A fractional calculus of variations for multiple integrals with application to vibrating string

A fractional calculus of variations for multiple integrals

with application to vibrating string

Ricardo Almeida,1,a兲Agnieszka B. Malinowska,1,2,b兲and Delfim F. M. Torres1,c兲

1_{Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal} 2_{Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok,}

Poland

共Received 4 December 2009; accepted 14 January 2010; published online 5 March 2010兲 We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string. © 2010 American Institute of Physics. 关doi:10.1063/1.3319559兴

I. INTRODUCTION

The fractional calculus 共FC兲 is one of the most interdisciplinary fields of mathematics, with many applications in physics and engineering. The history of FC goes back more than three centuries, when in 1695 the derivative of the order of ␣= 1/2 was described by Leibniz. Since then, many different forms of fractional operators were introduced: the Grunwald–Letnikov, Riemann–Liouville, Riesz, and Caputo fractional derivatives,31,35,39and the more recent notions of Klimek,32Cresson,13and Jumarie.29,28,26,30

FC is nowadays the realm of physicists and mathematicians, who investigate the usefulness of such noninteger order derivatives and integrals in different areas of physics and mathematics.11,23,31 It is a successful tool for describing complex quantum field dynamical sys-tems, dissipation, and long-range phenomena that cannot be well illustrated using ordinary differ-ential and integral operators.19,23,32,38Applications of FC are found, e.g., in classical and quantum mechanics, field theories, variational calculus, and optimal control.20,22,26

Although FC is an old mathematical discipline, the fractional vector calculus is at the very beginning. We mention the recent paper,42 where some fractional versions of the classical results of Green, Stokes, and Gauss are obtained via Riemann–Liouville and Caputo fractional operators. For the purposes of a multidimensional fractional calculus of variations 共FCVs兲, the Jumarie fractional integral and derivative seems, however, to be more appropriate.

The FCV started in 1996 with the work of Riewe.38 Riewe formulated the problem of the calculus of variations with fractional derivatives and obtained the respective Euler–Lagrange equations, combining both conservative and nonconservative cases. Nowadays, the FCV is a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. Investigations cover problems depending on Riemann–Liouville fractional derivatives 共see, e.g., Refs. 5, 19, and21兲, the Caputo fractional derivative共see, e.g., Refs.1,7, and8兲, the symmetric fractional derivative 共see, e.g., Ref.32兲, the

a兲_{Electronic mail: ricardo.almeida@ua.pt.} b兲_{Electronic mail: abmalinowska@ua.pt.} c兲_{Electronic mail: delfim@ua.pt.}

51, 033503-1

Jumarie fractional derivative共see, e.g., Refs.3 and26兲, and others.2,13,20 For applications of the FCV, we refer the reader to Refs.17,16,19,26,32,36,37, and40. Although the literature of FCV is already vast, much remains to be done.

Knowing the importance and relevance of multidimensional problems of the calculus of variations in physics and engineering,43it is at a first view surprising that a multidimensional FCV is a completely open research area. We are only aware of some preliminary results presented in Ref. 19, where it is claimed that an appropriate fractional variational theory involving multiple integrals would have important consequences in mechanical problems involving dissipative sys-tems with infinitely many degrees of freedom, but where a formal theory for that is missing. There is, however, a good reason for such omission in literature: most of the best well-known fractional operators are not suitable for a generalization of the FCV to the multidimensional case due to lack of good properties, e.g., an appropriate Leibniz rule.

The main aim of the present work is to introduce a FCV for multiple integrals. For that we make use of the recent Jumarie fractional integral and derivative,29,28,30extending such notions to the multidimensional case. The main advantage of using Jumarie’s approach lies in the following facts: the Leibniz rule for the Jumarie fractional derivative is equal to the standard one and, as we show, the fractional generalization of some fundamental multidimensional theorems of calculus is possible. We mention that Jumarie’s approach is also useful for the one-dimensional FCV, as recently shown in Ref.3 共see also Ref.27兲.

