Annals of Physicstarasov/PDF/AP2008.pdfV.E. Tarasov/Annals of Physics 323 (2008) 2756–2778 2757. where Cðaþ1Þ is the Gamma function. In these deﬁnitions, the Nishimoto fractional

$Page 1: Annals of Physicstarasov/PDF/AP2008.pdfV.E. Tarasov/Annals of Physics 323 (2008) 2756–2778 2757. where Cðaþ1Þ is the Gamma function. In these deﬁnitions, the Nishimoto fractional$
Annals of Physics 323 (2008) 2756–2778

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Annals of Physics

journal homepage: www.elsevier .com/locate /aop

Fractional vector calculus and fractional Maxwell’s equations

Vasily E. Tarasov *

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Leninskie gory, Moscow 119991, Russia

a r t i c l e i n f o

Article history:Received 18 January 2008Accepted 12 April 2008Available online 20 April 2008

PACS:45.10.Hj03.50.De41.20.�q

Keywords:Fractional vector calculusDerivatives and integrals of non-integerordersFractal mediaFractional electrodynamics

0003-4916/$ - see front matter � 2008 Elsevier Indoi:10.1016/j.aop.2008.04.005

* Fax: +7 499 939 0397.E-mail address: [email protected]

a b s t r a c t

The theory of derivatives and integrals of non-integer order goesback to Leibniz, Liouville, Grunwald, Letnikov and Riemann. Thehistory of fractional vector calculus (FVC) has only 10 years. Themain approaches to formulate a FVC, which are used in the physicsduring the past few years, will be briefly described in this paper.We solve some problems of consistent formulations of FVC byusing a fractional generalization of the Fundamental Theorem ofCalculus. We define the differential and integral vector operations.The fractional Green’s, Stokes’ and Gauss’s theorems are formu-lated. The proofs of these theorems are realized for simplestregions. A fractional generalization of exterior differential calculusof differential forms is discussed. Fractional nonlocal Maxwell’sequations and the corresponding fractional wave equations areconsidered.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The fractional calculus has a long history from 30 September 1695, when the derivative of ordera ¼ 1=2 has been described by Leibniz [1,2] (see also [6]). The theory of derivatives and integrals ofnon-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. There are manyinteresting books about fractional calculus and fractional differential equations [1–5] (see also[7,8]). Derivatives and integrals of fractional order, and fractional integro-differential equations havefound many applications in recent studies in physics (for example, see books [9–12], and reviews [13–15]).

The history of fractional vector calculus (FVC) is not so long. It has only 10 years and can be reducedto the papers [16–27]. The main approaches to formulate a FVC, which are used in the physics during

c. All rights reserved.

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V.E. Tarasov / Annals of Physics 323 (2008) 2756–2778 2757

the past few years, will be briefly described in this paper. There are some fundamental problems ofconsistent formulations of FVC that can be solved by using a fractional generalization of theFundamental Theorem of Calculus. Fractional vector calculus is very important to describe processesin fractal media (see for example [10]). A consistent FVC can be used in fractional electrodynamics[18–20,25] and fractional hydrodynamics [21,29].

In Section 2, we describe different approaches to formulate FVC, which are used in the physics dur-ing the past 10 years. The problems of consistent formulation of FVC are described in Section 3. A frac-tional generalization of the Fundamental Theorem of Calculus is considered in Section 4. In Section 5,the differential and integral vector operations are defined. In Sections 6–8, the fractional Green’s,Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplestregions. In Section 9, a fractional generalization of exterior calculus of differential forms is discussed.In Section 10, fractional nonlocal Maxwell’s equations and the corresponding fractional wave equa-tions are considered.

2. Approaches to fractional vector calculus

For Cartesian coordinates, fractional generalizations of the divergence or gradient operators can bedefined by

grada f ðxÞ ¼ esDas f ðxÞ; ð1Þ

diva FðxÞ ¼ Das FsðxÞ; ð2Þ

where Das are fractional (Liouville, Riemann–Liouville, Caputo, etc.) derivatives [1–5] of order a with

respect to xs, (s ¼ 1;2;3). Here es ðs ¼ 1;2;3Þ are orthogonal unit vectors, and FsðxÞ are componentsof the vector field

FðxÞ ¼ FsðxÞes ¼ Fxex þ Fyey þ Fzez: ð3Þ

The main problem of formulation of FVC appears, when we try to generalize the curl operator and theintegral theorems. In Cartesian coordinates, the usual (integer) curl operator for the vector field (3) isdefined by

curlF ¼ elelmnDmFn; ð4Þ

where Dm ¼ o=oxm, and elmn is Levi–Civita symbol, which is 1 if ði; j; kÞ is an even permutation ofð1;2;3Þ, ð�1Þ if it is an odd permutation, and 0 if any index is repeated. The Fourier transform ofthe curl operator is

FðcurlFðxÞÞ ¼ elelmnðikmÞeF nðkÞ; ð5Þ

where

eF nðkÞ ¼ F FnðxÞð Þ ¼Z þ1

�1d3xe�ikxFnðxÞ: ð6Þ

To define a generalization of (4), we can use a fractional integro-differentiation instead of the deriv-ative Dm.

2.1. Ben Adda’s fractional vector calculus

In the paper [16] (see also [17]), fractional generalizations of gradient, divergence and curl operatorfor analytic functions have been suggested in the form

grada f ðxÞ ¼ 1Cðaþ 1Þ esD

as f ðxÞ; ð7Þ

diva FðxÞ ¼ 1Cðaþ 1ÞD

as FsðxÞ; ð8Þ

curla F ¼ 1Cðaþ 1ÞelelmnDa

mFnðxÞ; ð9Þ

2758 V.E. Tarasov / Annals of Physics 323 (2008) 2756–2778

where Cðaþ 1Þ is the Gamma function. In these definitions, the Nishimoto fractional derivative [8](see also Section 22 of [2]) is used. This derivative is a generalization of the Cauchy’s differentiationformula.

Fractional generalizations of integral operations (flux and circulation), and generalizations ofGauss’s, Stokes’, Green’s integral theorems are not considered.

2.2. Engheta’s fractional vector calculus

In the paper [18] (see also [18–20]), a fractional generalization of curl operator has been suggestedin the form

curla F ¼ elelmnDamFnðxÞ; ð10Þ

where Dam are fractional Liouville derivatives [5] of order a with respect to xm, (m ¼ 1;2;3), that are

defined by

Damf ðxÞ :¼ lim

a!�1 aDaxm

f ðxmÞ: ð11Þ

Here aDax is the Riemann–Liouville derivative

aDaxf ðxÞ ¼ 1

Cðn� aÞon

oxn

Z x

a

f x0ð Þx� x0ð Þa�nþ1 dx0; ðn� 1 < a < nÞ: ð12Þ

The fractional Liouville derivative (11) can be defined through the Fourier transform by

DamFnðxÞ ¼ F�1 ikmð ÞaeF nðkÞ

� �¼ 1

2pð Þ3Z þ1

�1d3keikx ikmð Þa eF nðkÞ; ð13Þ

where eF nðkÞ is defined by (6) and ia ¼ expfiap sgn ðkÞ=2g. For this fractional curl operator, the frac-tional integral Stokes’ and Green’s theorems are not suggested. The problems of a generalization ofthese theorems will be considered in the next section.

In general, the fractional vector calculus must include generalizations of the differential operations(gradient, divergence, curl), the integral operations (flux, circulation), and the theorems of Gauss,Stokes and Green.

2.3. Meerschaert–Mortensen–Wheatcraft fractional vector calculus

In the paper of Meerschaert, Mortensen and Wheatcraft [21], a fractional generalization of curloperator has been suggested as

curlaF ¼ elelmnDmI1�an Fn; ð14Þ

where I1�an are fractional integrals of order ð1� aÞ with respect to xn, (n ¼ 1;2;3). Note that the inte-

gration I1�an in (14) is considered with the index n as the component Fn. The derivative Dm ¼ o=oxm in

Eq. (14) is considered with respect to xm, where m 6¼ n. Therefore expression (14) can be presented asthe usual (integer) curl operator

curlaF ¼ curlFðaÞ ð15Þ

for the field

FðaÞ ¼ enI1�an Fn: ð16Þ

Eq. (15) allows us to use the usual (integer) integral Stokes’ and Green’s theorems.In Eq. (14), the fractional integral Ian and the integer derivative Dm have antisymmetric indices, and

the components of (14) are

ðcurlaFÞx ¼ DyI1�az Fz � DzI

1�ay Fy; ð17Þ

ðcurlaFÞy ¼ DzI1�ax Fx � DxI1�a

z Fz; ð18ÞðcurlaFÞz ¼ DxI1�a

y Fy � DyI1�ax Fx: ð19Þ


It is easy to see that operator (14) has no fractional derivatives with respect to xm, (m ¼ 1;2;3), like asDa

m ¼ DmI1�am or CDa

m ¼ I1�am Dm. As a result, we have the usual (integer) vector calculus for new type of

fields as in (16). The suggested approach cannot be considered as a fractional generalization of vectorcalculus. It is important to define a curl operator with fractional derivatives in such a form that frac-tional generalizations of the integral theorems exist.

