A Fractional Calculus of Variations for Multiple Integrals with Application To

Home , Multiple integral

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Euler-Lagrange fractional the Stokes; fractional and integrals; multiple variations; of calculus fractional Keywords: 2010: Classification Subject Mathematics PACS: napiainw ics h rcinleuto fmotion of equation fractional the discuss we versi fraction application fractional and an provide equations, results Euler-Lagrange Main fractional Gauss, Jumarie. of sense the prahue h eetntoso imn-iuil frac Riemann-Liouville of notions recent the uses approach eitoueafatoa hoyo h aclso variatio of calculus the of theory fractional a introduce We 2 aut fCmue cec,Balso nvriyo Technology, of University Bia lystok Science, Computer of Faculty [email protected] 51.b 51.j 02.30.Xx. 45.10.Hj; 45.10.Db; iad Almeida Ricardo rcinlcluu;mdfidReanLovledrvtvsadintegr and derivatives Riemann-Liouville modified calculus; fractional ihApiaint irtn String Vibrating to Application with ora fMteaia Physics Mathematical of Journal 531Balso,Poland Bia lystok, 15-351 α efi .M Torres M. F. 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Agnieszka 91;2A3 26B20. 26A33; 49K10; [email protected] lntrlbudr odtos As conditions. boundary natural al (14/January/2010). 1 n fteterm fGenand Green of theorems the of ons inldrvtvsaditgasin integrals and derivatives tional favbaigstring. vibrating a of tio,ReanLovle Riesz, Riemann-Liouville, etnikov, sfrmlil nerl.Our integrals. multiple for ns d .. ncascladquantum and classical in e.g., nd, oin fKie 3] Cresson [32], Klimek of notions t 1,2,27]. 22, [19, l so hsc n mathematics and physics of as esoso h lsia results classical the of versions duigodnr differential ordinary using ed rm fGen Gauss, Green, of orems fmteais ihmany with mathematics, of uofatoa operators. fractional puto eddnmclsystems, dynamical field r hntrecenturies, three than ore luu sa h very the at is alculus iaeteusefulness the tigate ∗ 1 ,Portugal o, , 2 als; For the purposes of a multidimensional Fractional Calculus of Variations (FCV), 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 (see, e.g., [6, 20, 21]), the Caputo fractional derivative (see, e.g., [1, 8, 9]), the symmetric fractional derivative (see, e.g., [32]), the Jumarie fractional derivative (see, e.g., [5, 27]), and others [3, 14, 19]. For applications of the fractional calculus of variations we refer the reader to [16, 17, 20, 27, 32, 36, 37, 40]. 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 [43], it 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 [20], where it is claimed that an appropriate fractional variational theory involving multiple integrals would have important consequences in mechanical problems involving dissipative systems 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 the 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 fractional calculus of variations for multiple integrals. For that we make use of the recent Jumarie fractional integral and derivative [24, 25, 28], extending 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 [5] (see also [26]). The plan of the paper is as follows. In Section 2 some basic formulas of Jumarie’s fractional calculus are briefly reviewed. Then, in Section 3, the differential and integral vector operators are introduced, and fractional Green’s, Gauss’s and Stokes’ theorems formulated. Section 4 is devoted to the study of problems of fractional calculus of variations with multiple integrals. Our main results provide Euler-Lagrange necessary optimality type conditions for such problems (Theorems 3 and 5) as well as natural boundary conditions (Theorem 4). We end with Section 5 of applications and future perspectives.

2 Preliminaries

For an introduction to the classical fractional calculus we refer the reader to [31, 34, 35, 39]. In this section we briefly review the main notions and results from the recent fractional calculus proposed by Jumarie [25, 27, 28]. Let f : [0, 1] → R be a continuous function and α ∈ (0, 1). The Jumarie fractional derivative of f may be defined by x 1 d − f (α)(x)= (x − t) α(f(t) − f(0)) dt. (1) Γ(1 − α) dx Z0 One can obtain (1) as a consequence of a more basic definition, a local one, in terms of a fractional finite difference (cf. equation (2.2) of [28]):

∞ 1 α f (α)(x) = lim (−1)k f (x + (α − k)h) . h→0 hα k kX=0 Note that the Jumarie and the Riemann-Liouville fractional derivatives are equal if f(0) = 0. The advantage of deﬁnition (1) with respect to the classical deﬁnition of Riemann-Liouville is that the