The plan of the paper is as follows. In Sec. II some basic formulas of Jumarie’s FC are briefly reviewed. Then, in Sec. III the differential and integral vector operators are introduced, and fractional Green’s, Gauss’s, and Stokes’ theorems formulated. Section IV is devoted to the study of problems of FCVs with multiple integrals. Our main results provide Euler–Lagrange necessary optimality type conditions for such problems 共Theorems 3 and 5兲 as well as natural boundary conditions共Theorem 4兲. We end with Sec. V of applications and future perspectives.

II. PRELIMINARIES

For an introduction to the classical FC, we refer the reader to Refs.31,34,35, and39. In this section we briefly review the main notions and results from the recent FC proposed by Jumarie.28,26,30 Let f :关0,1兴→R be a continuous function and␣苸共0,1兲. The Jumarie fractional derivative of f may be defined by

f共␣兲共x兲 = 1 ⌫共1 −␣兲 d dx

冕

0 x 共x − t兲−␣_{共f共t兲 − f共0兲兲dt. 共1兲}

One can obtain共1兲as a consequence of a more basic definition, a local one, in terms of a fractional finite difference关cf. Eq. 共2.2兲 in Ref.30兴,

f共␣兲共x兲 = lim h→0 1 h␣

兺

_k=0 ⬁ 共− 1兲k

冉

␣ k

冊

f共x + 共␣− k兲h兲.

Note that the Jumarie and the Riemann–Liouville fractional derivatives are equal if f共0兲=0. The advantage of definition共1兲with respect to the classical definition of Riemann–Liouville is that the fractional derivative of a constant is now zero, as desired. An antiderivative of f, called the共dt兲␣ integral of f, is defined by

冕

0 x f共t兲共dt兲␣=␣

冕

0 x 共x − t兲␣−1_f共t兲dt.

The following equalities can be considered as fractional counterparts of the first and the second fundamental theorems of calculus and can be found in Refs.26and30,

d␣ dx␣

冕

₀ x f共t兲共dt兲␣=␣!f共x兲, 共2兲

冕

0 x f共␣兲共t兲共dt兲␣=␣!共f共x兲 − f共0兲兲, 共3兲

where␣!ª⌫共1+␣兲. The Leibniz rule for the Jumarie fractional derivative is equal to the standard one,

共f共x兲g共x兲兲共␣兲_{= f}共␣兲_{共x兲g共x兲 + f共x兲g}共␣兲_共x兲.

Here, we see another advantage of derivative共1兲: the fractional derivative of a product is not an infinite sum, in contrast to the Leibniz rule for the Riemann–Liouville fractional derivative共Ref. 35, p. 91兲.

One can easily generalize the previous definitions and results for functions with a domain 关a,b兴, f共␣兲共x兲 = 1 ⌫共1 −␣兲 d dx

冕

a x 共x − t兲−␣_{共f共t兲 − f共a兲兲dt} and

冕

a x f共t兲共dt兲␣=␣

冕

a x 共x − t兲␣−1_f共t兲dt.

III. FRACTIONAL INTEGRAL THEOREMS

In this section we introduce some useful fractional integral and fractional differential opera-tors. With them we prove fractional versions of the integral theorems of Green, Gauss, and Stokes. Throughout the text we assume that all integrals and derivatives exist.

A. Fractional operators

Let us consider a continuous function f = f共x₁, . . . , xn兲 defined on R=⌸i=1 n _关a

i, bi兴傺Rn. Let us

extend Jumarie’s fractional derivative and the共dt兲␣ integral to functions with n variables. For xi

苸关ai, bi兴, i=1, ... ,n, and␣苸共0,1兲, we define the fractional integral operator as

a_iIx␣_i关i兴 =␣

冕

a_i

x_i

共xi− t兲␣−1dt.

These operators act on f in the following way:

a_iIx␣_i关i兴f共x1, . . . ,xn兲 =␣

冕

a_i

x_i

f共x1, . . . ,xi−1,t,xi+1, . . . ,xn兲共xi− t兲␣−1dt, i = 1, . . . ,n.