2.4. Other approaches to fractional vector calculus

In the papers [25,29], fractional generalizations of integral operations and Gauss’s, Stokes’,Green’s theorems have been suggested. These generalizations are considered to describe frac-tional media by a continuous medium model. The differential operations are defined with re-spect to fractional powers of coordinates. These operations are connected with fractionalderivatives only by Fourier transforms (see [28]). As a result, an ”ideal” fractional vector calcu-lus is not suggested.

In the papers [26,27], fractional differential vector operations are considered by using fractionalgeneralizations of differential forms that are suggested in [22] (see also [23,24]). A fractional gradientis defined by an exact fractional 1-form. A fractional curl operator is described by a fractional exteriorderivative of a fractional differential 1-form. The Riemann–Liouville derivatives are used in [26], andthe fractional Caputo derivatives are used in [27]. We have

grada f ðxÞ ¼ esC0Da

xsf ðxÞ; ð20Þ

curlaF ¼ elelmnC0Da

xmFn; ð21Þ

where C0Da

xmis a fractional Caputo derivative with respect to xm:

Ca Da

xf ðxÞ ¼ 1Cðn� aÞ

Z x

a

1

x� x0ð Þa�nþ1 dx0onf x0ð Þo x0ð Þn

; ðn� 1 < a < nÞ: ð22Þ

The fractional generalizations of integral theorems (Gauss’s, Stokes’, Green’s theorems) are not consid-ered and the fractional integrals for differential forms are not defined.

3. Problems of fractional generalization of Green’s formula

Let us describe a main problem that appears when the curl operator and integral formulas are gen-eralized on a fractional case. For simplification, we consider a rectangular domain on R2 and integralformulas in Cartesian coordinates.

The Green’s formula in Cartesian coordinates is
ZoW
Fxdxþ Fydy� �

¼ZZ

Wdxdy DyFx � DxFy

� �; ð23Þ

where Fx ¼ Fxðx; yÞ and Fy ¼ Fyðx; yÞ are functions defined for all ðx; yÞ in the region W.Let W be the rectangular domain

W :¼ fðx; yÞ : a 6 x 6 b; c 6 y 6 dg

with the sides AB, BC, CD, DA, where the points A, B, C, D have coordinates

Aða; cÞ; Bða; dÞ; Cðb; dÞ; Dðb; cÞ:

These sides form a boundary oW of W. Then
ZoW
Fx dxþ Fy dy� �

¼Z

BCFx dxþ

ZDA

Fx dxþZ

ABFy dyþ

ZCD

Fy dy

¼Z b

aFxðx; dÞdxþ

Z a

bFxðx; cÞdxþ

Z d

cFyða; yÞdyþ

Z c

dFyðb; yÞdy

¼Z b

adx Fxðx; dÞ � Fxðx; cÞ½ � þ

Z d

cdy Fyða; yÞ � Fyðb; yÞ� �

: ð24Þ


The main step of proof of Green’s formula is to use the Newton-Leibniz formula
Z b
adx Dxf ðxÞ ¼ f ðbÞ � f ðaÞ: ð25Þ

The function f ðxÞ in (25) is absolutely continuous on ½a; b�. As a result, expression (24) can be presentedas
Z b
adx

Z d

cdyDyFxðx; yÞ

" #þZ d

cdy �

Z b

adxDxFyðx; yÞ

" #¼Z b

adxZ d

cdy DyFxðx; yÞ � DxFyðx; yÞdy� �

¼ZZ

Wdxdy DyFx � DxFy

� �:

To derive a fractional generalization of Green’s formula (23), we should have a generalization of theNewton-Leibniz formula (25) in the form

aIab aDaxf ðxÞ ¼ f ðbÞ � f ðaÞ; ð26Þ

where some integral and derivative of noninteger order are used. This generalization exists for spec-ified fractional integrals and derivatives, and does not exist for arbitrary taken type of the fractionalderivatives.

For the left Riemann–Liouville fractional integral and derivative (Lemma 2.5. of [5]), we have

aIabaDaxf ðxÞ ¼ f ðbÞ �

Xn

j¼1

ðb� aÞa�j

Cða� jþ 1Þ Dn�jx aIn�a

x f� �

ðaÞ; ð27Þ

where Dn�jx ¼ dn�j

=dxn�j are integer derivatives, and n� 1 < a < n. In particular, if 0 < a < 1, then

aIab aDaxf ðxÞ ¼ f ðbÞ � ðb� aÞa�1

CðaÞ aI1�ab f ðxÞ; ð28Þ

Obviously that Eq. (28) cannot be considered as a realization of (26). The left Riemann–Liouville frac-tional integral for x 2 ½a; b� is defined by

aIaxf ðxÞ :¼ 1CðaÞ

Z x

a

dx0

x� x0ð Þ1�aða > 0Þ: ð29Þ

The left Riemann–Liouville fractional integral for x 2 ½a; b� and n� 1a < n is defined by

aDax :¼ Dn

x aIn�ax f ðxÞ ¼ 1

Cðn� aÞon

oxn

Z x

a

dx0

x� x0ð Þa�nþ1 : ð30Þ

Note that Eq. (27) is satisfied if f ðxÞ is Lebesgue measurable functions on ½a; b� for which
Z b
af ðxÞdx <1;

and aIn�ab f ðxÞ of the right-hand side of (27) has absolutely continuous derivatives up to order ðn� 1Þ on

½a; b�. Properties (27) and (28) are connected with the definition of the Riemann–Liouville fractionalderivative, where the integer-order derivative acts on the fractional integral:

aDax ¼ Dn

x aIn�ax ; ðn� 1 < a < nÞ: ð31Þ

This definition gives that the left-hand side of (28) is

aIaxaDax ¼ aIaxDn

x aIn�ax ; ð32Þ

where the integer derivative Dnx is located between the fractional integrals. Since the operations Dn

x andaIax are not commutative

aIn�ax Dn

x � Dnx aIn�a

x 6¼ 0;

we get the additional terms, which cannot give the right-hand side of (26). This noncommutativity canbe presented as a nonequivalence of Riemann–Liouville and Caputo derivatives [4,5],


aCDa

xf ðxÞ ¼ aDaxf ðxÞ �

Xn�1

j¼0

ðx� aÞj�a

Cðj� aþ 1Þ ðDjxf ÞðaÞ; ðn� 1 < a < nÞ: ð33Þ

The left Caputo fractional derivative is defined by the equation (compare with (31))

aCDa

xf ðxÞ :¼ aIn�ax Dn

x f ðxÞ; ðn� 1 < a < nÞ: ð34Þ

The noncommutativity of Dnx and aIax in (32) does not allow us to use semi-group property (see Lemma

2.3 of [5] and Theorem 2.5 of [2]) of fractional integrals

aIaxaIbx ¼ aIaþbx ; ða > 0; b > 0Þ: ð35Þ

Note that Eq. (35) is satisfied at almost every point x 2 ½a; b� for f ðxÞ 2 Lpða; bÞ and a; b > 0. We denoteby Lpða; bÞ (1 < p <1) the set of those Lebesgue measurable functions on ½a; b� for which
Z b
adx jf ðxÞjp

!1=p

<1:

In general, the semi-group property

aDaxaDb

x ¼ aDaþbx ; ða > 0; b > 0Þ: ð36Þ

is not satisfied for fractional derivatives (see Property 2.4 in [5]). For some special cases, Eq. (36) can beused (see Theorem 2.5. in [2]). For example, the property (36) is satisfied for the functions

f ðxÞ 2 aIaþbx ðL1ða; bÞÞ;

i.e., Eq. (36) is valid for f ðxÞ if there exists a function gðxÞ 2 L1ða; bÞ such that

f ðxÞ ¼ aIaþbx gðxÞ:

The semi-group property for fractional derivatives is also valid if a ¼ 0, b ¼ 1 and f ðxÞ is infinitely dif-ferentiable (generalized) function on ½0;1Þ (see Sections 1.4.5. of [30] and 8.3. of [2]).

