Review Article A Review of Definitions for Fractional

Review ArticleA Review of Definitions for Fractional Derivatives and Integral

Edmundo Capelas de Oliveira1 and Joseacute Antoacutenio Tenreiro Machado2

1 Department of Applied Mathematics IMECC-UNICAMP 13083-859 Campinas SP Brazil2 Institute of Engineering Polytechnic of Porto Department of Electrical Engineering Rua Dr Antonio Bernardino de Almeida 4314200-072 Porto Portugal

Correspondence should be addressed to Jose Antonio Tenreiro Machado jtenreiromachadogmailcom

Received 30 January 2014 Accepted 18 May 2014 Published 10 June 2014

Academic Editor Riccardo Caponetto

Copyright copy 2014 E C de Oliveira and J A Tenreiro Machado This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics physics andengineering

1 Introduction

In 1695 lrsquoHopital sent a letter to Leibniz In his messagean important question about the order of the derivativeemerged What might be a derivative of order 12 In aprophetic answer Leibniz foresees the beginning of the areathat nowadays is named fractional calculus (FC) In fact FCis as old as the traditional calculus proposed independentlyby Newton and Leibniz [1ndash4]

In the classical calculus the derivative has an importantgeometric interpretation namely it is associated with theconcept of tangent in opposition to what occurs in the caseof FC This difference can be seen as a problem for the slowprogress of FCup to 1900After Leibniz it was Euler (1738) [3]that noticed the problem for a derivative of noninteger orderFourier (1822) [3 5] suggested an integral representationin order to define the derivative and his version can beconsidered the first definition for the derivative of arbitrary(positive) order Abel (1826) [3 5] solved an integral equationassociatedwith the tautochrone problemwhich is consideredto be the first application of FC Liouville (1832) [3 5]suggested a definition based on the formula for differentiatingthe exponential function This expression is known as thefirst Liouville definition The second definition formulatedby Liouville is presented in terms of an integral and isnow called the version by Liouville for the integration ofnoninteger orderAfter a series ofworks by Liouville themostimportant paper was published by Riemann [6] ten years

after his deathWe also note that both Liouville and Riemannformulations carry with them the so-called complementaryfunction a problem to be solved Grunwald [7] and Letnikov[8] independently developed an approach to nonintegerorder derivatives in terms of a convenient convergent seriesconversely to the Riemann-Liouville approach that is givenby an integral Letnikov showed that his definition coincideswith the versions formulated by Liouville for particularvalues of the order and by Riemann under a convenientinterpretation of the so-called noninteger order differenceHadamard (1892) [5] published a paper where the nonintegerorder derivative of an analytical function must be done interms of its Taylor series

After 1900 the FC experiences a fast development andin an attempt to formulate particular problems other def-initions were proposed We mention some of them Weyl[9] introduced a derivative in order to circumvent a prob-lem involving a particular class of functions the periodicfunctions Riesz [10 11] proved the mean value theorem forfractional integrals and introduced another formulation thatis associated with the Fourier transform Marchaud (1927)[3 5] introduced a new definition for noninteger order ofderivatives This definition coincides with the Liouville ver-sion for ldquosufficiently goodrdquo functions Erdelyi-Kober (1940)[3 5] presented a distinct definition for noninteger order ofintegration that is useful in applications involving integraland differential equations Caputo (1967) [12] formulateda definition more restrictive than the Riemann-Liouville

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 238459 6 pageshttpdxdoiorg1011552014238459

2 Mathematical Problems in Engineering

but more appropriate to discuss problems involving a frac-tional differential equation with initial conditions [13ndash21]

Due to the importance of the Caputo version we willcompare this approach with the Riemann-Liouville formula-tion The definition as proposed by Caputo inverts the orderof integral and derivative operators with the noninteger orderderivative of the Riemann-Liouville We summarize the dif-ference between these two formulations In the Caputo firstthe calculate derivative of integer order and after calculate theintegral of noninteger order In the Riemann-Liouville firstcalculate the integral of noninteger order and after calculatethe derivative of integer order It is important to cite that theCaputo derivative is useful to affront problems where initialconditions are done in the function and in the respectivederivatives of integer order

After the first congress at the University of New Havenin 1974 FC has developed and several applications emergedin many areas of scientific knowledge As a consequencedistinct approaches to solve problems involving the derivativewere proposed and distinct definitions of the fractionalderivative are available in the literature This paper presentsin a systematic form the existing formulations of fractionalderivatives and integralsWe shouldmention also that we canhave several alternative expressions for the same definitionTherefore we present only those more representative andwe cite particular papers [22ndash32] and books [33ndash40] that webelieve are the most relevant Furthermore the paper doesnot focus on the pros and cons of each definition and does notaddress the support of the function that is to be differentiatedor integrated

The paper is organized as follows Section 2 presents theadopted notation Sections 3 and 4 list the proposed defi-nitions of fractional derivatives and integrals respectivelyFinally Section 5 outlines some brief remarks

2 Notation

The following remarks clarify the notation used in the sequelin Sections 3 and 4

(i) Let 120572 isin C R(120572) isin (119899 minus 1 119899] 119899 isin N where R(sdot)denotes the real part of complex number

(ii) Let [119886 119887] be a finite interval in R 119896 isin N ] gt 0 andf(0) equiv 119891(0+) minus 119891(0minus)

(iii) The floor function denoted by lfloorsdotrfloor is defined as lfloor119909rfloor =max119911 isin Z 119911 le 119909

(iv) [120572] is the integer part of number 120572 and 120572 thefractional part 0 le 120572 lt 1 so that 120572 = [120572] + 120572

(v) Δ120572[119891(119909) minus 119891(1199090)] ≃ Γ(1 + 120572)Δ[119891(119909) minus 119891(119909

0)]

(vi) 120572(sdot sdot) is the variable fractional orderwith 0 lt 120572(119909 119905) lt1 and (119909 119905) isin [119886 119887] 120572(119909) is a continuous function on(0 1]