Let⌶=兵k1, . . . , ks其 be an arbitrary nonempty subset of 兵1, ... ,n其. We define the fractional multiple

integral operator over the region R_⌶=⌸_i=1s 关ak_i, xk_i兴 by

IR␣_⌶关k1, . . . ,ks兴 =a_k 1 Ix_k 1 ␣ _关k 1兴 ¯a_k s Ix_k s ␣ _关k s兴 =␣s

冕

a_k 1 x_k 1 ¯

冕

a_k s x_k s 共xk₁− tk₁兲␣−1¯ 共xk_s− tk_s兲␣−1dtk_s¯ dtk₁, which acts on f by

IR␣_⌶关k1, . . . ,ks兴f共x1, . . . ,xn兲 =␣s

冕

a_k 1 x_k 1 ¯

冕

a_k s x_k s f共␰1, . . . ,␰n兲共xk₁− tk₁兲␣−1¯ 共xk_s− tk_s兲␣−1dtk_s¯ dtk₁,

where␰j= tjif j苸⌶, and␰j= xj if j苸⌶, j=1, ... ,n. The fractional volume integral of f over the

whole domain R is given by

IR␣f =␣n

冕

a₁ b₁ ¯

冕

a_n b_n f共t₁, . . . ,tn兲共b1− t1兲␣−1¯ 共bn− tn兲␣−1dtn¯ dt1.

The fractional partial derivative operator with respect to the ith variable xi, i = 1 , . . . , n, of the

order of␣苸共0,1兲, is defined as follows:

a_iDx␣_i关i兴 = 1 ⌫共1 −␣兲 ⳵ ⳵xi

冕

a_i x_i 共xi− t兲−␣dt, which acts on f by a_iDx␣_i关i兴f共x1, . . . ,xn兲 = 1 ⌫共1 −␣兲 ⳵ ⳵xi

冕

a_i x_i 共xi− t兲−␣关f共x1, . . . ,xi−1,t,xi+1, . . . ,xn兲 − f共x1, . . . ,xi−1,ai,xi+1, . . . ,xn兲兴dt, i = 1, ... ,n.

We observe that the Jumarie fractional integral and the Jumarie fractional derivative can be obtained by putting n = 1, aIx␣关1兴f共x兲 =␣

冕

a x 共x − t兲␣−1_{f共t兲dt =}

冕

a x f共t兲共dt兲␣ and aDx␣关1兴f共x兲 = 1 ⌫共1 −␣兲 d dx

冕

a x 共x − t兲−␣_{共f共t兲 − f共a兲兲dt = f}共␣兲_共x兲.

Using these notations, formulas共2兲 and共3兲 can be presented as

aDx␣关1兴aIx␣关1兴f共x兲 =␣!f共x兲,

aIx␣关1兴aDx␣关1兴f共x兲 =␣!共f共x兲 − f共a兲兲. 共4兲

In the two dimensional case, we define the fractional line integral on⳵R, R =关a,b兴⫻关c,d兴, by I_⳵R␣ f = I_⳵R␣ 关1兴f + I_⳵R␣ 关2兴f , where I_⳵R␣关1兴f =aIb␣关1兴关f共b,c兲 − f共b,d兲兴 =␣

冕

a b 关f共t,c兲 − f共t,d兲兴共b − t兲␣−1_dt and I_⳵R␣关2兴f =cId␣关2兴关f共b,d兲 − f共a,d兲兴 =␣

冕

c d 关f共b,t兲 − f共a,t兲兴共d − t兲␣−1_dt.

B. Fractional differential vector operations

Let WX=关a,x兴⫻关c,y兴⫻关e,z兴, W=关a,b兴⫻关c,d兴⫻关e, f兴, and denote 共x1, x2, x3兲 by 共x,y,z兲.

We introduce the fractional nabla operator by

ⵜW␣_X= iaDx␣关1兴 + jcDy␣关2兴 + keDz␣关3兴,

where i, j, and k define a fixed right-handed orthonormal basis. If f :R3_{→R is a continuous}

function, then we define its fractional gradient as

GradW␣_Xf =ⵜW␣_Xf = iaDx␣关1兴f共x,y,z兲 + j cDy␣关2兴f共x,y,z兲 + keDz␣关3兴f共x,y,z兲.