In order to have a fractional generalization of the Newton-Leibniz formula of the form (26), wemust replace the left Riemann–Liouville derivative aDa

b in Eq. (26), where

aIax aDax ¼ aIab Dn

x aIn�ax

� �
by the left Caputo derivative a
CDax , such that the left-hand side of (26) is

aIaxaCDa

x ¼ aIax aIn�ax Dn

x

� �:

Then, we can use the semi-group property (35), and

aIax aCDa

xf ðxÞ ¼ aIax aIn�ax Dn

x f ðxÞ ¼ aInx Dn

x f ðxÞ:

In particular, if n ¼ 1 and 0 < a < 1, then

aIab aCDa

xf ðxÞ ¼ aI1b D1

x f ðxÞ ¼Z b

adx D1

x f ðxÞ ¼ f ðbÞ � f ðaÞ: ð37Þ

As a result, to generalize Gauss’s, Green’s and Stokes’ formulas for fractional case, we can use the equa-tion with the Riemann–Liouville integral and the Caputo derivative:

aIabaCDa

xf ðxÞ ¼ f ðbÞ � f ðaÞ: ð38Þ

This equation can be considered as a fractional analog of the Newton-Leibniz formula.

4. Fractional generalization of the fundamental theorem of calculus

The fundamental theorem of calculus (FTC) is the statement that the two central operations of cal-culus, differentiation and integration, are inverse operations: if a continuous function is first inte-grated and then differentiated, the original function is retrieved


D1x aI1

x f ðxÞ ¼ f ðxÞ: ð39Þ

An important consequence, sometimes called the second fundamental theorem of calculus, allows oneto compute integrals by using an antiderivative of the function to be integrated:

aI1b D1

x f ðxÞ ¼ f ðbÞ � f ðaÞ: ð40Þ

If we use the Riemann–Liouville integrals and derivatives [2,5], we cannot generalize (40) for frac-tional case, since

aIab aDaxf ðxÞ 6¼ f ðbÞ � f ðaÞ; ð41Þ

In this case, we have Eq. (27).The FTC states that the integral of a function f over the interval ½a; b� can be calculated by finding an

antiderivative F, i.e., a function, whose derivative is f. Integral theorems of vector calculus (Stokes’,Green’s, Gauss’s theorems) can be considered as generalizations of FTC.

The fractional generalization of the FTC for finite interval ½a; b� can be realized (see remarks afterproof of the theorem and Section 3.) in the following special form.

4.1. Fundamental theorem of fractional calculus

(1) Let f(x) be a real-valued function defined on a closed interval [a,b]. Let F(x) be the function definedfor x in [a,b] by

FðxÞ ¼ aIaxf ðxÞ; ð42Þ

where aIax is the fractional Riemann–Liouville integral

aIaxf ðxÞ :¼ 1CðaÞ

Z x

a

f x0ð Þx� x0ð Þ1�a

dx0; ð43Þ

then

aCDa

xFðxÞ ¼ f ðxÞ ð44Þ

for x 2 ða; bÞ, where aCDa

x is the Caputo fractional derivative

aCDa

xFðxÞ ¼ aIn�ax Dn

x FðxÞ ¼ 1Cðn� aÞ

Z x

a

dx0

x� x0ð Þ1þa�n

dnFðxÞdxn

; ðn� 1 < a < nÞ: ð45Þ

(2) Let f ðxÞ be a real-valued function defined on a closed interval ½a; b�. Let FðxÞ be a function such that

f ðxÞ ¼ aCDa

xFðxÞ ð46Þ

for all x in ½a; b�, then

aIabf ðxÞ ¼ FðbÞ � FðaÞ; ð47Þ

or, equivalently,

aIabaCDa

xFðxÞ ¼ FðbÞ � FðaÞ; ð0 < a < 1Þ: ð48Þ

As a result, we have the fractional analogs of Eqs. (39) and (40) in the form

aCDa

xaIaxf ðxÞ ¼ f ðxÞ; ða > 0Þ; ð49ÞaIaxa

CDaxFðxÞ ¼ FðxÞ � FðaÞ; ð0 < a < 1Þ; ð50Þ

where aIax is the Riemann–Liouville integral, and aCDa

x is the Caputo derivative.

Proof. The proof of this theorem can be realized by using Lemmas 2.21 and 2.22 of [5].(1) For real values of a > 0, the Caputo fractional derivative provides operation inverse to the

Riemann–Liouville integration from the left (see Lemma 2.21 of [5]),

aCDa

xaIaxf ðxÞ ¼ f ðxÞ; ða > 0Þ ð51Þ


for f ðxÞ 2 L1ða; bÞ or f ðxÞ 2 C½a; b�.(2) If f ðxÞ 2 ACn½a; b� or f ðxÞ 2 Cn½a; b�, then (see Lemma 2.22 of [5])

aIaxaCDa

xf ðxÞ ¼ f ðxÞ �Xn�1

j¼0

1j!ðx� aÞj Dj

xf� �

ðaÞ; ðn� 1 < a 6 nÞ; ð52Þ

where Cn½a; b� is a space of functions, which are n times continuously differentiable on ½a; b�. In partic-ular, if 0 < a 6 1 and f ðxÞ 2 AC½a; b� or f ðxÞ 2 C½a; b�, then

aIaxaCDa

xf ðxÞ ¼ f ðxÞ � f ðaÞ: ð53Þ

This equation can be considered as a fractional generalization of the Newton–Leibniz formula in theform (26).

Remark 1. In this theorem (see Eqs. (42)–(48)), the spaces L1½a; b� and AC½a; b� are used.(a) Here AC½a; b� is a space of functions FðxÞ, which are absolutely continuous on ½a; b�. It is known

that AC½a; b� coincides with the space of primitives of Lebesgue summable functions and therefore anabsolutely continuous function FðxÞ has a summable derivative D1

x ðxÞ almost everywhere on ½a; b�. IfFðxÞ 2 AC½a; b�, then the Caputo derivative ð0 < a < 1Þ exists almost everywhere on ½a; b� (see Theorem2.1 of [5]).

(b) We denote Lpða; bÞ the set of those Lebesgue measurable functions f on ½a; b� for which

kfkp ¼Z b

ajf ðxÞjp dx

!1=p

<1: ð54Þ

If f ðxÞ 2 Lpða; bÞ, where p > 1, then the fractional Riemann–Liouville integrations are bounded inLpða; bÞ, and the semi-group property

aIaxaIbx f ðxÞ ¼ aIaþbx f ðxÞ; ða > 0; b > 0Þ ð55Þ

are satisfied at almost every point x 2 ½a; b�. If aþ b > 1, then relation (55) holds at any point of ½a; b�(see Lemmas 2.1 and 2.3 in [5]).

Remark 2. For the Riemann–Liouville derivative aDax , the relation

aDax aIaxf ðxÞ ¼ f ðxÞ; ða > 0Þ ð56Þ

holds almost everywhere on ½a; b� for f ðxÞ 2 Lpða; bÞ (see Lemma 2.4 of [5]).

Remark 3. The Fundamental Theorem of Fractional Calculus (FTFC) uses the Riemann–Liouville inte-gration and the Caputo differentiation. The main property is that the Caputo fractional derivative pro-vides us an operation inverse to the Riemann–Liouville fractional integration from the left. It should benoted that consistent fractional generalizations of the FTC, the differential vector operations and theintegral theorems for other fractional integro-differentiation such as Riesz, Grunvald-Letnikov, Weyl,Nishimoto are open problems.

Remark 4. In the theorem, we use 0 < a 6 1. As a result, we obtain the fractional Green’s, Stokes’ andGauss’s theorems for 0 < a < 1. Eq. (49) is satisfied for a 2 Rþ. The Newton–Leibniz formula (50) holdsfor 0 < a 6 1. For a > 1, we have (52).

As a result, to generalize the Green’s, Stokes’ and Gauss’s theorems for a 2 Rþ, we can use Eq. (52)in the form

f ðbÞ � f ðaÞ ¼ aIab aCDa

xf ðxÞ þXn�1

j¼1

1j!ðb� aÞjf ðjÞðaÞ; ðn� 1 < a 6 nÞ; ð57Þ

where f ðjÞðxÞ ¼ Djxf ðxÞ. In particular, if 1 < a 6 2, then n ¼ 2 and

f ðbÞ � f ðaÞ ¼ aIab aCDa

xf ðxÞ þ ðb� aÞjf 0ðaÞ: ð58Þ


Remark 5. In the FTFC, we use the left fractional integrals and derivatives. The Newton–Leibniz for-mulas can be presented for the right fractional Riemann–Liouville integrals and the right fractionalCaputo derivatives in the form

xIab xCDa

bf ðxÞ ¼ f ðxÞ �Xn�1

j¼0

ð�1Þjf ðjÞðbÞj!