2 fractional derivative of a constant is now zero, as desired. An anti-derivative of f, called the (dt)α integral of f, is deﬁned by

x x − f(t)(dt)α = α (x − t)α 1f(t)dt. Z0 Z0 The following equalities can be considered as fractional counterparts of the ﬁrst and the second fundamental theorems of calculus and can be found in [27, 28]:

α x d α α f(t)(dt) = α!f(x), (2) dx Z0 x f (α)(t)(dt)α = α!(f(x) − f(0)), (3) Z0 where α! := Γ(1 + α). The Leibniz rule for the Jumarie fractional derivative is equal to the standard one: (f(x)g(x))(α) = f (α)(x)g(x)+ f(x)g(α)(x). Here we see another advantage of derivative (1): the fractional derivative of a product is not an inﬁnite sum, in opposite to the Leibniz rule for the Riemann-Liouville fractional derivative [35, p. 91]. One can easily generalize the previous deﬁnitions and results for functions with a domain [a,b]:

x 1 d − f (α)(x)= (x − t) α(f(t) − f(a)) dt Γ(1 − α) dx Za and x x − f(t)(dt)α = α (x − t)α 1f(t)dt. Za Za 3 Fractional integral theorems

In this section we introduce some useful fractional integral and fractional diﬀerential operators. 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.

3.1 Fractional operators n Rn Let us consider a continuous function f = f (x1,...,xn) deﬁned on R = Πi=1[ai,bi] ⊂ . Let us extend Jumarie’s fractional derivative and the (dt)α integral to functions with n variables. For xi ∈ [ai,bi], i =1,...,n, and α ∈ (0, 1), we deﬁne the fractional integral operator as

xi α α−1 ai Ixi [i]= α (xi − t) dt . Zai These operators act on f in the following way:

xi α α−1 − ai Ixi [i]f(x1,...,xn)= α f(x1,...,xi 1,t,xi+1,...,xn)(xi − t) dt, i =1, . . . , n . Zai

Let Ξ = {k1,...,ks} be an arbitrary nonempty subset of {1,...,n}. We deﬁne the fractional s multiple integral operator over the region RΞ = Πi=1[aki , xki ] by

α α α I [k ,...,ks]= a I [k ] ... a I [ks] RΞ 1 k1 xk1 1 ks xks xk1 xks s α−1 α−1 = α ··· (xk1 − tk1 ) ··· (xks − tks ) dtks ...dtk1 Zak1 Zaks

3 which acts on f by

α IRΞ [k1,...,ks]f(x1,...,xn) xk1 xks s α−1 α−1 = α ··· f(ξ1,...,ξn)(xk1 − tk1 ) ··· (xks − tks ) dtks ...dtk1 , Zak1 Zaks where ξj = tj if j ∈ Ξ, and ξj = xj if j∈ / Ξ, j =1,...,n. The fractional volume integral of f over the whole domain R is given by

b1 bn α n α−1 α−1 IRf = α ··· f(t1,...,tn)(b1 − t1) ··· (bn − tn) dtn ...dt1. Za1 Zan

The fractional partial derivative operator with respect to the ith variable xi, i = 1,...,n, of order α ∈ (0, 1) is deﬁned as follows:

xi α 1 ∂ −α ai Dxi [i]= (xi − t) dt , Γ(1 − α) ∂xi Zai which act on f by

α ai Dxi [i]f(x1,...,xn) xi 1 ∂ −α = (xi−t) [f(x1,...,xi−1,t,xi+1,...,xn) − f(x1,...,xi−1,ai, xi+1,...,xn)] dt , Γ(1 − α) ∂xi Zai i =1,...,n. We observe that the Jumarie fractional integral and the Jumarie fractional derivative can be obtained putting n = 1:

x x α α−1 α aIx [1]f(x)= α (x − t) f(t) dt = f(t) (dt) Za Za and x α 1 d −α (α) aDx [1]f(x)= (x − t) (f(t) − f(a)) dt = f (x). Γ(1 − α) dx Za Using these notations, formulae (2)–(3) can be presented as

α α aDx [1]aIx [1]f(x)= α!f(x)

α α aIx [1]aDx [1]f(x)= α!(f(x) − f(a)). (4) In the two dimensional case we deﬁne the fractional line integral on ∂R, R = [a,b] × [c, d], by