(vii) C(119886 119911+) is a closed contour in the complex planestarting at 120585 = 119886 encircling 120585 = 119911 once in the positivesense and returning to 120585 = 119886 120583 ] isin R0 with0 lt 120583 lt 1 and 0 le ] le 1

(viii) Consider 119911 isin C and 119896 isin R The so-called 119896-gammafunction denoted by Γ

119896(119911) is related to the classical

gamma function by means of Γ119896(119911) = 119896

119911119896minus1Γ(119911119896)

(ix) The so-called 119896-Pochhammer symbol yields (119911)119899119896

=

Γ119896(119909 + 119899119896)Γ

119896(119909)

(x) The 119896-fractionalHilfer derivative recovers as particu-lar cases the fractional Riemann-Liouville derivativeif ] = 0 and 119896 = 1 and the fractional Caputo derivativeif ] = 1 = 119896 [41]

3 Definitions of Fractional Derivatives

Liouville derivative

D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

minusinfin

(119909 minus 120585)minus120572

119891 (120585) d120585

minus infin lt 119909 lt +infin

(1)

Liouville left-sided derivative

D1205720+ [119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899int

119909

0

(119909 minus 120585)minus120572+119899minus1

119891 (120585) d120585

119909 gt 0

(2)

Liouville right-sided derivative

D120572minus[119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

d119899

d119909119899int

infin

119909


119891 (120585) d120585

119909 lt infin

(3)

Riemann-Liouville left-sided derivative

RLD120572119886+ [119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899int

119909

119886

(119909 minus 120585)119899minus120572minus1

119891 (120585) d120585

119909 ge 119886

(4)

Riemann-Liouville right-sided derivative

RLD120572119887minus [119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

d119899

d119909119899int

119887

119909


119891 (120585) d120585

119909 le 119887

(5)

Caputo left-sided derivative

lowastD120572119886+ [119891 (119909)] =

1

Γ (119899 minus 120572)

int

119909

119886

(119909 minus 120585)119899minus120572minus1 d119899

d120585119899[119891 (120585)] d120585

119909 ge 119886

(6)

Caputo right-sided derivative

lowastD120572119887minus [119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

int

119887

119909


d120585119899[119891 (120585)] d120585

119909 le 119887

(7)

Mathematical Problems in Engineering 3

Grunwald-Letnikov left-sided derivativeGLD120572119886+ [119891 (119909)]

= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 minus 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119909 minus 119886

(8)

Grunwald-Letnikov right-sided derivativeGLD120572119887minus [119891 (119909)]

= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 + 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119887 minus 119909

(9)

Weyl derivative

119909D120572infin[119891 (119909)] = D120572

minus[119891 (119909)] = (minus1)

119898

(

dd120585)

119899

[119909W120572infin[119891 (119909)]]

(10)

Marchaud derivative

D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

minusinfin

119891 (119909) minus 119891 (120585)

(119909 minus 120585)1+120572

d120585 (11)

Marchaud left-sided derivative

D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 minus 120585)

1205851+120572

d120585 (12)

Marchaud right-sided derivative

D120572minus[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 + 120585)

1205851+120572

d120585 (13)

Hadamard derivative [42]

D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

0

119891 (119909) minus 119891 (120585)

[ln (119909120585)]1+120572d120585120585

(14)

Chen left-sided derivative

D120572c [119891 (119909)] =1

Γ (1 minus 120572)

dd119909

int

119909

c(119909 minus 120585)

minus120572

119891 (120585) d120585

119909 gt c

(15)

Chen right-sided derivative

D120572c [119891 (119909)] = minus1

Γ (1 minus 120572)

dd119909

int

c

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

119909 lt c

(16)

Davidson-Essex derivative [15]

D1205720[119891 (119909)] =

1

Γ (1 minus 120572)

d119899+1minus119896

d119909119899+1minus119896

times int

119909

0

(119909 minus 120585)minus120572 d119896

d120585119896[119891 (120585)] d120585

(17)

Coimbra derivative [43ndash45]

D120572(119909)0

[119891 (119909)]

=

1

Γ (1 minus 120572 (119909))

times int

119909

0

(119909 minus 120585)minus120572(119909) d

d120585[119891 (120585)] d120585 + f (0) 119909

minus120572(119909)

(18)

Canavati derivative

119886D]119909[119891 (119909)] =

1

Γ (1 minus 120583)

dd119909

int

119909

0

(119909 minus 120585)120583 d119899

d120585119899[119891 (120585)] d120585

119899 = lfloor]rfloor 120583 = 119899 minus ]

(19)

Jumarie derivative 119899 = 1

D120572119909[119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899

times int

119909

0


[119891 (120585) minus 119891 (0)] d120585(20)

Riesz derivative

D120572119909[119891 (119909)] = minus

1

2 cos (1205721205872)1

Γ (120572)

d119899

d119909119899

sdot int

119909

minusinfin


119891 (120585) d120585

+int

infin

119909


119891 (120585) d120585

(21)

Cossar derivative

D120572minus[119891 (119909)] = minus

1

Γ (1 minus 120572)

lim119873rarrinfin

dd119909

int

119873

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

(22)

Local fractional Yang derivative [40]

D120572minus[119891 (119909)]

1003816100381610038161003816119909=1199090

= lim119909rarr119909

0

Δ120572[119891 (119909) minus 119891 (119909

0)]

(119909 minus 1199090)120572

(23)

Left Riemann-Liouville derivative of variable fractionalorder

119886D120572(sdotsdot)119909

[119891 (119909)] =

dd119909

int

119909

119886

(119909 minus 120585)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(24)

Right Riemann-Liouville derivative of variable fractionalorder

119909D120572(sdotsdot)119887

[119891 (119909)] =

dd119909

int

119887

119909

(120585 minus 119909)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(25)

Left Caputo derivative of variable fractional order

119886D120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(26)


Right Caputo derivative of variable fractional order

119909D120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(120585 minus 119909)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(27)

Caputo derivative of variable fractional order

lowastD120572(119909)119909

[119891 (119909)] =

1

Γ (1 minus 120572 (119909))

int

119909

0

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585) d120585

(28)