ðb� zÞj: ð59Þ

In particular, if 0 < a 6 1, then

xIabxCDa

bf ðxÞ ¼ f ðxÞ � f ðbÞ: ð60Þ

For Im ðaÞ ¼ 0, a > 0, f ðxÞ 2 L1ða; bÞ or f ðxÞ 2 C½a; b�, then

xCDa

bxIabf ðxÞ ¼ f ðxÞ: ð61Þ

As a result, fractional generalization of differential operations and integral theorems can be defined forthe right integrals and derivatives as well as for the left ones.

5. Definition of fractional vector operations

5.1. Fractional operators

To define fractional vector operations, we introduce the operators that correspond to the fractionalderivatives and integrals.

We define the fractional integral operator

aIax x0½ � :¼ 1CðaÞ

Z x

a

dx0

x� x0ð Þ1�a; ða > 0Þ; ð62Þ

which acts on a real-valued function f ðxÞ 2 L1½a; b� by

aIax x0½ �f x0ð Þ ¼ 1CðaÞ

Z x

a

f x0ð Þdx0

x� x0ð Þ1�a: ð63Þ

The Caputo fractional differential operator on ½a; b� can be defined by

aCDa

x x0½ � :¼ 1Cðn� aÞ

Z x

a

dx0


on

ox0n; ðn� 1 < a < nÞ; ð64Þ

such that the Caputo derivatives for f ðxÞ 2 ACn½a; b� is written as

aCDa

x x0½ �f x0ð Þ ¼ 1Cðn� aÞ

Z x

a

dx0


@nf x0ð Þ@x0n

; ðn� 1 < a < nÞ: ð65Þ

where It is easy to see that

aCDa

x x0½ � ¼ aIn�ax x0½ �D x0½ �; ðn� 1 < a < nÞ:

Using these notations, formulas (49) and (50) of the FTFC can be presented as

aCDa

x x0½ � aIax0 ½x00�f ðx00Þ ¼ f ðxÞ; ða > 0Þ; ð66ÞaIab½x� a

CDax x0½ �f x0ð Þ ¼ f ðbÞ � f ðaÞ; ð0 < a < 1Þ: ð67Þ

This form is more convenient than (49) and (50), since it allows us to take into account the variables ofintegration and the domain of the operators.

5.2. Definition of fractional differential vector operations

Let us define the fractional differential operators (grad, div, curl) such that fractional generaliza-tions of integral theorems (Green’s, Stokes’, Gauss’) can be realized. We use the Caputo derivatives


to defined these operators and we use the Riemann–Liouville integrals in the generalizations of theintegral theorems.

Let W be a domain of R3. Let f ðxÞ and FðxÞ be real-valued functions that have continuous derivativesup to order ðn� 1Þ on W, such that the ðn� 1Þ derivatives are absolutely continuous, i.e., f ; F 2 ACn½W�.We can define a fractional generalization of nabla operator by

raW ¼ CDa

W ¼ e1CDa

W ½x� þ e2CDa

W ½y� þ e3CDa

W ½z�; ðn� 1 < a < nÞ: ð68Þ
Here, we use the fractional Caputo derivatives CDa
W xm½ � with respect to coordinates xm. For theparallelepiped

W :¼ fa 6 x 6 b; c 6 y 6 d; g 6 z 6 hg;
we have
CDaW ½x� ¼ a

CDab½x�; CDa

W ½y� ¼ cCDa

d½y�; CDaW ½z� ¼ g

CDah½z�: ð69Þ

The right-hand sides of these equations the Caputo derivatives are used.

(1) If f ¼ f ðx; y; zÞ is ðn� 1Þ times continuously differentiable scalar field such that Dn�1xl

f are abso-lutely continuous, then we define its fractional gradient as the following

GradaW f ¼ CDa

W f ¼ elCDa

W xl½ �f ðx; y; zÞ¼ e1

CDaW ½x�f ðx; y; zÞ þ e2

CDaW ½y�f ðx; y; zÞ þ e3

CDaW ½z�f ðx; y; zÞ: ð70Þ

(2) If Fðx; y; zÞ is ðn� 1Þ times continuously differentiable vector field such that Dn�1xl

Fl are absolutelycontinuous, then we define its fractional divergence as a value of the expression

DivaW F ¼ CDa

W ; F� �

¼ CDaW xl½ �Flðx; y; zÞ

¼ CDaW ½x�Fxðx; y; zÞ þ CDa

W ½y�Fyðx; y; zÞ þ CDaW ½z�Fzðx; y; zÞ: ð71Þ

(3) The fractional curl operator is defined by

CurlaW F ¼ CDaW ; F

� �¼ elelmk

CDaW xm½ �Fk

¼ e1CDa

W ½y�Fz � CDaW ½z�Fy

� �þþe2

CDaW ½z�Fx � CDa

W ½x�Fz� �

þ e3CDa

W ½x�Fy � CDaW ½y�Fx

� �; ð72Þ

where Fk ¼ Fkðx; y; zÞ 2 ACn½W�, ðk ¼ 1;2;3Þ.

Note that these fractional differential operators are nonlocal. As a result, the fractional gradient,divergence and curl depend on the region W.

5.3. Relations for fractional differential vector operations

(a) The first relation for the scalar field f ¼ f ðx; y; zÞ is

CurlaW GradaW f ¼ el elmn

CDaW xm½ � CDa

W xn½ �f ¼ 0; ð73Þ

where elmn is Levi–Civita symbol, i.e. it is 1 if ði; j; kÞ is an even permutation of ð1;2;3Þ, ð�1Þ if it is anodd permutation, and 0 if any index is repeated.

(b) The second relation,

DivaW Grada

W f ðx; y; zÞ ¼ CDaW xl½ � CDa

W xl½ �f ðx; y; zÞ ¼X3

l¼1

CDaW xl½ �

� �2f ðx; y; zÞ: ð74Þ

Using notation (68),

DivaW Grada

W ¼ CDaW

� �2 ¼ CDaW ;

CDaW

� �: ð75Þ

In the general case,

CDaW xl½ �

� �2 6¼ CD2aW xl½ �: ð76Þ


It is obvious from

aCDa

x

� �2 ¼ aIn�ax Dn

x aIn�ax Dn

x ¼ aIn�ax aIn�a

x Dnx Dn

x þ aIn�ax ½D

nx ; aIn�a

x �Dnx ¼ aD2a

x þ aIn�ax Dn

x ; aIn�ax

� �Dn

x ;

where

Dnx ; aIn�a

x

� �:¼ Dn

x aIn�ax � aIn�a

x Dnx ¼ aDa

x � aCDa

x 6¼ 0:

(c) It is easy to prove the following relation,

DivaW CurlaW Fðx; y; zÞ ¼ CDa

W xl½ �elmnCDa

W xm½ �Fnðx; y; zÞ ¼ elmnCDa

W xl½ � CDaW xm½ �Fnðx; y; zÞ ¼ 0; ð77Þ

where we use antisymmetry of elmn with respect to m and n.(d) There exists a relation for the double curl operation in the form

CurlaW CurlaW Fðx; y; zÞ ¼ elelmnCDa

W xm½ �enpqCDa

W xp� �

Fqðx; y; zÞ¼ elelmnenpq

CDaW xm½ � CDa

W xp� �

Fqðx; y; zÞ: ð78Þ

Using

elmnelpq ¼ dmpdnq � dmqdnp; ð79Þ

we get

CurlaW CurlaW Fðx; y; zÞ ¼ GradaW Diva

W Fðx; y; zÞ � CDaW

� �2Fðx; y; zÞ; ð80Þ

(e) In the general case,

aCDa

x x0½ � f x0ð Þg x0ð Þð Þ 6¼ aCDa

x x0½ �f x0ð Þ� �

gðxÞ þ aCDa

x x0½ �g x0ð Þ� �

f ðxÞ: ð81Þ

For example (see Theorem 15.1. from [2]),

aDax x0½ � f x0ð Þg x0ð Þð Þ ¼

X1j¼0

Cðaþ 1ÞCðjþ 1ÞCða� jþ 1Þ aDa�j

x x0½ �f x0ð Þ� �

DjxgðxÞ

� �; ð82Þ

if f ðxÞ and gðxÞ are analytic functions on ½a; b�. As a result, we have

GradaWðfgÞ 6¼ ðGrada

W f Þg þ ðGradaW gÞf ; ð83Þ

DivaWðf FÞ 6¼ ðGrada

W f ; FÞ þ f DivaW F: ð84Þ

These relations state that we cannot use the Leibniz rule in a fractional generalization of the vectorcalculus.