If F =关Fx, Fy, Fz兴:R3→R3is a continuous vector field, then we define its fractional divergence and

fractional curl by

DivW␣_XF =ⵜW␣_Xⴰ F =aDx␣关1兴Fx共x,y,z兲 +cDy␣关2兴Fy共x,y,z兲 +eDz␣关3兴Fz共x,y,z兲

and

CurlW␣_XF =ⵜW␣_X⫻ F = i共cDy␣关2兴Fz共x,y,z兲 −eDz␣关3兴Fy共x,y,z兲兲 + j共eDz␣关3兴Fx共x,y,z兲

−aDx␣关1兴Fz共x,y,z兲兲 + k共aD␣x关1兴Fy共x,y,z兲 −cDy␣关2兴Fx共x,y,z兲兲.

Note that these fractional differential operators are nonlocal. Therefore, the fractional gradient, divergence, and curl depend on the region WX.

For F :R3_→R3_{and f , g :R}3_{→R, it is easy to check the following relations:}

共i兲 Div_W X ␣ _{共fF兲= f Div} W_X ␣ _{F + FⴰGrad} W_X ␣ _f,

共ii兲 CurlW␣_X共GradW␣_X f兲=关0,0,0兴,

共iii兲 DivW␣_X共CurlW␣_XF兲=0,

共iv兲 GradW␣_X共fg兲=g GradW␣_X f + f GradW␣_Xg, and

共v兲 Div_W X ␣ _共Grad W_X ␣ _f兲= aDx␣关1兴aDx␣关1兴f +cDy␣关2兴cDy␣关2兴f +eDz␣关3兴eDz␣关3兴f.

In general, let us recall that共DW␣_X兲2⫽DW2␣_X共see Ref. 30兲.

A fractional flux of the vector field F across⳵W is a fractional oriented surface integral of the field such that

共I_⳵W␣ ,F兲 = I_⳵W␣ 关2,3兴Fx共x,y,z兲 + I_⳵W␣ 关1,3兴Fy共x,y,z兲 + I_⳵W␣ 关1,2兴Fz共x,y,z兲,

where

I_⳵W␣ 关1,2兴f共x,y,z兲 =aIb␣关1兴cId␣关2兴关f共b,d, f兲 − f共b,d,e兲兴,

I_⳵W␣ 关1,3兴f共x,y,z兲 =aIb␣关1兴eI␣f关3兴关f共b,d, f兲 − f共b,c, f兲兴,

and

I_⳵W␣ 关2,3兴f共x,y,z兲 =cId␣关2兴eI␣f关3兴关f共b,d, f兲 − f共a,d, f兲兴.

C. Fractional theorems of Green, Gauss, and Stokes

We now formulate the fractional formulae of Green, Gauss, and Stokes. Analogous results via Caputo fractional derivatives and Riemann–Liouville fractional integrals were obtained by Tara-sov in Ref.42.

Theorem 1: (Fractional Green’s theorem for a rectangle) Let f and g be two continuous

I_⳵R␣ 关1兴f + I_⳵R␣ 关2兴g = 1 ␣!IR ␣_关 aDb␣关1兴g −cDd␣关2兴f兴. Proof: We have I_⳵R␣ 关1兴f + I_⳵R␣ 关2兴g =aIb␣关1兴关f共b,c兲 − f共b,d兲兴 +cId␣关2兴关g共b,d兲 − g共a,d兲兴. By Eq.共4兲, f共b,c兲 − f共b,d兲 = − 1 ␣!cId ␣_关2兴 cDd␣关2兴f共b,d兲, g共b,d兲 − g共a,d兲 = 1 ␣!aIb ␣_关1兴 aDb␣关1兴g共b,d兲. Therefore, I_⳵R␣ 关1兴f + I_⳵R␣ 关2兴g = −aIb␣关1兴 1 ␣!cId ␣_关2兴 cDd␣关2兴f共b,d兲 +cId␣关2兴 1 ␣!aIb ␣_关1兴 aDb␣关1兴g共b,d兲 = 1 ␣!IR ␣_关 aDb␣关1兴g −cDd␣关2兴f兴. 䊐 Theorem 2: (Fractional Gauss’s theorem for a parallelepiped) Let F =共Fx, Fy, Fz兲 be a

con-tinuous vector field in a domain that contains W =关a,b兴⫻关c,d兴⫻关e, f兴. If the boundary of W is a closed surface⳵W, then

共I_⳵W␣ ,F兲 = 1

␣!IW

␣ _Div

W

␣ _{F. 共5兲}

Proof: The result follows by direct transformations:

共I_⳵W␣ ,F兲 = I_⳵W␣ 关2,3兴Fx+ I_⳵W␣ 关1,3兴Fy+ I_⳵W␣ 关1,2兴Fz=cId␣关2兴eI␣f关3兴共Fx共b,d, f兲 − Fx共a,d, f兲兲 +aIb␣关1兴eI␣f关3兴共Fy共b,d, f兲 − Fy共b,c, f兲兲 +aIb␣关1兴cId␣关2兴共Fz共b,d, f兲 − Fz共b,d,e兲兲 = 1 ␣!aIb ␣_关1兴 cId␣关2兴eI␣f关3兴共aDb␣关1兴Fx共b,d, f兲 +cDd␣关2兴Fy共b,d, f兲 +eD␣f关3兴Fz共b,d, f兲兲 = 1 ␣!IW ␣_共 aDb␣关1兴Fx+cD␣d关2兴Fy+eDf␣关3兴Fz兲 = 1 ␣!IW ␣ _Div W ␣ _F. 䊐 Let S be an open, oriented, and nonintersecting surface bounded by a simple and closed curve

⳵S. Let F =关Fx, Fy, Fz兴 be a continuous vector field. Divide up S by sectionally curves into N

subregions S1, S2, . . . , SN. Assume that for small enough subregions, each Sjcan be approximated

by a plane rectangle Aj bounded by curves C1, C2, . . . , CN. Apply Green’s theorem to each

indi-vidual rectangle Aj. Then, summing over the subregions,

兺

_j 1 ␣!IAj ␣_共ⵜ A_j ␣ _{⫻ F兲 =}

兺

j I_⳵A j ␣ _F. Furthermore, letting N→⬁

兺

_j 1 ␣!IAj ␣_共ⵜ A_j ␣ _{⫻ F兲 →} 1 ␣!共IS ␣_,Curl S ␣_F兲, while

兺

_j I_⳵A j ␣ _{F→ I} ⳵S ␣_F.

We conclude with the fractional Stokes formula 1 ␣!共IS ␣_,Curl S ␣_{F兲 = I} ⳵S ␣_F.

IV. FCVs WITH MULTIPLE INTEGRALS

Consider a function w = w共x,y兲 with two variables. Assume that the domain of w contains the rectangle R =关a,b兴⫻关c,d兴 and that w is continuous on R. We introduce the variational functional defined by

J共w兲 = IR␣L共x,y,w共x,y兲,aD␣x关1兴w共x,y兲,cDy␣关2兴w共x,y兲兲 ª␣2

冕

a b

冕

c d L共x,y,w,aDx␣关1兴w,cDy␣关2兴w兲共b − x兲␣−1_{共d − y兲}␣−1_{dydx. 共6兲}

We assume that the Lagrangian L is at least of class C1_{. Observe that, using the notation of the}

共dt兲␣_{integral as presented in Ref. 30,共6兲 can be written as}

J共w兲 =

冕

a b

冕

c d

L共x,y,w共x,y兲,aDx␣关1兴w共x,y兲,cD␣y关2兴w共x,y兲兲共dy兲␣共dx兲␣. 共7兲

Consider the following FCV problem, which we address as problem共P兲.

Problem (P): Minimize 共or maximize兲 functional J defined by共7兲 with respect to the set of continuous functions w共x,y兲 such that w兩⳵R=␸共x,y兲 for some given function␸.

The continuous functions w共x,y兲 that assume the prescribed values w兩⳵R=␸共x,y兲 at all points

of the boundary curve of R are said to be admissible. In order to prove necessary optimality conditions for problem 共P兲, we use a two dimensional analog of fractional integration by parts. Lemma 1 provides the necessary fractional rule.

Lemma 1: Let F, G, and h be continuous functions whose domains contain R. If h⬅0 on⳵R, then

冕

a b

冕

c d

关G共x,y兲aDx␣关1兴h共x,y兲 − F共x,y兲cDy␣关2兴h共x,y兲兴共b − x兲␣−1共d − y兲␣−1dydx

= −

冕

a b

冕

c d

关共aDx␣关1兴G共x,y兲 −cDy␣关2兴F共x,y兲兲h共x,y兲兴共b − x兲␣−1共d − y兲␣−1dydx.