α α α I∂Rf = I∂R[1]f + I∂R[2]f where

α α I∂R[1]f = aIb [1][f(b,c) − f(b, d)] b − = α [f(t,c) − f(t, d)] (b − t)α 1 dt Za and

α α I∂R[2]f = cId [2][f(b, d) − f(a, d)] d − = α [f(b,t) − f(a,t)] (d − t)α 1 dt. Zc

4 3.2 Fractional diﬀerential vector operations

Let WX = [a, x] × [c,y] × [e,z], W = [a,b] × [c, d] × [e,f], and denote (x1, x2, x3) by (x,y,z). We introduce the fractional nabla operator by

α α α α ∇WX = iaDx [1] + jcDy [2] + keDz [3] , where the i, j, k define a fixed right-handed orthonormal basis. If f : R3 → R is a continuous function, then we define its fractional gradient as

α α α α α GradWX f = ∇WX f = iaDx [1]f(x,y,z)+ jcDy [2]f(x,y,z)+ keDz [3]f(x,y,z) .

3 3 If F = [Fx, Fy, Fz]: R → R is a continuous vector ﬁeld, then we deﬁne its fractional divergence and fractional curl by

α α α α α DivWX F = ∇WX ◦ F = aDx [1]Fx(x,y,z)+ cDy [2]Fy(x,y,z)+ eDz [3]Fz(x,y,z) and

α α α α CurlWX F = ∇WX × F = i cDy [2]Fz(x,y,z) − eDz [3]Fy(x,y,z) α α α α + j (eDz [3]Fx(x,y,z) − aDx [1]Fz(x,y,z)) + k aDx [1]Fy(x,y,z) − cDy [2]Fx(x,y,z) . Note that these fractional differential operators are non-local. Therefore, the fractional gradient, divergence, and curl, depend on the region WX . For F : R3 → R3 and f,g : R3 → R it is easy to check the following relations: α α α (i) DivWX (fF )= fDivWX F + F ◦ GradWX f, α α (ii) CurlWX (GradWX f) = [0, 0, 0], α α (iii) DivWX (CurlWX F ) = 0, α α α (iv) GradWX (fg)= gGradWX f + fGradWX g, α α α α α α α α (v) DivWX (GradWX f)= aDx [1]aDx [1]f + cDy [2]cDy [2]f + eDz [3]eDz [3]f. α 2 2α Let us recall that in general (DWX ) 6= DWX (see [28]). A fractional flux of the vector field F across ∂W is a fractional oriented surface integral of the field such that

α α α α (I∂W , F )= I∂W [2, 3]Fx(x,y,z)+ I∂W [1, 3]Fy(x,y,z)+ I∂W [1, 2]Fz(x,y,z) , where α α α I∂W [1, 2]f(x,y,z)= aIb [1]cId [2][f(b,d,f) − f(b,d,e)] , α α α I∂W [1, 3]f(x,y,z)= aIb [1]eIf [3][f(b,d,f) − f(b,c,f)] , and α α α I∂W [2, 3]f(x,y,z)= cId [2]eIf [3][f(b,d,f) − f(a,d,f)] .

3.3 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 Tarasov in [42]. Theorem 1 (Fractional Green’s theorem for a rectangle). Let f and g be two continuous functions whose domains contain R = [a,b] × [c, d] ⊂ R2. Then, 1 Iα [1]f + Iα [2]g = Iα [ Dα[1]g − Dα[2]f] . ∂R ∂R α! R a b c d

5 Proof. We have α α α α I∂R [1]f + I∂R[2]g = aIb [1][f(b,c) − f(b, d)] + cId [2][g(b, d) − g(a, d)] . By equation (4), 1 f(b,c) − f(b, d)= − Iα[2] Dα[2]f(b, d) , α! c d c d 1 g(b, d) − g(a, d)= Iα[1] Dα[1]g(b, d) . α! a b a b Therefore, 1 1 Iα [1]f + Iα [2]g = − Iα[1] Iα[2] Dα[2]f(b, d)+ Iα[2] Iα[1] Dα[1]g(b, d) ∂R ∂R a b α! c d c d c d α! a b a b 1 = Iα [ Dα[1]g − Dα[2]f] . α! R a b c d