5.4. Fractional integral vector operations

In this section, we define fractional generalizations of circulation, flux and volume integral.Let F ¼ Fðx; y; zÞ be a vector field such that

Fðx; y; zÞ ¼ e1Fxðx; y; zÞ þ e2Fyðx; y; zÞ þ e3Fzðx; y; zÞ:

If Fx, Fy, Fx are absolutely integrable real-valued functions on R3, i.e., Fx, Fy, Fx 2 L1ðR3Þ, then we candefine the following fractional integral vector operations of order a > 0.

(1) A fractional circulation is a fractional line integral along a line L that is defined by

EaLðFÞ ¼ Ia

L; F� �

¼ IaL½x�Fx þ IaL½y�Fy þ IaL½z�Fz: ð85Þ

For a ¼ 1, we get

E1L ðFÞ ¼ I1

L ; F� �

¼Z

LðdL; FÞ ¼

ZL

Fxdxþ Fydyþ Fzdz� �

; ð86Þ

where dL ¼ e1dxþ e2dyþ e3dz.(2) A fractional flux of the vector field F across a surface S is a fractional surface integral of the field,

such that

UaSðFÞ ¼ Ia

S; F� �

¼ IaS½y; z�Fx þ IaS½z; x�Fy þ IaS½x; y�Fz: ð87Þ


For a ¼ 1, we get

U1S ðFÞ ¼ ðI

1S ; FÞ ¼

ZZSðdS; FÞ ¼

Z ZS

Fxdydzþ Fydzdxþ Fzdxdy� �

; ð88Þ

where dS ¼ e1 dydzþ e2dzdxþ e3 dxdy.(3) A fractional volume integral is a triple fractional integral within a region W in R3 of a scalar field

f ¼ f ðx; y; zÞ,

VaWðf Þ ¼ IaW ½x; y; z�f ðx; y; zÞ ¼ IaW ½x�I

aW ½y�I

aW ½z�f ðx; y; zÞ: ð89Þ

For a ¼ 1, we have

V1Wðf Þ :¼

ZZZW

dVf ðx; y; zÞ ¼ZZZ

Wdxdydzf ðx; y; zÞ: ð90Þ

This is the usual volume integral for the function f ðx; y; zÞ.

6. Fractional Green’s formula

Green’s theorem gives the relationship between a line integral around a simple closed curve oWand a double integral over the plane region W bounded by oW . The theorem statement is the follow-ing. Let oW be a positively oriented, piecewise smooth, simple closed curve in the plane and let W be aregion bounded by oW . If Fx and Fy have continuous partial derivatives on an open region containingW, then
Z
oWFxdxþ Fydy� �

¼Z Z

WDyFx � DxFy� �

dxdy: ð91Þ

A fractional generalization of the Green’s formula (91) is presented by the following statement.

Theorem. (Fractional Green’s Theorem for a Rectangle)Let Fxðx; yÞ and Fyðx; yÞ be absolutely continuous (or continuously differentiable) real-valued functions

in a domain that includes the rectangle

W :¼ fðx; yÞ : a 6 x 6 b; c 6 y 6 dg: ð92Þ

Let the boundary of W be the closed curve oW. Then

IaoW ½x�Fxðx; yÞ þ IaoW ½y�Fyðx; yÞ ¼ IaW ½x; y� CDaoW ½y�Fxðx; yÞ � CDa

oW ½x�Fyðx; yÞ� �

; ð93Þ

where 0 < a 6 1.

Proof. To prove Eq. (93), we change the double fractional integral IaW ½x; y� to the repeated fractionalintegrals IaW ½x� IaW ½y�, and then employ the Fundamental Theorem of Fractional Calculus.

Let W be the rectangular domain (92) with the sides AB, BC, CD, DA, where the points A, B, C, D havecoordinates

Aða; cÞ; Bða; dÞ; Cðb; dÞ; Dðb; cÞ:

These sides form the boundary oW of W.For the rectangular region W defined by a 6 x 6 b, c 6 y 6 d, the repeated integral is

IaW ½x� IaW ½y� ¼ aIab½x� cIad½y�;

and Eq. (93) is

aIab½x� Fxðx; dÞ � Fxðx; cÞð Þ þ cIad½y� Fyða; yÞ � Fyðb; yÞ� �

¼ aIab½x� cIad½y� cCDa

y y0½ �Fx x; y0ð Þ � aCDa

x x0½ �Fy x0; yð Þ� �

: ð94Þ


To prove of the fractional Green’s formula, we realize the following transformations

IaoW ; F

� �¼ IaoW ½x�Fx þ IaoW ½y�Fy ¼ IaBC ½x�Fx þ IaDA½x�Fx þ IaAB½y�Fy þ IaCD½y�Fy

¼ aIab½x�Fxðx; dÞ � aIab½x�Fxðx; cÞ þ cIad½y�Fyða; yÞdy � cIad½y�Fyðb; yÞ¼ aIab½x� Fxðx; dÞ � Fxðx; cÞ½ � þ cIad½y� Fyða; yÞ � Fyðb; yÞ

� �: ð95Þ

The main step of the proof of Green’s formula is to use the fractional Newton–Leibniz formula

Fxðx; dÞ � Fxðx; cÞ ¼ cIad½y�cCDay y0½ � Fðx; y0Þ; Fyða; yÞ � Fyðb; yÞ ¼ �aIab½x�aCDa

x x0½ �F x0; yð Þ: ð96Þ

As a result, expression (95) can be presented as

aIab½x� cIad½y�cCDay y0½ � Fx x; y0ð Þ

n oþ cIad½y� �aIab½x�aCDa

x x0½ � Fy x0; yð Þ�

¼ aIab½x�cIad½y� cCDa

y y0½ � Fx x; y0ð Þ � aCDa

x x0½ � Fy x0; yð Þ� �

¼ IaW ½x; y� cCDa

y y0½ � Fx x; y0ð Þ � aCDa

x x0½ � Fy x0; yð Þ� �

:

This is the left-hand side of Eq. (94). This ends the proof.

Remark 1. In this fractional Green’s theorem, we use the rectangular region W. If the region can beapproximated by a set of rectangles, the fractional Green’s formula can also be proved. In this case,the boundary oW is presented by paths each consisting of horizontal and vertical line segments, lyingin W.

Remark 2. To define the double integral and the theorem for nonrectangular regions R, we can con-sider the function fðx; yÞ, that is defined in the rectangular region W such that R �W and

fðx; yÞ ¼Fðx; yÞ; ðx; yÞ 2 R;

0; ðx; yÞ 2W=R:

ð97Þ

As a result, we define a fractional double integral over the nonrectangular region R, through thefractional double integral over the rectangular region W:

IaR½x; y� Fðx; yÞ ¼ Ia

W ½x; y� fðx; yÞ: ð98Þ

To define double integrals over nonrectangular regions, we can use a fairly general method to calculatethem. For example, we can do this for special regions called elementary regions. Let R be a set of allpoints ðx; yÞ such that

a 6 x 6 b; u1ðxÞ 6 y 6 u2ðxÞ:

Then, the double integrals for such regions can be calculated by

IaR½x; y� Fðx; yÞ ¼ aIa

b½x�u1ðxÞIau2ðxÞ½y� Fðx; yÞ: ð99Þ

It is easy to consider the following examples.

(1) u1ðxÞ ¼ 0, y ¼ u2ðxÞ ¼ x2, Fðx; yÞ ¼ xþ y.(2) u1ðxÞ ¼ x3, u1ðxÞ ¼ x2, Fðx; yÞ ¼ xþ y.(3) u1ðxÞ ¼ 0, y ¼ u2ðxÞ ¼ x, Fðx; yÞ ¼ xy.

The fractional integrals can be calculated by using the relations

aIax½x�ðx� aÞb ¼ Cðbþ 1ÞCðbþ aþ 1Þ ðx� aÞbþa

; ð100Þ

where a > 0, b > 0. For other relations see Table 9.1 in [2]. To calculate the Caputo derivatives, we canuse this table and the equation

aCDa

x x0½ �f x0ð Þ ¼ aDax x0½ �f x0ð Þ �

Xn�1

k¼0

f ðkÞðaÞCðk� aþ 1Þ ; n� 1 < a 6 n: ð101Þ


Note that the Mittag–Leffler function Ea½ðx0 � aÞa� is not changed by the Caputo derivative

aCDa

x x0½ � Ea ðx0 � aÞa½ � ¼ Ea ðx� aÞa½ �: ð102Þ

This equation is a fractional analog of the well-known property of exponential function of the formD1

x expðx� aÞ ¼ expðx� aÞ. Therefore the Mittag–Leffler function can be considered as a fractional ana-log of exponential function.