Modified Riemann-Liouville fractional derivative

D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

0

(119909 minus 120585)minus120572

[119891 (120585) minus 119891 (0)] d120585

(29)

Osler fractional derivative [46]

119886D120572119911119891 (119911) =

Γ (120572 + 1)

2120587119894

int

C(119886119911+)

119891 (120585)

(120585 minus 119911)1+120572

d120585 (30)

119896-fractional Hilfer derivative [41]

119896D120583]119891 (119909) = I](1minus120583)

119896

dd119909

I(1minus120583)(1minus])119896

119891 (119909) (31)

4 Definitions of Fractional Integrals

Riemann-Liouville left-sided integral

RLI120572119886+ [119891 (119909)] =

1

Γ (120572)

int

119909

119886

(119909 minus 120585)120572minus1

119891 (120585) d120585 119909 ge 119886 (32)

Riemann-Liouville right-sided integral

RLI120572119887minus [119891 (119909)] =

1

Γ (120572)

int

119887

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 le 119887 (33)

Hadamard integral

I120572+[119891 (119909)] =

1

Γ (120572)

int

119909

0

119891 (120585)

[ln (120585119909)]1minus120572sdot

d120585120585

119909 gt 0 120572 gt 0

(34)

Weyl integral

119909W120572infin[119891 (119909)] =

1

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 (35)

Chen left-sided integral

I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (36)

Chen right-sided integral

I120572c [119891 (119909)] =1

Γ (120572)

int

c

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 lt c (37)

Cossar integral [47]

I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (38)

Erdelyi (left-sided) integral

I120572120590120578[119891 (119909)] =

120590119909minus120590(120572+120578)

Γ (120572)

int

119909

0

(119909120590

minus 120585120590

)120572minus1

120585120590120578+120590minus1

119891 (120585) d120585

(39)

Erdelyi (right-sided) integral

I120572120590120578[119891 (119909)] =

120590119909120590120572

Γ (120572)

int

infin

119909

(120585120590

minus 119909120590

)120572minus1

120585120590(1minus120572minus120578)minus1

119891 (120585) d120585

(40)

Kober (left-sided) integral

I1205721120578[119891 (119909)] =

119909minus120572minus120578

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

120585120578

119891 (120585) d120585 (41)

Kober (right-sided) integral

I1205721120578[119891 (119909)] =

119909120578

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

120585minus120572minus120578

119891 (120585) d120585 (42)

Local fractional Yang integral

119886I120572119887[119891 (119909)] =

1

Γ (1 + 120572)

int

119887

119886

119891 (120585) (d120585)120572 (43)

Left Riemann-Liouville integral of variable fractional order

119886I120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(120585 minus 119909)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(44)

Right Riemann-Liouville integral of variable fractional order

119909I120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(119909 minus 120585)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(45)

119896-fractional Hilfer integral

I120572119896119891 (119909) =

1

119896Γ119896(120572)

int

119909

0

(119909 minus 120585)120572119896minus1

119891 (120585) d120585 (46)

5 Some Remarks

Remark 1 If D120572 is any fractional derivative the Miller-Rosssequential derivative of order 119896120572 119896 isin Z is given by [3]

D120572

= 119863120572

D119896120572

= 119863120572

D(119896minus1)120572

(47)

Remark 2 Whatever the definition employed I0119891(119909) =

D0119891(119909) = 119891(119909)

Remark 3 Some authors do not distinguish the definitionemployed bymeans of a superscript (GL RL C and L) but usedifferent fonts for the operator instead (D 119863 DD andD)The particular correspondence between fonts and definitionsvaries Very often no indication at all is given save perhapsin the accompanying text and the reader is presumed tounderstand from the context which particular definition isintended


Remark 4 In the literature several alternative notations foroperator D may be found

D120572119886+119891 (119909) = (D120572

119886+119891) (119909) =

119886D120572119909119891 (119909) =

119886Iminus120572119909119891 (119909)

= D120572119909minus119886

119891 (119909) =

d120572119891 (119909)d(119909 minus 119886)120572

D120572119887minus119891 (119909) = (D120572

119887minus119891) (119909) =

119909D120572119887119891 (119909) =

119909Iminus120572119887119891 (119909)

= D120572119887minus119909

119891 (119909) =

d120572119891 (119909)d(119887 minus 119909)120572

(48)

Only one of the two operators I and D needs to be used sinceit is all a matter of changing the sign of 120572 In practice D is theone more often used

Remark 5 In the expressions for the right and left Liouvillefractional derivatives (2) and (3) respectively some authorshave a slight distinct expression instead of 0+ just + and atthe lower limit minusinfin

Remark 6 We can mention the ldquodifference of fractionalorderrdquo discussed by Bosanquet [48] and the ldquoRuscheweyhDerivativerdquo presented in [42 49ndash51]

Remark 7 The authorsrsquo intention is not to discuss pros andcons of the list of definitions of fractional derivatives andintegrals in Sections 3 and 4 Having in mind that the readercan find benefits in applying the correct definition for hisherspecific research interest it can be said that the most useddefinitions are the Riemann-Liouville (eg in calculus) theCaputo (eg in physics and numerical integration) and theGrunwald-Letnikov (eg in signal processing engineeringand control)The problem of initialization plays an importantrole in applied sciences and consequently various definitionsare occasionally adopted within the scope of specific topicsbut the overall problem remains to be clarified

Remark 8 The paper does not focus on particular rela-tions involving explicit parameters intervals or constantsassociated with the distinct derivatives For example wecan mention that for R(120572) = 0 with 120572 = 0 the Liouvillefractional derivatives are of purely imaginary order Also for120572 = 119899 isin N we recover the derivative of integer order Forexample D119899

+[119891(119909)] = 119891

(119899)(119909) and D119899

minus[119891(119909)] = (minus1)

119899

119891(119899)(119909)

Conflict of Interests

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

References

[1] M M Dzherbashyan and A B Nersesian ldquoAbout applicationof some integro-differential operatorsrdquoDoklady Akademii Nauk(Proceedings of the Russian Academy of Sciences) vol 121 no 2pp 210ndash213 1958

