GREEN'S THEOREM FOR GENERALIZED FRACTIONAL DERIVATIVES

arXiv:1205.4851v2 [math.CA] 26 Oct 2012

GREEN’S THEOREM FOR GENERALIZED FRACTIONAL DERIVATIVES T. ODZIJEWICZ 1 , A. B. MALINOWSKA 2 , D. F. M. TORRES

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Abstract. We study three types of generalized partial fractional operators. An extension of Green’s theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann–Liouville or Caputo. MSC 2010 : Primary 26B20; Secondary 35R11 Key Words and Phrases: fractional calculus, generalized operators, Green’s theorem

1. Introduction In 1828, the English mathematician George Green (1793-1841), who up to his forties was working as a baker and a miller, published an essay where he introduced a formula connecting the line integral around a simple closed curve with a double integral. Within years, this result turned out to be useful in many fields of mathematics, physics and engineering [4, 6, 15, 17]. Generalizations of Green’s theorem have chosen different directions, and are known as the Kelvin–Stokes and the Gauss–Ostrogradsky theorems. In this paper, in contrast with previous works, we want to state a Green’s theorem for generalized partial fractional derivatives. Notions of generalized fractional derivatives were introduced in [1, 8], and then developed in [11, 12]. A fractional version of the Green theorem has been already showed for Riemann–Liouville integrals and Caputo derivatives [18], and for fractional operators in the sense of Jumarie [3]. However, generalized fractional operators have never been considered. Our result may be useful in the theory of fractional calculus (see, e.g., [7, 9, 14, 16]), in particular This is a preprint of a paper whose final and definite form will appear at www.springerlink.com: Fract. Calc. Appl. Anal. 16 (2013), no. 1, in press. 1

2

T. ODZIJEWICZ, A.B. MALINOWSKA, D.F.M. TORRES

for the two-dimensional fractional calculus of variations, where the derivation of Euler–Lagrange equations uses, as a key step in the proof, Green’s theorem [3, 5, 10, 13]. The paper is organized as follows. In Section 2 a common review of ordinary and partial generalized fractional calculus is given. Our results are then formulated and proved in Section 3: we show the two-dimensional integration by parts formula for generalized Riemann–Liouville partial fractional integrals (Theorem 3.1) and Green’s theorem for generalized partial fractional derivatives (Theorem 3.2). 2. Basic Notions In this section we give definitions of generalized ordinary and partial fractional operators. By the choice of a certain kernel, these operators can be reduced to the standard fractional integrals and derivatives. For more on the subject, we refer the reader to [1, 2, 8, 11, 12]. 2.1. Generalized fractional operators. Definition 2.1 (Generalized fractional integral). The operator KPα is given by (KPα f ) (t)

:= p

Zt

kα (t, τ )f (τ )dτ + q

a

Zb

kα (τ, t)f (τ )dτ,

t

where P = ha, t, b, p, qi is the parameter set (p-set for brevity), t ∈ [a, b], p, q are real numbers, and kα (t, τ ) is a kernel which may depend on α. The operator KPα is referred as the operator K (K-op for simplicity) of order α and p-set P . Theorem 2.1 (Theorem 2.3 of [11]). Let kα be a difference kernel, i.e., kα (t, τ ) = kα (t − τ ) and kα ∈ L1 ([a, b]). Then, KPα : L1 ([a, b]) → L1 ([a, b]) is well defined, bounded and linear operator. The K-op reduces to the classical left or right Riemann–Liouville fractional integral (see, e.g., [7, 14]) for a suitably chosen kernel kα (t, τ ) and 1 (t − τ )α−1 . If P = ha, t, b, 1, 0i, then p-set P . Indeed, let kα (t − τ ) = Γ(α) (KPα f ) (t)

1 = Γ(α)

Zt a

(t − τ )α−1 f (τ )dτ =: (a Iαt f ) (t)

GREEN’S THEOREM FOR GENERALIZED FRACTIONAL . . .

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is the left Riemann–Liouville fractional integral of order α; if P = ha, t, b, 0, 1i, then Zb 1 α (τ − t)α−1 f (τ )dτ =: (t Iαb f ) (t) (KP f ) (t) = Γ(α) t

is the right Riemann–Liouville fractional integral of order α. Definition 2.2 (Generalized Riemann–Liouville derivative). Let P be a given parameter set. The operator AαP , 0 < α < 1, is defined for d functions f such that KP1−α f ∈ AC ([a, b]) by AαP := dt ◦ KP1−α , where D denotes the standard derivative. We refer to AαP as operator A (A-op) of order α and p-set P . Definition 2.3 (Generalized Caputo derivative). Let P be a given parameter set. The operator BPα , α ∈ (0, 1), is defined for functions f such d and is referred as the operator B that f ∈ AC ([a, b]) by BPα := KP1−α ◦ dt (B-op) of order α and p-set P . Let k1−α (t − τ ) = (AαP f ) (t)

1 Γ(1−α) (t

− τ )−α , α ∈ (0, 1). If P = ha, t, b, 1, 0i, then

1 d = Γ(1 − α) dt

Zt

(t − τ )−α f (τ )dτ =: (a Dαt f ) (t)

a

is the standard left Riemann–Liouville fractional derivative of order α while Zt 1 α α (BP f ) (t) = (t − τ )−α f ′ (τ )dτ =: C a Dt f (t) Γ(1 − α) a

is the standard left Caputo fractional derivative of order α; if P = ha, t, b, 0, 1i, then Zb d 1 α (τ − t)−α f (τ )dτ =: (t Dαb f ) (t) − (AP f ) (t) = − Γ(1 − α) dt t

is the standard right Riemann–Liouville fractional derivative of order α while Zb 1 α α − (BP f ) (t) = − (τ − t)−α f ′ (τ )dτ =: C t Db f (t) Γ(1 − α) t

is the standard right Caputo fractional derivative of order α.