Theorem 2 (Fractional Gauss’s theorem for a parallelepiped). Let F = (Fx, Fy, Fz) be a continuous vector ﬁeld in a domain that contains W = [a,b] × [c, d] × [e,f]. If the boundary of W is a closed surface ∂W , then 1 (Iα , F )= Iα Divα F. (5) ∂W α! W W Proof. The result follows by direct transformations: α α α α (I∂W , F )= I∂W [2, 3]Fx + I∂W [1, 3]Fy + I∂W [1, 2]Fz α α α α = cId [2]eIf [3](Fx(b,d,f) − Fx(a,d,f)) + aIb [1]eIf [3](Fy(b,d,f) − Fy(b,c,f)) α α + aIb [1]cId [2](Fz(b,d,f) − Fz(b,d,e)) 1 = Iα[1] Iα[2] Iα[3]( Dα[1]F (b,d,f)+ Dα[2]F (b,d,f)+ Dα[3]F (b,d,f)) α! a b c d e f a b x c d y e f z 1 = Iα ( Dα[1]F + Dα[2]F + Dα[3]F ) α! W a b x c d y e f z 1 = Iα Divα F. α! W W

Let S be an open, oriented, and nonintersecting surface, bounded by a simple and closed curve ∂S. Let F = [Fx, Fy, Fz] be a continuous vector ﬁeld. Divide up S by sectionally curves into N subregions S1, S2, ..., SN . Assume that for small enough subregions each Sj can be approximated by a plane rectangle Aj bounded by curves C1, C2, ..., CN . Apply Green’s theorem to each individual rectangle Aj . Then, summing over the subregions, 1 Iα (∇α × F )= Iα F. α! Aj Aj ∂Aj Xj Xj Furthermore, letting N → ∞ 1 1 Iα (∇α × F ) → (Iα, CurlαF ) α! Aj Aj α! S S Xj while α α I∂Aj F → I∂S F. Xj We conclude with the fractional Stokes formula 1 (Iα, CurlαF )= Iα F. α! S S ∂S

6 4 Fractional calculus of variations with multiple integrals

Consider a function w = w(x, y) with two variables. Assume that the domain of w contains the rectangle R = [a,b] × [c, d] and that w is continuous on R. We introduce the variational functional deﬁned by

α α α J(w)= IRL x, y, w(x, y), aDx [1]w(x, y), cDy [2]w(x, y) b d (6) 2 α α α−1 α−1 := α L x, y, w, aD [1]w, cD [2]w (b − x) (d − y) dydx. Z Z x y a c We assume that the lagrangian L is at least of class C1. Observe that, using the notation of the (dt)α integral as presented in [28], (6) can be written as

b d α α α α J(w)= L x, y, w(x, y), aD [1]w(x, y), cD [2]w(x, y) (dy) (dx) . (7) Z Z x y a c Consider the following FCV problem, which we address as problem (P ).

Problem (P): minimize (or maximize) functional J deﬁned by (7) with respect to the set of continuous functions w(x, y) such that w|∂R = ϕ(x, y) for some given function ϕ.

The continuous functions w(x, y) that assume the prescribed values w|∂R = ϕ(x, y) at all points of the boundary curve of R are said to be admissible. In order to prove necessary optimality conditions for problem (P ) we use a two dimensional analogue 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

b d α α α−1 α−1 [G(x, y)aDx [1]h(x, y) − F (x, y)cDy [2]h(x, y)](b − x) (d − y) dydx Za Zc b d α α α−1 α−1 = − [(aDx [1]G(x, y) − cDy [2]F (x, y))h(x, y)](b − x) (d − y) dydx . Za Zc Proof. By choosing f = F · h and g = G · h in Green’s formula, we obtain 1 Iα [1](Fh)+ Iα [2](Gh)= Iα[ Dα[1]G · h + G · Dα[1]h − Dα[2]F · h − F · Dα[2]h] , ∂R ∂R α! R a b a b c d c d which is equivalent to 1 1 Iα [G · Dα[1]h − F · Dα[2]h]= Iα [1](Fh)+ Iα [2](Gh) − Iα [( Dα[1]G − Dα[2]F )h] . α! R a b c d ∂R ∂R α! R a b c d In addition, since h ≡ 0 on ∂R, we deduce that

α α α α α α IR [G · aDb [1]h − F · cDd [2]h]= −IR [(aDb [1]G − cDd [2]F )h] .