[2] M M Dzherbashyan and A B Nersesian ldquoThe criterionof the expansion of the functions to Dirichlet seriesrdquo Izvestiya

Akademii Nauk Armyanskoi SSR Seriya Fiziko-Matemati-cheskih Nauk vol 11 no 5 pp 85ndash108 1958

[3] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[4] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary OrderAcademic Press New York NY USA 1974

[5] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[6] B Riemann Versuch Einer Allgemeinen Auffassung der Integra-tion und Differentiation Gesammelte MathematischeWerke undWissenschaftlicher Nachlass Teubner Leipzig 1876 Dover NewYork NY USA 1953

[7] A K Grunwald ldquoUber ldquobegrenzterdquo derivationen und derenanwendungrdquo Zeitschrift fur Mathematik und Physik vol 12 pp441ndash480 1867

[8] A V Letnikov ldquoTheory of differentiation with an arbitraryindexrdquo Sbornik Mathematics vol 3 pp 1ndash66 1868 (Russian)

[9] H Weyl ldquoBemerkungen zum begriff des differentialquotien-ten gebrochener ordung vierteljahresschrrdquo NaturforschendeGesellschaft in Zurich vol 62 pp 296ndash302 1917

[10] M Riesz ldquoLrsquointegrale de Riemann-Liouville et le probleme deCauchyrdquo Acta Mathematica vol 81 no 1 pp 1ndash222 1949

[11] M Riesz ldquoLrsquointegrale de Riemann-Liouville et le problemede Cauchy pour lrsquoequation des ondesrdquo Bulletin de la SocieteMathematique de France vol 67 pp 153ndash170 1939

[12] M Caputo ldquoLinear models of dissipation whose q is almostfrequency independent-iirdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[13] R Figueiredo Camargo A O Chiacchio and E Capelas deOliveira ldquoDifferentiation to fractional orders and the fractionaltelegraph equationrdquo Journal ofMathematical Physics vol 49 no3 Article ID 033505 2008

[14] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific Singapore 2010

[15] M Davison and C Essex ldquoFractional differential equations andinitial value problemsrdquo The Mathematical Scientist vol 23 no2 pp 108ndash116 1998

[16] G Jumarie ldquoOn the solution of the stochastic differentialequation of exponential growth driven by fractional BrownianmotionrdquoAppliedMathematics Letters vol 18 no 7 pp 817ndash8262005

[17] G Jumarie ldquoAn approach to differential geometry of fractionalorder via modified Riemann-Liouville derivativerdquo Acta Mathe-matica Sinica vol 28 no 9 pp 1741ndash1768 2012

[18] G Jumarie ldquoOn the derivative chain-rules in fractional calculusvia fractional difference and their application to systems mod-ellingrdquo Central European Journal of Physics vol 11 no 6 pp617ndash633 2013

[19] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Amsterdam TheNetherlands 2006

[20] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andapplications Springer London UK 2010

[21] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations to


Methods ofTheir Solution vol 198 ofMathematics in Science andEngineering Academic Press San Diego Calif USA 1999

[22] E C Grigoletto and E C deOliveira ldquoFractional versions of thefundamental theorem of calculusrdquo Applied Mathematics vol 4pp 23ndash33 2013

[23] G H Hardy ldquoFractional versions of the fundamental theoremof calculusrdquo The Journal of the London Mathematical Society Ivol 20 no 1 pp 48ndash57 2013

[24] Q A Naqvi and M Abbas ldquoComplex and higher orderfractional curl operator in electromagneticsrdquo Optics Communi-cations vol 241 no 4ndash6 pp 349ndash355 2004

[25] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquosformulardquo Applied Mathematics and Computation vol 186 no1 pp 286ndash293 2007

[26] E Capelas de Oliveira and J Vaz Jr ldquoTunneling in fractionalquantum mechanicsrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 18 Article ID 185303 2011

[27] H T C Pedro M H Kobayashi J M C Pereira and C FM Coimbra ldquoVariable order modeling of diffusive-convectiveeffects on the oscillatory flow past a sphererdquo Journal of Vibrationand Control vol 14 no 9-10 pp 1659ndash1672 2008

[28] B Ross S G Samko and E R Love ldquoFunctions that have nofirst order derivative might have fractional derivatives of allorders less than onerdquo Real Analysis Exchange vol 20 no 2 pp140ndash157 1994-1995

[29] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquoAnnals of Physics vol 323 no 3 pp 2756ndash2778 2008

[30] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Fields and Media NonlinearPhysical Science Springer Heidelberg Germany 2011

[31] J J Trujillo M Rivero and B Bonilla ldquoOn a Riemann-Liouvillegeneralized Taylorrsquos formulardquo Journal of Mathematical Analysisand Applications vol 231 no 1 pp 255ndash265 1999

[32] D Valerio J J Trujillo M Rivero J T Machado and DBaleanu ldquoFractional calculus a survey of useful formulasrdquoTheEuropean Physical Journal Special Topics vol 222 no 8 pp1827ndash1846 2013

[33] G A Anastassiou Fractional Differentiation InequalitiesSpringer Dordrecht The Netherlands 2009

[34] 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 Singa-pore 2012

[35] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 Longman Scientific amp Technical Harlow NY USA1994

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

[37] A M Mathai and H J Haubold Special Functions for AppliedScientists Springer New York NY USA 2008

[38] D Valerio and J S da Costa An Introduction to FractionalControl IET Stevenage UK 2012

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

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

[41] G A Dorrego and R A Cerutti ldquoThe k-fractional Hilferderivativerdquo International Journal of Mathematical Analysis vol7 no 11 pp 543ndash550 2013

[42] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals and Derivatives Theory and Applications Gordon andBreach Science Publishers AmsterdamThe Netherlands 1993

[43] C F M Coimbra ldquoMechanics with variable-order differentialoperatorsrdquo Annalen der Physik vol 12 no 11-12 pp 692ndash7032003

