 (119909) lowast 119892 (119909) =1


0

119891 (119905) 119892 (119909 minus 119905) (119889119905)120572 (38)


LFS120572119891 (119909) lowast 119892 (119909) =

119871120572119891 (119909) lowast 119892 (119909)

119911120572

=119871120572119891 (1119909) 119871

120572119892 (1119909)

119911120572

= 119911120572119865120572(119911) 119866120572(119911)

(39)

where

119865120572(119911) =

119871120572119891 (1119909)

119911120572

119866120572(119911) =

119871120572119892 (1119909)

119911120572

(40)







119889120572119891 (119909)

119889119909120572= 119891 (119909) (41)


119891 (0) = 5 (42)


119865120572(119911) minus 119891 (0)

119911120572= 119865120572(119911) (43)

where

LFS120572119891 (119909) = 119865

120572(119911) (44)


119865120572(119911) =

5

1 minus 119911120572 (45)


119891 (119909) = 5119864120572(119909120572) (46)



0 02 04 06 08 1

4

6

8

10

12

14

16

18

20

22

x

f(x)



119889120572119891 (119909)

119889119909120572+ 119891 (119909) =

119909120572

Γ (1 + 120572)(47)


119891 (0) = minus1 (48)


119865120572(119911) minus 119891 (0)

119911120572+ 119865120572(119911) = 119911120572 (49)

so that

119865120572(119911) = 119911120572 minus 1 (50)


119891 (119909) =119909120572

Γ (1 + 120572)minus 1 (51)



1198892120572119891 (119909)

1198891199092120572= 119891 (119909) (52)


119891(120572) (0) = 0

119891 (0) = 2(53)


1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)] = 119865

120572(119911) (54)

0 02 04 06 08 1

minus1

minus08

minus04

minus02

0

02

04

x

f(x)

minus06


0 02 04 06 08 1

2

25

3

35

4

45

5

x

f(x)


which leads to

119865120572(119911) =

119891 (0) + 119911120572119891(120572) (0)

1 minus 1199112120572=

2

1 minus 1199112120572 (55)


119891 (119909) = 2cosh120572(119909120572) (56)


5 Conclusions






References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119886 119886 119886 is a constant119909120572

Γ (1 + 120572)119911120572

infin

sum119896=0

119886119896119909120572119896

infin

sum119896=0

Γ (1 + 119896120572) 119886119896119911120572119896

119864120572(119886119909120572)

1

1 minus 119886119911120572119864120572(119909120572) =

infin

sum119896=0

119909120572119896

Γ (1 + 119896120572)

sin120572(119886119909120572)

119886119911120572

1 + 11988621199112120572sin120572119909120572 =

infin

sum119896=0

(minus1)119896

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cos120572(119886119909120572)

1

1 + 11988621199112120572cos120572119909120572 =

infin

sum119896=0

(minus1)119896

1199092120572119896

Γ (1 + 2120572119896)

sinh120572(119886119909120572)

119886119911120572

1 minus 11988621199112120572sinh120572119909120572 =

infin

sum119896=0

119909120572(2119896+1)

Γ [1 + 120572 (2119896 + 1)]

cosh120572(119886119909120572)

1

1 minus 11988621199112120572cosh120572119909120572 =

infin

sum119896=0

1199092120572119896

Γ (1 + 2120572119896)


1198711205720119868(120572)119909

119891 (119909) =1

119904120572119871120572119891 (119909) (34)

so that

LFS120572119861 (119909) =

119871120572119861 (1119909)

119911120572= 119871120572119891(

1

119909) = 119911120572119865

120572(119911)

(35)where

119861 (119909) =0119868(120572)119909



119865120572(119911) and LFS

120572119892(119909) = 119866

120572(119911) then one has

LFS120572119891 (119909) lowast 119892 (119909) = 119911120572119865

120572(119911) 119866120572(119911) (37)

where

119891 (119909) lowast 119892 (119909) =1


0

119891 (119905) 119892 (119909 minus 119905) (119889119905)120572 (38)


LFS120572119891 (119909) lowast 119892 (119909) =

119871120572119891 (119909) lowast 119892 (119909)

119911120572

=119871120572119891 (1119909) 119871

120572119892 (1119909)

119911120572

= 119911120572119865120572(119911) 119866120572(119911)

(39)

where

119865120572(119911) =

119871120572119891 (1119909)

119911120572

119866120572(119911) =

119871120572119892 (1119909)

119911120572

(40)







119889120572119891 (119909)

119889119909120572= 119891 (119909) (41)


119891 (0) = 5 (42)


119865120572(119911) minus 119891 (0)

119911120572= 119865120572(119911) (43)

where

LFS120572119891 (119909) = 119865

120572(119911) (44)


119865120572(119911) =

5

1 minus 119911120572 (45)


119891 (119909) = 5119864120572(119909120572) (46)



0 02 04 06 08 1

4

6

8

10

12

14

16

18

20

22

x

f(x)



119889120572119891 (119909)

119889119909120572+ 119891 (119909) =

119909120572

Γ (1 + 120572)(47)


119891 (0) = minus1 (48)


119865120572(119911) minus 119891 (0)

119911120572+ 119865120572(119911) = 119911120572 (49)

so that

119865120572(119911) = 119911120572 minus 1 (50)


119891 (119909) =119909120572

Γ (1 + 120572)minus 1 (51)



1198892120572119891 (119909)

1198891199092120572= 119891 (119909) (52)


119891(120572) (0) = 0

119891 (0) = 2(53)


1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)] = 119865

120572(119911) (54)

0 02 04 06 08 1

minus1

minus08

minus04

minus02

0

02

04

x

f(x)

minus06


0 02 04 06 08 1

2

25

3

35

4

45

5

x

f(x)


which leads to

119865120572(119911) =

119891 (0) + 119911120572119891(120572) (0)

1 minus 1199112120572=

2

1 minus 1199112120572 (55)


119891 (119909) = 2cosh120572(119909120572) (56)


5 Conclusions






References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

4

6

8

10

12

14

16

18

20

22

x

f(x)



119889120572119891 (119909)

119889119909120572+ 119891 (119909) =

119909120572

Γ (1 + 120572)(47)


119891 (0) = minus1 (48)


119865120572(119911) minus 119891 (0)

119911120572+ 119865120572(119911) = 119911120572 (49)

so that

119865120572(119911) = 119911120572 minus 1 (50)


119891 (119909) =119909120572

Γ (1 + 120572)minus 1 (51)



1198892120572119891 (119909)

1198891199092120572= 119891 (119909) (52)


119891(120572) (0) = 0

119891 (0) = 2(53)


1

1199112120572[119865120572(119911) minus 119891 (0) minus 119911120572119891(120572) (0)] = 119865

120572(119911) (54)

0 02 04 06 08 1

minus1

minus08

minus04

minus02

0

02

04

x

f(x)

minus06


0 02 04 06 08 1

2

25

3

35

4

45

5

x

f(x)


which leads to

119865120572(119911) =

119891 (0) + 119911120572119891(120572) (0)

1 minus 1199112120572=

2

1 minus 1199112120572 (55)


119891 (119909) = 2cosh120572(119909120572) (56)


5 Conclusions






References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article local fractional sumudu transform with application...

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