The lemma is proved. 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

α α ∂3L − aDx [1]∂4L − cDy [2]∂5L =0 , (8) where by ∂iL, i =1,..., 5, we denote the usual partial derivative of L(·, ·, ·, ·, ·) with respect to its i-th argument.

7 Proof. Let h be a continuous function on R such that h ≡ 0 on ∂R, and consider an admissible variation w + ǫh, for ǫ taking values on a suﬃcient small neighborhood of zero. Let

j(ǫ)= J(w + ǫh).

Then j′(0) = 0, i.e.,

b d 2 α α α−1 α−1 α ∂3Lh + ∂4L aD [1]h + ∂5L cD [2]h (b − x) (d − y) dydx =0 . Z Z x y a c Using Lemma 1, we obtain

b d 2 α α α−1 α−1 α ∂3L − aD [1]∂4L − cD [2]∂5L h(b − x) (d − y) dydx =0 . Z Z x y a c Since h is an arbitrary function, by the fundamental lemma of calculus of variations we deduce equation (8). Let us consider now the situation where we do not impose admissible functions w to be of ﬁxed values on ∂R.

Problem (P′): minimize (or maximize) 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 (P ′). Then w is a solution of the fractional diﬀerential equation (8) and satisﬁes the following equations:

α α 1. ∂4L(a,y,w(a,y), aDa [1]w(a,y), cDy [2]w(a,y))=0 for all y ∈ [c, d]; α α 2. ∂4L(b,y,w(b,y), aDb [1]w(b,y), cDy [2]w(b,y))=0 for all y ∈ [c, d]; α α 3. ∂5L(x, c, w(x, c), aDx [1]w(x, c), cDc [2]w(x, c)) =0 for all x ∈ [a,b]; α α 4. ∂5L(x, d, w(x, d), aDx [1]w(x, d), cDd [2]w(x, d)) =0 for all x ∈ [a,b]. Proof. Proceeding as in the proof of Theorem 3 (see also Lemma 1), we obtain

b d 2 α α α−1 α−1 0= α ∂3Lh + ∂4L aD [1]h + ∂5L cD [2]h (b − x) (d − y) dydx Z Z x y a c b d 2 α α α−1 α−1 = α ∂3L − aDx [1]∂4L − cDy [2]∂5L h (b − x) (d − y) dydx Za Zc α α + α!I∂R[2](∂4Lh) − α!I∂R[1](∂5Lh) , (9) 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 (9) vanishes and by the fundamental lemma of the calculus of variations we deduce equation (8). With this result, equation (9) takes the form

d α α α−1 0 = ∂4L(b,y,w(b,y), aDb [1]w(b,y), cDy [2]w(b,y)) h(b,y)(d − y) dy Zc d α α α−1 − ∂4L(a,y,w(a,y), aDa [1]w(a,y), cDy [2]w(a,y)) h(a,y)(d − y) dy Zc b (10) α α α−1 − ∂5L(x, c, w(x, c), aDx [1]w(x, c), cDc [2]w(x, c)) h(x, c)(b − x) dx Za b α α α−1 + ∂5L(x, d, w(x, d), aDx [1]w(x, d), cDd [2]w(x, d)) h(x, d)(b − x) dx . Za

8 Since h is an arbitrary function, we can consider the subclass of functions for which h ≡ 0 on

[a,b] ×{c} ∪ [a,b] ×{d}∪{b}× [c, d].

For such h equation (10) reduce to

d α α α−1 0= ∂4L(a,y,w(a,y), aDa [1]w(a,y), cDy [2]w(a,y)) h(a,y)(d − y) dy. Zc By the fundamental lemma of calculus of variations, we obtain

α α ∂4L(a,y,w(a,y), aDa [1]w(a,y), cDy [2]w(a,y)) = 0 for all y ∈ [c, d] .