[44] L E S Ramirez and C F M Coimbra ldquoOn the selection andmeaning of variable order operators for dynamic modelingrdquoInternational Journal of Differential Equations vol 2010 ArticleID 846107 16 pages 2010

[45] L E S Ramirez and C F M Coimbra ldquoOn the variableorder dynamics of the nonlinear wake caused by a sedimentingparticlerdquo Physica D Nonlinear Phenomena vol 240 no 13 pp1111ndash1118 2011

[46] T J Osler ldquoLeibniz rule for fractional derivatives generalizedand an application to infinite seriesrdquo SIAM Journal on AppliedMathematics vol 18 pp 658ndash674 1970

[47] J Cossar ldquoA theorem on Cesaro summabilityrdquo Journal of theLondon Mathematical Society I vol 16 pp 56ndash68 1941

[48] L S Bosanquet ldquoNote on convexity theoremsrdquo Journal of theLondon Mathematical Society I vol 18 no 4 pp 146ndash148 1943

[49] A A Lupas ldquoSome differential subordinations using Rus-cheweyh derivative and Salagean operatorrdquo Advances in Differ-ence Equations vol 2013 article 150 2013

[50] M M Meerschaert J Mortensen and S W WheatcraftldquoFractional vector calculus for fractional advection-dispersionrdquoPhysica A StatisticalMechanics and Its Applications vol 367 no8 pp 181ndash190 2006

[51] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the AmericanMathematical Society vol 49 no 1 pp 109ndash115 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


but more appropriate to discuss problems involving a frac-tional differential equation with initial conditions [13ndash21]

Due to the importance of the Caputo version we willcompare this approach with the Riemann-Liouville formula-tion The definition as proposed by Caputo inverts the orderof integral and derivative operators with the noninteger orderderivative of the Riemann-Liouville We summarize the dif-ference between these two formulations In the Caputo firstthe calculate derivative of integer order and after calculate theintegral of noninteger order In the Riemann-Liouville firstcalculate the integral of noninteger order and after calculatethe derivative of integer order It is important to cite that theCaputo derivative is useful to affront problems where initialconditions are done in the function and in the respectivederivatives of integer order

After the first congress at the University of New Havenin 1974 FC has developed and several applications emergedin many areas of scientific knowledge As a consequencedistinct approaches to solve problems involving the derivativewere proposed and distinct definitions of the fractionalderivative are available in the literature This paper presentsin a systematic form the existing formulations of fractionalderivatives and integralsWe shouldmention also that we canhave several alternative expressions for the same definitionTherefore we present only those more representative andwe cite particular papers [22ndash32] and books [33ndash40] that webelieve are the most relevant Furthermore the paper doesnot focus on the pros and cons of each definition and does notaddress the support of the function that is to be differentiatedor integrated

The paper is organized as follows Section 2 presents theadopted notation Sections 3 and 4 list the proposed defi-nitions of fractional derivatives and integrals respectivelyFinally Section 5 outlines some brief remarks

2 Notation

The following remarks clarify the notation used in the sequelin Sections 3 and 4

(i) Let 120572 isin C R(120572) isin (119899 minus 1 119899] 119899 isin N where R(sdot)denotes the real part of complex number

(ii) Let [119886 119887] be a finite interval in R 119896 isin N ] gt 0 andf(0) equiv 119891(0+) minus 119891(0minus)

(iii) The floor function denoted by lfloorsdotrfloor is defined as lfloor119909rfloor =max119911 isin Z 119911 le 119909

(iv) [120572] is the integer part of number 120572 and 120572 thefractional part 0 le 120572 lt 1 so that 120572 = [120572] + 120572

(v) Δ120572[119891(119909) minus 119891(1199090)] ≃ Γ(1 + 120572)Δ[119891(119909) minus 119891(119909

0)]

(vi) 120572(sdot sdot) is the variable fractional orderwith 0 lt 120572(119909 119905) lt1 and (119909 119905) isin [119886 119887] 120572(119909) is a continuous function on(0 1]

(vii) C(119886 119911+) is a closed contour in the complex planestarting at 120585 = 119886 encircling 120585 = 119911 once in the positivesense and returning to 120585 = 119886 120583 ] isin R0 with0 lt 120583 lt 1 and 0 le ] le 1

(viii) Consider 119911 isin C and 119896 isin R The so-called 119896-gammafunction denoted by Γ

119896(119911) is related to the classical

gamma function by means of Γ119896(119911) = 119896

119911119896minus1Γ(119911119896)

(ix) The so-called 119896-Pochhammer symbol yields (119911)119899119896

=

Γ119896(119909 + 119899119896)Γ

119896(119909)

(x) The 119896-fractionalHilfer derivative recovers as particu-lar cases the fractional Riemann-Liouville derivativeif ] = 0 and 119896 = 1 and the fractional Caputo derivativeif ] = 1 = 119896 [41]

3 Definitions of Fractional Derivatives

Liouville derivative

D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

minusinfin

(119909 minus 120585)minus120572

119891 (120585) d120585

minus infin lt 119909 lt +infin

(1)

Liouville left-sided derivative

D1205720+ [119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899int

119909

0


119891 (120585) d120585

119909 gt 0

(2)

Liouville right-sided derivative

D120572minus[119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

d119899

d119909119899int

infin

119909


119891 (120585) d120585

119909 lt infin

(3)

Riemann-Liouville left-sided derivative

RLD120572119886+ [119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899int

119909

119886


119891 (120585) d120585

119909 ge 119886

(4)

Riemann-Liouville right-sided derivative

RLD120572119887minus [119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

d119899

d119909119899int

119887

119909


119891 (120585) d120585

119909 le 119887

(5)

Caputo left-sided derivative

lowastD120572119886+ [119891 (119909)] =

1

Γ (119899 minus 120572)

int

119909

119886


d120585119899[119891 (120585)] d120585

119909 ge 119886

(6)

Caputo right-sided derivative

lowastD120572119887minus [119891 (119909)] =

(minus1)119899

Γ (119899 minus 120572)

int

119887

119909


d120585119899[119891 (120585)] d120585

119909 le 119887

(7)



= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 minus 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119909 minus 119886

(8)


= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 + 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119887 minus 119909

(9)

Weyl derivative

119909D120572infin[119891 (119909)] = D120572

minus[119891 (119909)] = (minus1)

119898

(

dd120585)

119899

[119909W120572infin[119891 (119909)]]

(10)

Marchaud derivative

D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

minusinfin

119891 (119909) minus 119891 (120585)

(119909 minus 120585)1+120572

d120585 (11)


D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 minus 120585)

1205851+120572

d120585 (12)


D120572minus[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 + 120585)

1205851+120572

d120585 (13)


D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

0

119891 (119909) minus 119891 (120585)

[ln (119909120585)]1+120572d120585120585

(14)


D120572c [119891 (119909)] =1

Γ (1 minus 120572)

dd119909

int

119909

c(119909 minus 120585)

minus120572

119891 (120585) d120585

119909 gt c

(15)


D120572c [119891 (119909)] = minus1

Γ (1 minus 120572)

dd119909

int

c

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

119909 lt c

(16)


D1205720[119891 (119909)] =

1

Γ (1 minus 120572)

d119899+1minus119896

d119909119899+1minus119896

times int

119909

0

(119909 minus 120585)minus120572 d119896

d120585119896[119891 (120585)] d120585

(17)


D120572(119909)0

[119891 (119909)]

=

1

Γ (1 minus 120572 (119909))

times int

119909

0

(119909 minus 120585)minus120572(119909) d

d120585[119891 (120585)] d120585 + f (0) 119909

minus120572(119909)

(18)

Canavati derivative

119886D]119909[119891 (119909)] =

1

Γ (1 minus 120583)

dd119909

int

119909

0

(119909 minus 120585)120583 d119899

d120585119899[119891 (120585)] d120585


(19)


D120572119909[119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899

times int

119909

0


[119891 (120585) minus 119891 (0)] d120585(20)

Riesz derivative

D120572119909[119891 (119909)] = minus

1

2 cos (1205721205872)1

Γ (120572)

d119899

d119909119899

sdot int

119909

minusinfin


119891 (120585) d120585

+int

infin

119909


119891 (120585) d120585

(21)

Cossar derivative

D120572minus[119891 (119909)] = minus

1

Γ (1 minus 120572)

lim119873rarrinfin

dd119909

int

119873

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

(22)


D120572minus[119891 (119909)]

1003816100381610038161003816119909=1199090

= lim119909rarr119909

0

Δ120572[119891 (119909) minus 119891 (119909

0)]

(119909 minus 1199090)120572

(23)


119886D120572(sdotsdot)119909

[119891 (119909)] =

dd119909

int

119909

119886

(119909 minus 120585)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(24)


119909D120572(sdotsdot)119887

[119891 (119909)] =

dd119909

int

119887

119909

(120585 minus 119909)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(25)


119886D120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(26)



119909D120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(120585 minus 119909)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(27)


lowastD120572(119909)119909

[119891 (119909)] =

1

Γ (1 minus 120572 (119909))

int

119909

0

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585) d120585

(28)


D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

0

(119909 minus 120585)minus120572

[119891 (120585) minus 119891 (0)] d120585

(29)


119886D120572119911119891 (119911) =

Γ (120572 + 1)

2120587119894

int

C(119886119911+)

119891 (120585)

(120585 minus 119911)1+120572

d120585 (30)


119896D120583]119891 (119909) = I](1minus120583)

119896

dd119909

I(1minus120583)(1minus])119896

119891 (119909) (31)



RLI120572119886+ [119891 (119909)] =

1

Γ (120572)

int

119909

119886

(119909 minus 120585)120572minus1

119891 (120585) d120585 119909 ge 119886 (32)


RLI120572119887minus [119891 (119909)] =

1

Γ (120572)

int

119887

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 le 119887 (33)

Hadamard integral

I120572+[119891 (119909)] =

1

Γ (120572)

int

119909

0

119891 (120585)

[ln (120585119909)]1minus120572sdot

d120585120585

119909 gt 0 120572 gt 0

(34)

Weyl integral

119909W120572infin[119891 (119909)] =

1

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 (35)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (36)


I120572c [119891 (119909)] =1

Γ (120572)

int

c

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 lt c (37)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (38)


I120572120590120578[119891 (119909)] =

120590119909minus120590(120572+120578)

Γ (120572)

int

119909

0

(119909120590

minus 120585120590

)120572minus1

120585120590120578+120590minus1

119891 (120585) d120585

(39)


I120572120590120578[119891 (119909)] =

120590119909120590120572

Γ (120572)

int

infin

119909

(120585120590

minus 119909120590

)120572minus1


119891 (120585) d120585

(40)


I1205721120578[119891 (119909)] =

119909minus120572minus120578

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

120585120578

119891 (120585) d120585 (41)


I1205721120578[119891 (119909)] =

119909120578

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

120585minus120572minus120578

119891 (120585) d120585 (42)


119886I120572119887[119891 (119909)] =

1

Γ (1 + 120572)

int

119887

119886

119891 (120585) (d120585)120572 (43)


119886I120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(120585 minus 119909)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(44)


119909I120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(119909 minus 120585)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(45)


I120572119896119891 (119909) =

1

119896Γ119896(120572)

int

119909

0

(119909 minus 120585)120572119896minus1

119891 (120585) d120585 (46)

5 Some Remarks


D120572

= 119863120572

D119896120572

= 119863120572

D(119896minus1)120572

(47)


D0119891(119909) = 119891(119909)




D120572119886+119891 (119909) = (D120572

119886+119891) (119909) =

119886D120572119909119891 (119909) =

119886Iminus120572119909119891 (119909)

= D120572119909minus119886

119891 (119909) =

d120572119891 (119909)d(119909 minus 119886)120572

D120572119887minus119891 (119909) = (D120572

119887minus119891) (119909) =

119909D120572119887119891 (119909) =

119909Iminus120572119887119891 (119909)