Proof: By choosing f = F · h and g = G · h in Green’s formula, we obtain

I_⳵R␣ 关1兴共Fh兲 + I_⳵R␣关2兴共Gh兲 = 1

␣!IR

␣_关

aDb␣关1兴G · h + G ·aDb␣关1兴h −cDd␣关2兴F · h − F ·cDd␣关2兴h兴,

1 ␣!IR ␣_{关G ·} aDb␣关1兴h − F ·cDd␣关2兴h兴 = I⳵R␣关1兴共Fh兲 + I⳵R␣ 关2兴共Gh兲 − 1 ␣!IR ␣_关共 aDb␣关1兴G −cDd␣关2兴F兲h兴.

In addition, since h⬅0 on⳵R, we deduce that

IR␣关G ·aDb␣关1兴h − F ·cDd␣关2兴h兴 = − IR␣关共aDb␣关1兴G −cDd␣关2兴F兲h兴.

The lemma is proved. 䊐

Theorem 3: (Fractional Euler–Lagrange equation) Let w be a solution to problem (P). Then,

w is a solution of the fractional partial differential equation

⳵3L −aDx␣关1兴⳵4L −cD␣y关2兴⳵5L = 0, 共8兲

where by⳵iL, i = 1 , . . . , 5, we denote the usual partial derivative of L共·, · , · , · ,·兲 with respect to its

ith argument.

Proof: Let h be a continuous function on R such that h⬅0 on⳵R and consider an admissible variation w +⑀h for⑀taking values on a sufficient small neighborhood of zero. Let

j共⑀兲 = J共w +⑀h兲. Then, j

⬘

共0兲=0, i.e., ␣2

冕

a b

冕

c d 共⳵3Lh +⳵4LaDx␣关1兴h +⳵5LcDy␣关2兴h兲共b − x兲␣−1共d − y兲␣−1dydx = 0.

Using Lemma 1, we obtain

␣2

冕

a b

冕

c d 共⳵3L −aDx␣关1兴⳵4L −cDy␣关2兴⳵5L兲h共b − x兲␣−1共d − y兲␣−1dydx = 0.

Since h is an arbitrary function, by the fundamental lemma of calculus of variations, we deduce

Eq.共8兲. 䊐

Let us consider now the situation where we do not impose admissible functions w to be of fixed values on⳵R.

Problem 共P

⬘

兲 : Minimize 共or maximize兲 J among the set of all continuous curves w whose domain contains R.

Theorem 4: (Fractional natural boundary conditions) Let w be a solution to problem共P

⬘

兲 .

Then, w is a solution of the fractional differential equation(8) and satisfies the following equa-tions:

共1兲 ⳵4L共a,y,w共a,y兲,aDa␣关1兴w共a,y兲,cDy␣关2兴w共a,y兲兲=0 for all y苸关c,d兴;

共2兲 ⳵4L共b,y,w共b,y兲,aDb␣关1兴w共b,y兲,cDy␣关2兴w共b,y兲兲=0 for all y苸关c,d兴;

共3兲 ⳵5L共x,c,w共x,c兲,aDx␣关1兴w共x,c兲,cDc␣关2兴w共x,c兲兲=0 for all x苸关a,b兴; and

共4兲 ⳵5L共x,d,w共x,d兲,aDx␣关1兴w共x,d兲,cDd␣关2兴w共x,d兲兲=0 for all x苸关a,b兴.

Proof: Proceeding as in the proof of Theorem 3 共see also Lemma 1兲, we obtain

0 =␣2

冕

a b

冕

c d 共⳵3Lh +⳵4L aDx␣关1兴h +⳵5LcDy␣关2兴h兲共b − x兲␣−1共d − y兲␣−1dydx =␣2

冕

a b

冕

c d 共⳵3L − aDx␣关1兴⳵4L − cDy␣关2兴⳵5L兲h共b − x兲␣−1共d − y兲␣−1dydx +␣!I⳵R␣关2兴共⳵4Lh兲 −␣!I⳵R␣关1兴共⳵5Lh兲, 共9兲 where h is an arbitrary continuous function. In particular, the above equation holds for h⬅0 on

⳵R. For such h the second member of共9兲vanishes and by the fundamental lemma of the calculus of variations we deduce Eq.共8兲. With this result, Eq.共9兲takes the form