7. Fractional Stokes’ formula

We shall restrict ourselves to the consideration of a simple surface. If we denote the boundary ofthe simple surface W by oW and if F is a smooth vector field defined on W, then the Stokes’ theoremasserts that
Z
oWðF;dLÞ ¼

ZWðcurlF;dSÞ: ð103Þ

The right-hand side of this equation is the surface integral of curlF over W, whereas the left-hand sideis the line integral of F over the line oW . Thus the Stokes’ theorem is the assertion that the line integralof a vector field over the boundary of the surface W is the same as the integral over the surface of thecurl of F.

For Cartesian coordinates, Eq. (103) givesZ ZZ
oW
Fxdxþ Fydyþ Fzdz� �

¼W

dydz DyFz � DzFy� �

þ dzdx DzFx � DxFz½ � þ dxdy DxFy � DyFx� ��

:

ð104Þ

Let F ¼ Fðx; y; zÞ be a vector field such that
Fðx; y; zÞ ¼ e1Fxðx; y; zÞ þ e2Fyðx; y; zÞ þ e3Fzðx; y; zÞ:
where Fx, Fy, Fx are absolutely continuous (or continuously differentiable) real-valued functions on R3.Then the fractional generalization of the Stokes’ formula (104) can be written as

IaoW ; F

� �¼ Ia

W ;CurlaoW F� �

: ð105Þ

Here we use the notations
Ia
L ¼ IaoW ¼ emIaoW xm½ � ¼ e1IaoW ½x� þ e2IaoW ½y� þ e3IaoW ½z�; ð106Þ

such that

IaoW ; F

� �¼ IaoW ½x�Fx þ IaoW ½y�Fy þ IaoW ½z�Fz: ð107Þ

The integral (106) can be considered as a fractional line integral.In the right-hand side of (105), Ia

W is a fractional surface integral over S ¼W such that

IaS ¼ Ia

W ¼ e1IaW ½y; z� þ e2IaW ½z; x� þ e3IaW ½x; y�: ð108Þ

The fractional curl operation is

CurlaW F ¼ elelmnCDa

W xm½ �Fn

¼ e1CDa


� �þþe2

CDaW ½z�Fx � CDa

W ½x�Fz� �

þ e3CDa


� �: ð109Þ

For a ¼ 1, Eq. (109) gives the well-known expression

Curl1W F ¼ curlF ¼ elelmnDxm Fn ¼ e1ðDyFz � DzFyÞ þ þe2ðDzFx � DxFzÞ þ e3ðDxFy � DyFxÞ: ð110Þ

The right-hand side of Eq. (105) means

IaW ;CurlaW F

� �¼ IaW ½y; z� CDa


� �þþIaW ½z; x� CDa

W ½z�Fx � CDaW ½x�Fz

� �þ IaW ½x; y� CDa


� �: ð111Þ

This integral can be considered as a fractional surface integral.


8. Fractional Gauss’s formula

Let us give the basic theorem regarding the Gauss’s formula in a fractional case.

Theorem. (Fractional Gauss’s Theorem for a Parallelepiped)Let Fxðx; y; zÞ, Fyðx; yÞ, Fzðx; y; zÞ be continuously differentiable real-valued functions in a domain that

includes the parallelepiped

W :¼ fðx; y; zÞ : a 6 x 6 b; c 6 y 6 d; g 6 z 6 hg: ð112Þ

If the boundary of W be a closed surface oW, then

IaoW ; F

� �¼ IaW Diva

W F: ð113Þ

This equation can be called the fractional Gauss’s formula.

Proof. For Cartesian coordinates, we have the vector field F ¼ Fxe1 þ Fye2 þ Fze3, and

IaW ¼ IaW ½x; y; z�; IaoW ¼ e1IaoW ½y; z� þ e2IaoW ½x; z� þ e3IaoW ½x; y�: ð114Þ

Then

IaoW ; F

� �¼ IaoW ½y; z�Fx þ IaoW ½x; z�Fy þ IaoW ½x; y�Fz; ð115Þ

and

IaW DivaW F ¼ IaW ½x; y; z� CDa

oW ½x�Fx þ CDaoW ½y�Fy þ CDa

oW ½z�Fz� �

: ð116Þ

If W is the parallelepiped

W :¼ fa 6 x 6 b; c 6 y 6 d; g 6 z 6 hg; ð117Þ

then the integrals (114) are

IaW ½x; y; z� ¼ aIab½x�cIad½y� gIah½z�; ð118Þ

and

IaoW ½y; z� ¼ cIad½y�gIah½z�; ð119ÞIaoW ½x; z� ¼ aIab½x�gIah½z�; ð120ÞIaoW ½x; y� ¼ aIab½x�cIad½y�: ð121Þ

As a result, we can realize the following transformations

IaoW ; F�

� �¼ IaoW ½y; z�Fx þ IaoW ½z; x�Fy þ IaoW ½x; y�Fz

¼ cIad½y�gIah½z� Fxðb; y; zÞ � Fxða; y; zÞf g þ aIab½x�gIah½z� Fyðx; d; zÞ � Fyðx; c; zÞ�

þ aIab½x�cIad½y� Fzðx; y; gÞ � Fzðx; y;hÞf g

¼ aIab½x�cIad½y�gIah½z� aCDa

x x0½ �Fx x0; y; zð Þ þ cCDa

y y0½ �Fy x; y0; zð Þ þ gCDa

z z0½ �Fz x; y; z0ð Þn o

¼ IaWCDa

W ; F� �

¼ IaW DivaW F:

This ends the proof of the fractional Gauss’s formula for parallelepiped region.

Remark.To define the triple integral and the theorem for non-parallelepiped regions R, we consider the

function f ðx; y; zÞ, that is defined in the parallelepiped region W such that R �W such that

f ðx; y; zÞ ¼Fðx; y; zÞ; ðx; y; zÞ 2 R;

0; ðx; y; zÞ 2W=R:

ð122Þ


Then we have

IaR½x; y; z�Fðx; y; zÞ ¼ Ia

W ½x; y; z�f ðx; y; zÞ: ð123Þ

As a result, we define a fractional triple integral over the non-parallelepiped region R, through the frac-tional triple integral over the parallelepiped region W.

9. Fractional differential forms

9.1. Brief description of different approaches

A fractional generalization of differential has been presented by Ben Adda in [16,17]. A fractionalgeneralization of the differential forms has been suggested by Cottrill-Shepherd and Naber in [22](see also [23,24]). The application of fractional differential forms to dynamical systems are consideredin [26,27]. Fractional integral theorems are not considered.

(1) In the papers [16,17], the fractional differential for analytic functions is defined by

daf ¼ 1Cð1þ aÞdxjN

axj

f ðxÞ; ð124Þ

where Naxj

are Nishimoto fractional derivatives [8] (see also Section 22 of [2]), which is a generalizationof the Cauchy’s differentiation formula.

(2) In the paper [22] (see also [23,26,24]), an exterior fractional differential is defined through theRiemann–Liouville derivatives by

da ¼Xn

j¼1

ðdxjÞa0Daxj: ð125Þ

In two dimensions n ¼ 2,

da ¼ ðdxÞa0Dax þ ðdyÞa0Da

y;

such that

dax ¼ ðdxÞa x1�a

Cð2� aÞ þ ðdyÞa xy�a

Cð1� aÞ ; ð126Þ

day ¼ ðdxÞa yx�a

Cð1� aÞ þ ðdyÞa y1�a


where we use

0Daxj

1 ¼x�a

j

Cð1� aÞ : ð128Þ

(3) In the paper [27], an exterior fractional differential is defined through the fractional Caputo deriv-atives in the form

da ¼Xn

j¼1

ðdxjÞa0CDa

xj: ð129Þ

For two dimensions ðx; yÞ, we have

da ¼ ðdxÞa0CDa

x þ ðdyÞa0CDa

y;

such that

dax ¼ ðdxÞa x1�a


day ¼ ðdyÞa y1�a



(compare with (126) and (127)). Eq. (129) can be rewritten as

da ¼Xn

j¼1

Cð2� aÞxa�1j daxj0

CDaxj: ð132Þ

This relation is used in [27] as a fractional exterior differential.