= D120572119887minus119909

119891 (119909) =

d120572119891 (119909)d(119887 minus 119909)120572

(48)






+[119891(119909)] = 119891

(119899)(119909) and D119899

minus[119891(119909)] = (minus1)

119899

119891(119899)(119909)



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 minus 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119909 minus 119886

(8)


= limℎrarr0

1

ℎ120572

lfloor119899rfloor

sum

119896=0

(minus1)119896Γ (120572 + 1) 119891 (119909 + 119896ℎ)

Γ (119896 + 1) Γ (120572 minus 119896 + 1)

119899ℎ = 119887 minus 119909

(9)

Weyl derivative

119909D120572infin[119891 (119909)] = D120572

minus[119891 (119909)] = (minus1)

119898

(

dd120585)

119899

[119909W120572infin[119891 (119909)]]

(10)

Marchaud derivative

D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

minusinfin

119891 (119909) minus 119891 (120585)

(119909 minus 120585)1+120572

d120585 (11)


D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 minus 120585)

1205851+120572

d120585 (12)


D120572minus[119891 (119909)] =

120572

Γ (1 minus 120572)

int

infin

0

119891 (119909) minus 119891 (119909 + 120585)

1205851+120572

d120585 (13)


D120572+[119891 (119909)] =

120572

Γ (1 minus 120572)

int

119909

0

119891 (119909) minus 119891 (120585)

[ln (119909120585)]1+120572d120585120585

(14)


D120572c [119891 (119909)] =1

Γ (1 minus 120572)

dd119909

int

119909

c(119909 minus 120585)

minus120572

119891 (120585) d120585

119909 gt c

(15)


D120572c [119891 (119909)] = minus1

Γ (1 minus 120572)

dd119909

int

c

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

119909 lt c

(16)


D1205720[119891 (119909)] =

1

Γ (1 minus 120572)

d119899+1minus119896

d119909119899+1minus119896

times int

119909

0

(119909 minus 120585)minus120572 d119896

d120585119896[119891 (120585)] d120585

(17)


D120572(119909)0

[119891 (119909)]

=

1

Γ (1 minus 120572 (119909))

times int

119909

0

(119909 minus 120585)minus120572(119909) d

d120585[119891 (120585)] d120585 + f (0) 119909

minus120572(119909)

(18)

Canavati derivative

119886D]119909[119891 (119909)] =

1

Γ (1 minus 120583)

dd119909

int

119909

0

(119909 minus 120585)120583 d119899

d120585119899[119891 (120585)] d120585


(19)


D120572119909[119891 (119909)] =

1

Γ (119899 minus 120572)

d119899

d119909119899

times int

119909

0


[119891 (120585) minus 119891 (0)] d120585(20)

Riesz derivative

D120572119909[119891 (119909)] = minus

1

2 cos (1205721205872)1

Γ (120572)

d119899

d119909119899

sdot int

119909

minusinfin


119891 (120585) d120585

+int

infin

119909


119891 (120585) d120585

(21)

Cossar derivative

D120572minus[119891 (119909)] = minus

1

Γ (1 minus 120572)

lim119873rarrinfin

dd119909

int

119873

119909

(120585 minus 119909)minus120572

119891 (120585) d120585

(22)


D120572minus[119891 (119909)]

1003816100381610038161003816119909=1199090

= lim119909rarr119909

0

Δ120572[119891 (119909) minus 119891 (119909

0)]

(119909 minus 1199090)120572

(23)


119886D120572(sdotsdot)119909

[119891 (119909)] =

dd119909

int

119909

119886

(119909 minus 120585)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(24)


119909D120572(sdotsdot)119887

[119891 (119909)] =

dd119909

int

119887

119909

(120585 minus 119909)minus120572(120585119909)

119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(25)


119886D120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(26)



119909D120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(120585 minus 119909)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(27)


lowastD120572(119909)119909

[119891 (119909)] =

1

Γ (1 minus 120572 (119909))

int

119909

0

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585) d120585

(28)


D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

0

(119909 minus 120585)minus120572

[119891 (120585) minus 119891 (0)] d120585

(29)


119886D120572119911119891 (119911) =

Γ (120572 + 1)

2120587119894

int

C(119886119911+)

119891 (120585)

(120585 minus 119911)1+120572

d120585 (30)


119896D120583]119891 (119909) = I](1minus120583)

119896

dd119909

I(1minus120583)(1minus])119896

119891 (119909) (31)



RLI120572119886+ [119891 (119909)] =

1

Γ (120572)

int

119909

119886

(119909 minus 120585)120572minus1

119891 (120585) d120585 119909 ge 119886 (32)


RLI120572119887minus [119891 (119909)] =

1

Γ (120572)

int

119887

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 le 119887 (33)

Hadamard integral

I120572+[119891 (119909)] =

1

Γ (120572)

int

119909

0

119891 (120585)

[ln (120585119909)]1minus120572sdot

d120585120585

119909 gt 0 120572 gt 0

(34)

Weyl integral

119909W120572infin[119891 (119909)] =

1

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 (35)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (36)


I120572c [119891 (119909)] =1

Γ (120572)

int

c

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 lt c (37)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (38)


I120572120590120578[119891 (119909)] =

120590119909minus120590(120572+120578)

Γ (120572)

int

119909

0

(119909120590

minus 120585120590

)120572minus1

120585120590120578+120590minus1

119891 (120585) d120585

(39)


I120572120590120578[119891 (119909)] =

120590119909120590120572

Γ (120572)

int

infin

119909

(120585120590

minus 119909120590

)120572minus1


119891 (120585) d120585

(40)


I1205721120578[119891 (119909)] =

119909minus120572minus120578

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

120585120578

119891 (120585) d120585 (41)


I1205721120578[119891 (119909)] =

119909120578

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

120585minus120572minus120578

119891 (120585) d120585 (42)


119886I120572119887[119891 (119909)] =

1

Γ (1 + 120572)

int

119887

119886

119891 (120585) (d120585)120572 (43)


119886I120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(120585 minus 119909)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(44)