0 =

冕

c d

⳵4L共b,y,w共b,y兲,aDb␣关1兴w共b,y兲,cDy␣关2兴w共b,y兲兲h共b,y兲共d − y兲␣−1dy

−

冕

c d

⳵4L共a,y,w共a,y兲,aDa␣关1兴w共a,y兲,cDy␣关2兴w共a,y兲兲h共a,y兲共d − y兲␣−1dy

−

冕

a b ⳵5L共x,c,w共x,c兲,aDx␣关1兴w共x,c兲,cDc␣关2兴w共x,c兲兲h共x,c兲共b − x兲␣−1dx +

冕

a b ⳵5L共x,d,w共x,d兲,aDx␣关1兴w共x,d兲,cDd␣关2兴w共x,d兲兲h共x,d兲共b − x兲␣−1dx. 共10兲

Since h is an arbitrary function, we can consider the subclass of functions for which h⬅0 on 共关a,b兴 ⫻ 兵c其兲 艛 共关a,b兴 ⫻ 兵d其兲 艛 共兵b其 ⫻ 关c,d兴兲.

For such h, Eq.共10兲reduces to 0 =

冕

c d

⳵4L共a,y,w共a,y兲,aDa␣关1兴w共a,y兲,cDy␣关2兴w共a,y兲兲h共a,y兲共d − y兲␣−1dy .

By the fundamental lemma of calculus of variations, we obtain

⳵4L共a,y,w共a,y兲,aDa␣关1兴w共a,y兲,cDy␣关2兴w共a,y兲兲 = 0 for all y 苸 关c,d兴.

The other natural boundary conditions are proved similarly by appropriate choices of h. 䊐 We can generalize Lemma 1 and Theorem 3 to the three dimensional case in the following way.

Lemma 2: Let A, B, C, and ␩ be continuous functions whose domains contain the parallel-epiped W. If␩⬅0 on⳵W, then

IW␣共A ·aDb␣关1兴␩+ B ·cDd␣关2兴␩+ C ·eD␣f关3兴␩兲 = − IW␣共关aDb␣关1兴A +cDd␣关2兴B +eD␣f关3兴C兴␩兲.

共11兲 Proof: By choosing Fx=␩A, Fy=␩B, and Fz=␩C in 共5兲, we obtain the three dimensional

analog of integrating by parts,

IW␣共A ·aDb␣关1兴␩+ B ·cDd␣关2兴␩+ C ·eD␣f关3兴␩兲 = − IW␣共关aDb␣关1兴A +cDd␣关2兴B +eD␣f关3兴C兴␩兲

+␣!共I_⳵W␣ ,关␩A,␩B,␩C兴兲.

In addition, if we assume that␩⬅0 on⳵W, we have formula共11兲. 䊐

Theorem 5: (Fractional Euler–Lagrange equation for triple integrals) Let w = w共x,y,z兲 be a

continuous function whose domain contains W =关a,b兴⫻关c,d兴⫻关e, f兴. Consider the functional J共w兲 = IW␣L共x,y,z,w共x,y,z兲,aDx␣关1兴w共x,y,z兲,cDy␣关2兴w共x,y,z兲,eDz␣关3兴w共x,y,z兲兲

=

冕

a b

冕

c d

冕

e f L共x,y,z,w,aDx␣关1兴w,cDy␣关2兴w,eDz␣关3兴w兲共dz兲␣共dy兲␣共dx兲␣

defined on the set of continuous curves such that their values on⳵W take prescribed values. Let L be at least of class C1 . If w is a minimizer (or maximizer) of J, then w satisfies the fractional partial differential equation,

⳵4L −aDb␣关1兴⳵5L −cDd␣关2兴⳵6L −eDf␣关3兴⳵7L = 0.