9.2. Definition of a fractional exterior differential

A definition of fractional differential forms must be correlated with a possible generalization of thefractional integration of differential forms. To derive fractional analogs of differential forms and itsintegrals, we consider a simplest case that is an exact 1-form on the interval L ¼ ½a; b�. It allows usto use the fractional Newton–Leibniz formula.

In order to define an integration of fractional differential forms, we can use the fractional Riemann–Liouville integrals. Then a fractional exterior derivative must be defined through the Caputo fractionalderivative.

Eq. (67) of FTFC means that
Z x
a

dx0

CðaÞ x� x0ð Þ1�a aCDa

x0 x00½ �f x00ð Þ ¼ f ðxÞ � f ðaÞ; ð0 < a < 1Þ: ð133Þ

Using

dx0 ¼ dx0ð Þ1�a dx0ð Þa; ð0 < a < 1Þ;

Eq. (133) can be presented in the form
Z x
a

dx0ð Þ1�a

CðaÞ x� x0ð Þ1�adx0ð Þaa

CDax0 x00½ �f x00ð Þ

� �¼ f ðxÞ � f ðaÞ; ð0 < a < 1Þ: ð134Þ

The expression in the big brackets of (134) can be considered as a fractional differential of the functionf ðxÞ. As a result, we have
bIa
L½x�aDaxf ðxÞ ¼ f ðbÞ � f ðaÞ; ð0 < a < 1Þ; ð135Þ

where L ¼ ½a; b�, and the fractional integration for differential forms is defined by the operator

bIaL½x� :¼

Z b

a

ðdxÞ1�a

CðaÞðb� xÞ1�a: ð136Þ

The exact fractional differential 0-form is a fractional differential of the function

aDaxf ðxÞ :¼ ðdxÞaa

CDax x0½ �f x0ð Þ: ð137Þ

Eq. (135) can be considered as a fractional generalization of the integral for differential 1-form.As a result, the fractional exterior derivative is defined as

aDax :¼ dxmð Þaa

CDaxm

x0m� �

: ð138Þ

Then the fractional differential 1-form is

xðaÞ ¼ dxmð Þa FmðxÞ: ð139Þ

The exterior derivative of this form gives

aDaxxðaÞ ¼ dxmð Þa ^ dxnð Þaa

CDaxn

x0½ �Fm x0ð Þ: ð140Þ

To prove the proposition (140), we use the rule

DaxðfgÞ ¼

X1s¼0

akð Þ a

CDa�sx f

� �Ds

xg;

and the relation [5]


Ds½x� dxð Þa ¼ 0 ðs P 1Þ;

for integer s, where

akð Þ ¼

ð�1Þk�1aCðk� aÞCð1� aÞCðkþ 1Þ :

For example, we have

da ðdxmÞaFm½ � ¼X1s¼0

dxnð Þa ^ akð Þ aDa�s

xnx0n� �

Fm x0ð Þ� �

Ds xn½ � dxmð Þa ¼ dxnð Þa ^ dxmð Þa a0ð ÞDa

xnx0n� �

Fm x0ð Þ

¼ Daxn½x0n�Fm x0ð Þ

� �dxnð Þa ^ dxmð Þa:

Using the equation (see Property 2.16 in [5])

aDax x0½ � x0 � að Þb ¼ Cðbþ 1Þ

Cðbþ 1� aÞ ðx� aÞb�a; ð141Þ

where n� 1 < a < n, and b > n, and

aDax x0½ � x0 � að Þk ¼ 0 ðk ¼ 0;1;2; . . . ;n� 1Þ; ð142Þ

we obtain

adaxðx� aÞa ¼ ðdxÞaa

CDax x0½ �x0 ¼ ðdxÞa Cðaþ 1Þ: ð143Þ

Then

ðdxÞa ¼ 1Cðaþ 1Þ ada

xðx� aÞa; ð144Þ

and the fractional exterior derivative (138) is presented as

adax :¼ 1

Cðaþ 1Þ adax xm � amð Þaa

CDaxm

x0m� �

: ð145Þ

The fractional differential 1-form (139) can be written as

xðaÞ ¼ 1Cðaþ 1Þ ada

x xm � amð ÞaFmðxÞ: ð146Þ

RemarkUsing the suggested definition of fractional integrals and differential forms, it is possible to define a

fractional integration of n-form over the hypercube ½0;1�n. Unfortunately, a generalization of this frac-tional integral, which uses the mapping / of the region W � Rn into ½0;1�n, has a problem. For the inte-ger case, we use the equation

D1x f ð/ðxÞÞ ¼ D1

/f� �

D1x/

� �: ð147Þ

For the fractional case, the chain rule for differentiation (the fractional derivative of composite func-tions) is more complicated (see Section 2.7.3. [4]). As a result, a consistent definition of fractional inte-gration of differential form for arbitrary manifolds is an open question.

9.3. Differential vector operations through the differential forms

To define a fractional divergence of the field F, we can consider the 2-form

x2 ¼ Fzdx ^ dy þ Fydz ^ dxþ Fx dy ^ dz: ð148Þ

Then the fractional exterior derivative of this form is

dx2 ¼ DxFx þ DyFy þ DzFz� �

dx ^ dy ^ dz ¼ divFdx ^ dy ^ dz: ð149Þ


To define a fractional generalization of the curl operation for F, we can use the 1-form

x1 ¼ Fxdxþ Fy dy þ Fzdz: ð150Þ

Then the fractional exterior derivative of this 1-form is

dx1 ¼ DxFy � DyFx� �

dx ^ dy þ DyFz � DzFy� �

dy ^ dzþ DxFz � DzFxð Þdx ^ dz: ð151Þ

To define the fractional gradient, we consider the 0-form

x0 ¼ f ðx; y; zÞ ð152Þ

Then the fractional exterior derivative of f gives

dx0 ¼ Dxf dxþ Dyf dy þ Dzf dz ¼X3

k¼1

ðgrad f Þkdxk: ð153Þ

It is not hard to obtain fractional generalizations of these definitions.

10. Fractional nonlocal Maxwell’s equations

10.1. Local Maxwell’s equations

The behavior of electric fields (E;D), magnetic fields (B;H), charge density (qðt; rÞ), and current den-sity (jðt; rÞ) is described by the Maxwell’s equations

divDðt; rÞ ¼ qðt; rÞ; ð154ÞcurlEðt; rÞ ¼ �otBðt; rÞ; ð155ÞdivBðt; rÞ ¼ 0; ð156ÞcurlHðt; rÞ ¼ jðt; rÞ þ otDðt; rÞ: ð157Þ

Here r ¼ ðx; y; zÞ is a point of the domain W. The densities qðt; rÞ and jðt; rÞ describe an external sources.We assume that the external sources of electromagnetic field are given.

The relations between electric fields (E;D) for the medium can be realized by

Dðt; rÞ ¼ e0

Z þ1

�1e r; r0ð ÞE t; r0ð Þdr0; ð158Þ

where e0 is the permittivity of free space. Homogeneity in space gives eðr; r0Þ ¼ eðr � r0Þ. Eq. (158)means that the displacement D is a convolution of the electric field E at other space points. A local casecorresponds to the Dirac delta-function permittivity eðrÞ ¼ edðrÞ. Then Eq. (158) givesDðt; rÞ ¼ e0eEðt; rÞ.

Analogously, we have nonlocal equation for the magnetic fields (B;H).

10.2. Caputo derivative in electrodynamics

Let us demonstrate a possible way of appearance of the Caputo derivative in the classical electro-dynamics. If we have

Dðt; xÞ ¼Z þ1

�1e x� x0ð ÞE t; x0ð Þdx0; ð159Þ

then

D1x Dðt; xÞ ¼

Z þ1

�1D1

x eðx� x0Þh i

E t; x0ð Þdx0 ¼ �Z þ1

�1D1

x0 e x� x0ð Þh i

E t; x0ð Þdx0: ð160Þ

Using the integration by parts, we get

D1x Dðt; xÞ ¼

Z þ1

�1e x� x0ð ÞD1

x0E t; x0ð Þdx0: ð161Þ


Consider the kernel eðx� x0Þ of integral (161) in the interval ð0; xÞ such that

eðx� x0Þ ¼e x� x0ð Þ; 0 < x0 < x;

0; x0 > x; x0 < 0;

ð162Þ

with the power-like function

e x� x0ð Þ ¼ 1Cð1� aÞ

1x� x0ð Þa ; ð0 < a < 1Þ: ð163Þ

Then Eq. (161) gives the relation

D1x Dðt; xÞ ¼ 0

CDaxEðt; xÞ; ð0 < a < 1Þ ð164Þ

with the Caputo fractional derivatives 0CDa

x.