119909I120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(119909 minus 120585)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(45)


I120572119896119891 (119909) =

1

119896Γ119896(120572)

int

119909

0

(119909 minus 120585)120572119896minus1

119891 (120585) d120585 (46)

5 Some Remarks


D120572

= 119863120572

D119896120572

= 119863120572

D(119896minus1)120572

(47)


D0119891(119909) = 119891(119909)




D120572119886+119891 (119909) = (D120572

119886+119891) (119909) =

119886D120572119909119891 (119909) =

119886Iminus120572119909119891 (119909)

= D120572119909minus119886

119891 (119909) =

d120572119891 (119909)d(119909 minus 119886)120572

D120572119887minus119891 (119909) = (D120572

119887minus119891) (119909) =

119909D120572119887119891 (119909) =

119909Iminus120572119887119891 (119909)

= D120572119887minus119909

119891 (119909) =

d120572119891 (119909)d(119887 minus 119909)120572

(48)






+[119891(119909)] = 119891

(119899)(119909) and D119899

minus[119891(119909)] = (minus1)

119899

119891(119899)(119909)



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909D120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(120585 minus 119909)minus120572(120585119909) d

d120585119891 (120585)

d120585Γ [1 minus 120572 (120585 119909)]

(27)


lowastD120572(119909)119909

[119891 (119909)] =

1

Γ (1 minus 120572 (119909))

int

119909

0

(119909 minus 120585)minus120572(120585119909) d

d120585119891 (120585) d120585

(28)


D120572 [119891 (119909)] = 1

Γ (1 minus 120572)

dd119909

int

119909

0

(119909 minus 120585)minus120572

[119891 (120585) minus 119891 (0)] d120585

(29)


119886D120572119911119891 (119911) =

Γ (120572 + 1)

2120587119894

int

C(119886119911+)

119891 (120585)

(120585 minus 119911)1+120572

d120585 (30)


119896D120583]119891 (119909) = I](1minus120583)

119896

dd119909

I(1minus120583)(1minus])119896

119891 (119909) (31)



RLI120572119886+ [119891 (119909)] =

1

Γ (120572)

int

119909

119886

(119909 minus 120585)120572minus1

119891 (120585) d120585 119909 ge 119886 (32)


RLI120572119887minus [119891 (119909)] =

1

Γ (120572)

int

119887

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 le 119887 (33)

Hadamard integral

I120572+[119891 (119909)] =

1

Γ (120572)

int

119909

0

119891 (120585)

[ln (120585119909)]1minus120572sdot

d120585120585

119909 gt 0 120572 gt 0

(34)

Weyl integral

119909W120572infin[119891 (119909)] =

1

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 (35)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (36)


I120572c [119891 (119909)] =1

Γ (120572)

int

c

119909

(120585 minus 119909)120572minus1

119891 (120585) d120585 119909 lt c (37)


I120572c [119891 (119909)] =1

Γ (120572)

int

119909

c(119909 minus 120585)

120572minus1

119891 (120585) d120585 119909 gt c (38)


I120572120590120578[119891 (119909)] =

120590119909minus120590(120572+120578)

Γ (120572)

int

119909

0

(119909120590

minus 120585120590

)120572minus1

120585120590120578+120590minus1

119891 (120585) d120585

(39)


I120572120590120578[119891 (119909)] =

120590119909120590120572

Γ (120572)

int

infin

119909

(120585120590

minus 119909120590

)120572minus1


119891 (120585) d120585

(40)


I1205721120578[119891 (119909)] =

119909minus120572minus120578

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

120585120578

119891 (120585) d120585 (41)


I1205721120578[119891 (119909)] =

119909120578

Γ (120572)

int

infin

119909

(120585 minus 119909)120572minus1

120585minus120572minus120578

119891 (120585) d120585 (42)


119886I120572119887[119891 (119909)] =

1

Γ (1 + 120572)

int

119887

119886

119891 (120585) (d120585)120572 (43)


119886I120572(sdotsdot)119909

[119891 (119909)] = int

119909

119886

(120585 minus 119909)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(44)


119909I120572(sdotsdot)119887

[119891 (119909)] = int

119887

119909

(119909 minus 120585)120572(120585119909)minus1

119891 (120585)

d120585Γ [120572 (120585 119909)]

(45)


I120572119896119891 (119909) =

1

119896Γ119896(120572)

int

119909

0

(119909 minus 120585)120572119896minus1

119891 (120585) d120585 (46)

5 Some Remarks


D120572

= 119863120572

D119896120572

= 119863120572

D(119896minus1)120572

(47)


D0119891(119909) = 119891(119909)




D120572119886+119891 (119909) = (D120572

119886+119891) (119909) =

119886D120572119909119891 (119909) =

119886Iminus120572119909119891 (119909)

= D120572119909minus119886

119891 (119909) =

d120572119891 (119909)d(119909 minus 119886)120572

D120572119887minus119891 (119909) = (D120572

119887minus119891) (119909) =

119909D120572119887119891 (119909) =

119909Iminus120572119887119891 (119909)

= D120572119887minus119909

119891 (119909) =

d120572119891 (119909)d(119887 minus 119909)120572

(48)






+[119891(119909)] = 119891

(119899)(119909) and D119899

minus[119891(119909)] = (minus1)

119899

119891(119899)(119909)



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











D120572119886+119891 (119909) = (D120572

119886+119891) (119909) =

119886D120572119909119891 (119909) =

119886Iminus120572119909119891 (119909)

= D120572119909minus119886

119891 (119909) =

d120572119891 (119909)d(119909 minus 119886)120572

D120572119887minus119891 (119909) = (D120572

119887minus119891) (119909) =

119909D120572119887119891 (119909) =

119909Iminus120572119887119891 (119909)

= D120572119887minus119909

119891 (119909) =

d120572119891 (119909)d(119887 minus 119909)120572

(48)






+[119891(119909)] = 119891

(119899)(119909) and D119899

minus[119891(119909)] = (minus1)

119899

119891(119899)(119909)



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

Review Article A Review of Definitions for Fractional