The other natural boundary conditions are proved similarly, by appropriate choices of h. We can generalize Lemma 1 and Theorem 3 to the three dimensional case in the following way. Lemma 2. Let A, B, C, and η be continuous functions whose domains contain the parallelepiped W . If η ≡ 0 on ∂W , then

α α α α α α α α IW (A · aDb [1]η + B · cDd [2]η + C · eDf [3]η)= −IW ( aDb [1]A + cDd [2]B + eDf [3]C η) . (11) Proof. By choosing Fx = ηA, Fy = ηB, and Fz = ηC in (5), we obtain the three dimensional analogue of integrating by parts:

α α α α IW (A · aDb [1]η + B · cDd [2]η + C · eDf [3]η) α α α α α = −IW ( aDb [1]A + cDd [2]B + eDf [3]C η)+ α!(I∂W , [ηA,ηB,ηC]). In addition, if we assume that η ≡ 0 on ∂W , we have formula (11). Theorem 5 (Fractional Euler-Lagrange equation for triple integrals). Let w = w(x,y,z) be a continuous function whose domain contains W = [a,b] × [c, d] × [e,f]. Consider the functional

α α α α J(w)= IW L x,y,z,w(x,y,z), aDx [1]w(x,y,z), cDy [2]w(x,y,z), eDz [3]w(x,y,z) b d f α α α α α α = L x, y, z, w, aD [1]w, cD [2]w, eD [3]w (dz) (dy) (dx) Z Z Z x y z a c e 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

α α α ∂4L − aDb [1]∂5L − cDd [2]∂6L − eDf [3]∂7L =0.

Proof. A proof can be done similarly to the proof of Theorem 3 where instead of using Lemma 1 we apply Lemma 2.

5 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 (string, membranes, etc.). Let us consider a flexible elastic string stretched under constant tension τ 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, t1 ≤ t ≤ t2, whose equilibrium position is characterized by its distance x from the end of the string at x = 0 by the function w = w(x, t). Thus w(x, t), with 0 ≤ x ≤ L, describes the shape of the string during the course of the vibration. Assume a

9 distribution of mass along the string of density σ = σ(x). Then the function that describes the actual motion of the string is one which renders

t2 L 1 2 2 J(w)= (σwt − τwx) dx dt 2 Zt1 Z0 an extremum with respect to functions w(x, t) which describe the actual conﬁguration at t = t1 and t = t2 and which vanish, for all t, at x = 0 and x = L (see [43, p. 95] for more details). 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 α dynamics do not depend on the usual partial derivatives but on some fractional derivatives 0Dx [1]w α and t1 Dt [2]w. For example, we can assume that there is some coarse graining phenomenon – see details in [26, 29]. 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

1 α α 2 α 2 J(w)= I (σ ( 1 D [2]w) − τ ( D [1]w) ), 2 R t t 0 x where R = [0,L]×[t1,t2], an extremum. Note that we recover the classical problem of the vibrating string when α → 1−. Applying Theorem 3 we obtain the fractional equation of motion for the vibrating string: α α σ α α D [1] D [1]w = 1 D [2] 1 D [2]w . 0 x 0 1 τ t t t 2 This equation becomes the classical equation of the vibrating string (cf., e.g., [43, p. 97]) if α → 1−. We remark that the fractional operators are non-local, therefore they are suitable for con- structing models possessing 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 derivatives 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 fractional calculus of variations with multiple integrals as was introduced in this paper, e.g., in describing non-local properties of physical systems in mechanics (see, e.g., [11, 12, 32, 36, 41]) or electrodynamics (see, e.g., [10, 42]). We end with some open problems for further investigations. It has been recognized that fractional calculus is useful in the study of scaling in physical systems [13, 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 irregular signals and functions on a fractal set [4, 33]. Fractional calculus appear naturally, e.g., when working with fractal sets and coarse-graining spaces [29, 30], and fractal patterns of deformation and vibration in porous media and heterogeneous materials [2]. The importance of vibrating strings to the fractional calculus has been given in [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, are questions 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 fractional calculus of variations with multiple integrals in terms of other type of fractional operators. An interesting open question consists to generalize the fractional Noether-type theorems obtained in [7, 21, 22] to the case of several independent variables.

Acknowledgments

Work supported by the Centre for Research on Optimization and Control (CEOC) from the “Funda¸cão para a Ciência e a Tecnologia” (FCT), cofinanced by the European Community Fund

10 FEDER/POCI 2010. Agnieszka Malinowska is also supported by Bialys tok University of Technol- ogy, via a project of the Polish Ministry of Science and Higher Education “Wsparcie miedzynar- odowej mobilnosci naukowcow”. The authors are grateful to an anonymous referee for useful remarks and references.

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