10.3. Fractional nonlocal Maxwell’s equations

Fractional nonlocal differential Maxwell’s equations have the form

Diva1W Eðt; rÞ ¼ g1qðt; rÞ; ð165Þ

Curla2W Eðt; rÞ ¼ �otBðt; rÞ; ð166Þ

Diva3W Bðt; rÞ ¼ 0; ð167Þ

g2Curla4W Bðt; rÞ ¼ jðt; rÞ þ g�1

3 otEðt; rÞ; ð168Þ

where as, (s ¼ 1;2;3;4), can be integer or fractional.Fractional integral Maxwell’s equations, which use integrals of noninteger orders, have been sug-

gested in [25] to describe fractional distributions of electric charges and currents.In the general form, the fractional integral Maxwell’s equations can be presented in the form

Ia1oW ;Eðt; rÞ

� �¼ g1Ia1

Wqðt; rÞ; ð169Þ

Ia2oS ;Eðt; rÞ

� �¼ � d

dtðIa2

S ;Bðt; rÞÞ; ð170Þ

Ia3oW ;Bðt; rÞ

� �¼ 0; ð171Þ

g2 Ia4oS ;Bðt; rÞ

� �¼ Ia4

S ; jðt; rÞ� �

þ g�13

ddt

Ia4S ;Eðt; rÞ

� �: ð172Þ

These fractional differential and integral equations can be used to describes an electromagnetic field ofmedia that demonstrate fractional nonlocal properties. The suggested equations can be considered asa special case of nonlocal electrodynamics (see [31–35]).

Fractional coordinate derivatives are connected with nonlocal properties of the media. For exam-ple, a power-law long-range interaction in the 3-dimensional lattice in the continuous limit can givea fractional equation [40].

10.4. Fractional conservation law for electric charge

Let us derive a conservation law equation for density of electric charge in the region W from thefractional nonlocal Maxwell’s equations.

The time derivative of (169) is

Diva1WotEðt; rÞ ¼ g1otqðt; rÞ: ð173Þ

Substitution of (168) into (173) gives

g3Diva1W g2Curla4

W Bðt; rÞ � jðt; rÞ� �

¼ g1otqðt; rÞ: ð174Þ

If a1 ¼ a4, then

Diva1W Curla4

W Bðt; rÞ ¼ 0; ð175Þ

and we have the law

g1otqðt; rÞ þ g3Diva1W jðt; rÞ ¼ 0: ð176Þ


This fractional equation is a differential form of charge conservation law for fractional nonlocalelectrodynamics.

If a1 ¼ a4, we can define the fractional integral characteristics such as

QWðtÞ ¼ g1Ia1W ½x; y; z�qðt; x; y; zÞ; ð177Þ

which can be called the total fractional nonlocal electric charge, and

JoW ðtÞ ¼ g3 Ia1oW ; j

� �¼ g3 Ia1

oW ½y; z�jx þ Ia1oW ½z; x�jy þ Ia1

oW ½x; y�jz

� �ð178Þ

is a fractional nonlocal current. Then the fractional nonlocal conservation law is

ddt

QWðtÞ þ JoWðtÞ ¼ 0: ð179Þ

This integral equation describes the conservation of the electric charge in the nonlocal electrodynam-ics for the case a1 ¼ a4.

10.5. Fractional waves

Let us derive wave equations for electric and magnetic fields in a region W from the fractional non-local Maxwell’s equations with j ¼ 0 and q ¼ 0.

The time derivative of Eq. (166) is

o2t B ¼ �Curla2

WotE ð180Þ

Substitution of (172) and j ¼ 0 into (180) gives

o2t B ¼ �g2g3Curla2

W Curla4W Bðt; rÞ: ð181Þ

Using (80) and (167) for a2 ¼ a3 ¼ a4, we get

o2t B ¼ g2g3ðCDa

WÞ2B: ð182Þ

As a result, we obtain

o2t B� v2ðCDa

WÞ2B ¼ 0; ð183Þ

where v2 ¼ g2g3. This is the fractional wave equation for the magnetic field B. Analogously, Eqs. (166)and (172) gives the fractional wave equation for electric field

o2t E� v2ðCDa

WÞ2E ¼ 0: ð184Þ

The solution Bðt; rÞ of Eq. (183) is a linear combination of the solutions Bþðt; rÞ and B�ðt; rÞ of theequations

otBþðt; rÞ � vCDaW Bþðt; rÞ ¼ 0; ð185Þ

otB�ðt; rÞ þ vCDaW B�ðt; rÞ ¼ 0: ð186Þ

As a result, we get the fractional extension of D’Alembert expression that is considered in [36].For the boundary conditions

limjtj!1

Bðt; rÞ ¼ 0; Bðt;0Þ ¼ GðtÞ; ð187Þ

the general solution of Eqs. (185) and (186) is given [5] by

Bm�ðt; rÞ ¼1

2p

Z þ1

�1dxEa;1 �ivxxa

m

� � eGmðxÞe�ixt ; ð188Þ

where eGmðxÞ ¼ F ½GmðtÞ�, and Ea;b½z� is the biparametric Mittag–Leffler function [5]. Here B�mðt; rÞ, andGmðtÞ are components of B�ðt; rÞ and GðtÞ. For one-dimensional case, Bxðx; y; z; tÞ ¼ uðx; tÞ, By ¼ Bz ¼ 0,and we can consider the fractional partial differential equation

D2t uðx; tÞ � v2

0D2ax uðx; tÞ ¼ 0; x 2 R; x > 0; v > 0; ð189Þ


with the conditions

Dkxuð0; tÞ ¼ fkðtÞ; ð190Þ

where k ¼ 0 for 0 < a 6 1=2, and k ¼ 1 for 1=2 < a 6 1. If 0 < 2a < 2 and v > 0, the system of Eqs.(189) and (190) is solvable (Theorem 6.3. of [5]), and the solution uðx; tÞ is given by

uðx; tÞ ¼Xn�1

k¼0

Z þ1

�1G2a

k ðx� y; tÞfkðyÞdy; n� 1 < a 6 nð Þ; ð191Þ

where

G2ak ðx; tÞ ¼

12

vxk�a/ �a; kþ 1� a; vjtjx�að Þ: ð192Þ

Here /ð�a; kþ 1� a; vjtjx�aÞ is the Wright function [5].Note that the solutions of equations as (185) and (186) are based primary on the use of Laplace

transforms for equations with the Caputo 0CDa

x derivatives. This leaves certain problems [5] withthe fractional derivatives a

CDax for a 2 R.

11. Conclusions

Let us note some possible extensions of the fractional vector calculus.

(1) It is very important to prove the suggested fractional integral theorems for a general form ofdomains and boundaries.

(2) It is interesting to generalize the formulations of fractional integral theorems for a > 1.(3) A proof of fractional theorems for differential forms can be interesting to formulate a fractional

generalization of differential geometry.

In the fundamental theorem of fractional calculus (FTFC) we use the Riemann–Liouville integrationand the Caputo differentiation. The main property is that the Caputo fractional derivative provides usan operation inverse to the Riemann–Liouville fractional integration from the left. Note that a frac-tional generalization of the differential vector operations and the integral theorems for the fractionalintegro-differentiation of Riesz, Grunvald-Letnikov, Weyl, Nishimoto is an open problem.

There are the following possible applications of the fractional variational calculus (FVC).

(a) A fractional nonlocal electrodynamics that is characterized by the power law non-locality canbe formulated by using the FVC.

(b) Nonlocal properties in classical dynamics can be described by the FVC and by possible fractionalgeneralizations of symplectic geometry and Poisson algebra. In general, fractional differentialforms and fractional integral theorems for these forms can be used to describe classicaldynamics.

(c) A possible dynamics of fractional gradient and Hamiltonian dynamical systems can be describedby the FVC.

(d) The continuum mechanics of fluids and solids with nonlocal properties (with a nonlocal inter-action of medium particles) can be described by the FVC.

The fractional derivatives in equations can be connected with a long-range power-law interactionof the systems [37,38,40]. The nonlocal properties of electrodynamics can be considered [39] as a re-sult of dipole–dipole interactions with a fractional power-law screening that is connected with theintegro-differentiation of non-integer order. For noninteger derivatives with respect to coordinates,we have the power-like tails as the important property of the solutions of the fractional equations.

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Annals of Physicstarasov/PDF/AP2008.pdfV.E. Tarasov/Annals of Physics 323 (2008) 2756–2778 2757. where Cðaþ1Þ is the Gamma function. In these deﬁnitions, the Nishimoto fractional