Proof: A proof can be done similar to the proof of Theorem 3, where instead of using Lemma

V. APPLICATIONS AND POSSIBLE EXTENSIONS

In classical mechanics, functionals that depend on functions of two or more variables arise in a natural way, e.g., in mechanical problems involving systems with infinitely many degrees of freedom共string, membranes, etc.兲. Let us consider a flexible elastic string stretched under constant tension␶along the x axis with its end points fixed at x = 0 and x = L. Let us denote the transverse displacement of the particle at time t, t₁ⱕtⱕt₂, whose equilibrium position is characterized by its distance x from the end of the string at x = 0 by the function w = w共x,t兲. Thus, w共x,t兲, with 0 ⱕxⱕL, describes the shape of the string during the course of the vibration. Assume a distribution of mass along the string of density␴=␴共x兲. Then, the function that describes the actual motion of the string is one that renders

J共w兲 =1 2

冕

t₁ t₂

冕

0 L 共␴wt2−␶wx2兲dxdt

an extremum with respect to functions w共x,t兲, which describes the actual configuration at t=t1and

t = t2 and which vanishes, for all t, at x = 0 and x = L共see Ref. 43, p. 95, for more details兲.

We discuss the description of the motion of the string within the framework of the fractional differential calculus. One may assume that, due to some constraints of physical nature, the dy-namics does not depend on the usual partial derivatives but on some fractional derivatives

0Dx␣关1兴w andt₁Dt␣关2兴w. For example, we can assume that there is some coarse-graining

phenom-enon共see details in Refs. 27 and25兲. In this condition, one is entitled to assume again that the actual motion of the system, according to the principle of Hamilton, is such as to render the action function

J共w兲 =1₂IR␣共␴共t₁Dt␣关2兴w兲2−␶共0Dx␣关1兴w兲2兲,

where R =关0,L兴⫻关t1, t2兴, an extremum. Note that we recover the classical problem of the vibrating

string when ␣→1−_{. Applying Theorem 3, we obtain the fractional equation of motion for the}

vibrating string,

0Dx␣关1兴0D1␣关1兴w = ␴

␶t₁Dt␣关2兴t₁D2␣关2兴w.

This equation becomes the classical equation of the vibrating string共cf., e.g., Ref. 43, p. 97兲 if

␣→1−_.

We remark that the fractional operators are nonlocal, therefore they are suitable for construct-ing models possessconstruct-ing memory effect. In the above example, we discussed the application of the fractional differential calculus to the vibrating string. We started with a variational formulation of the physical process in which we modify the Lagrangian density by replacing integer order de-rivatives with fractional ones. Then, the action integral in the sense of Hamilton was minimized and the governing equation of the physical process was obtained in terms of fractional derivatives. Similarly, many others physical fields can be derived from a suitably defined action functional. This gives several possible applications of the FCVs with multiple integrals as was introduced in this paper, e.g., in describing nonlocal properties of physical systems in mechanics共see, e.g., Refs. 10,11,32,36, and 41兲 or electrodynamics 共see, e.g., Refs.9 and42兲.

We end with some open problems for further investigations. It has been recognized that FC is useful in the study of scaling in physical systems.14,15 In particular, there is a direct connection between local fractional differentiability properties and the dimensions of Holder exponents of nowhere differentiable functions, which provide a powerful tool to analyze the behavior of irregu-lar signals and functions on a fractal set.4,33FC appear naturally, e.g., when working with fractal sets and coarse-graining spaces,25,24 and fractal patterns of deformation and vibration in porous media and heterogeneous materials.12The importance of vibrating strings to the FC has been given in Ref.18, where it is shown that a fractional Brownian motion can be identified with a string. The usefulness of our fractional theory of the calculus of variations with multiple integrals in physics,

to deal with fractal and coarse-graining spaces, porous media, and Brownian motions, is a question to be studied. It should be possible to prove the theorems obtained in this work for a general form of domains and boundaries and to develop a FCVs with multiple integrals in terms of other type of fractional operators. An interesting open question consists of generalizing the fractional Noether-type theorems obtained in Refs.6,21, and22to the case of several independent variables. ACKNOWLEDGMENTS

This work was supported by the Centre for Research on Optimization and Control 共CEOC兲 from the “Fundação para a Ciência e a Tecnologia”共FCT兲, cofinanced by the European Commu-nity Fund 共Grant No. FEDER/POCI 2010兲. One of the authors 共A.M.兲 was also supported by Białystok University of Technology via a project of the Polish Ministry of Science and Higher Education “Wsparcie miedzynarodowej mobilnosci naukowcow.” The authors are grateful to an anonymous referee for useful remarks and references.

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