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2.2. Generalized partial fractional operators. Let α be a real number from the interval (0, 1), ∆n = [a1 , b1 ]×· · ·×[an , bn ], n ∈ N, be a subset of Rn , t = (t1 , . . . , tn ) be a point in ∆n and p = (p1 , . . . , pn ), q = (q1 , . . . , qn ) ∈ Rn . Generalized partial fractional integrals and derivatives are a natural generalization of the corresponding one-dimensional generalized fractional integrals and derivatives. Definition 2.4 (Generalized partial fractional integral). Let function f = f (t1 , . . . , tn ) be continuous on the set ∆n . The generalized partial Riemann–Liouville fractional integral of order α with respect to the ith variable ti is given by Zti α KPt f (t) := pi kα (ti , τ )f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ i

ai

+ qi

Zbi

kα (τ, ti )f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ,

ti

where Pti = hai , ti , bi , pi , qi i. We refer to KPαt as the partial operator K i (partial K-op) of order α and p-set Pti . Definition 2.5 (Generalized partial Riemann–Liouville derivatives). f ∈ C 1 (∆n ). The generalized partial Let Pti = hai , ti , bi , pi , qi i and KP1−α ti Riemann–Liouville fractional derivative of order α with respect to the ith variable ti is given by ∂ 1−α α ◦ KPt f (t) APt f (t) := i i ∂ti  t Zi ∂  = pi k1−α (ti , τ )f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ ∂ti ai

+qi

Zbi

ti



k1−α (τ, ti )f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ  .

The operator AαPt is referred as the partial operator A (partial A-op) of i order α and p-set Pti . Definition 2.6 (Generalized partial Caputo derivative). Let Pti = hai , ti , bi , pi , qi i and f ∈ C 1 (∆n ). The generalized partial Caputo fractional


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derivative of order α with respect to the ith variable ti is given by ∂ 1−α α f (t) BPt f (t) := KPt ◦ i i ∂ti Zti ∂ = pi k1−α (ti , τ ) f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ ∂τ ai

+ qi

Zbi

k1−α (τ, ti )

∂ f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ ∂τ

ti

and is referred as the partial operator B (partial B-op) of order α and p-set Pti . Similarly as in the one-dimensional case [1, 11, 12], the generalized partial operators K, A and B here introduced give the standard partial fractional integrals and derivatives for particular kernels and p-sets. The leftand right-sided Riemann–Liouville partial fractional integrals with respect to the ith variable ti are obtained by choosing the kernel kα (ti , τ ) =

1 (ti − τ )α−1 Γ(α)

and p-sets Lti = hai , ti , bi , 1, 0i and Rti = hai , ti , bi , 0, 1i, respectively: α ai Iti f

(t) = KLαt f (t) i

1 = Γ(α)

Zti

(ti − τ )α−1 f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ,

ai

α α f (t) f (t) = K I ti bi Rt i

1 = Γ(α)

Zbi

(τ − ti )α−1 f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ.

ti

The standard left- and right-sided partial Riemann–Liouville and Caputo fractional derivatives with respect to the ith variable ti are obtained with 1 (ti − τ )−α : if Pti = hai , ti , bi , 1, 0i, the choice of kernel k1−α (ti , τ ) = Γ(1−α)

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then α ai Dti f

(t) = AαPt f (t) i

1 ∂ = Γ(1 − α) ∂ti

Zti

(ti − τ )−α f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ

ai

and C α ai Dti f

(t) = BPαt f (t) i

1 = Γ(1 − α)

Zti

(ti − τ )−α

∂ f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ ; ∂τ

ai

if Pti = hai , ti , bi , 0, 1i, then α α ti Dbi f (t) = − APt f (t) i

∂ 1 =− Γ(1 − α) ∂ti

Zbi

(τ − ti )−α f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ

ti

and C α ti Dbi f

(t) = − BPαt f (t) i

1 =− Γ(1 − α)

Zbi

(τ − ti )−α

∂ f (t1 , . . . , ti−1 , τ, ti+1 , . . . , tn )dτ. ∂τ

ti

Remark 2.1. In Definitions 2.4, 2.5 and 2.6, all the variables, except ti , are kept fixed. That choice of fixed values determines a function ft1 ,...,ti−1 ,ti+1 ,...,tn : [ai , bi ] → R of one variable ti : ft1 ,...,ti−1 ,ti+1 ,...,tn (ti ) = f (t1 , . . . , ti−1 , ti , ti+1 , . . . , tn ). By Definitions 2.1, 2.2, 2.3 and 2.4, 2.5, 2.6, we have α α KPt ft1 ,...,ti−1 ,ti+1 ,...,tn (ti ) = KPt f (t1 , . . . , ti−1 , ti , ti+1 , . . . , tn ), i i α α APt ft1 ,...,ti−1 ,ti+1 ,...,tn (ti ) = APt f (t1 , . . . , ti−1 , ti , ti+1 , . . . , tn ), i i α α BPt ft1 ,...,ti−1 ,ti+1 ,...,tn (ti ) = BPt f (t1 , . . . , ti−1 , ti , ti+1 , . . . , tn ). i

i

Therefore, as in the classical integer order case, computation of partial generalized fractional operators is reduced to the computation of one-variable generalized fractional operators.


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3. Green’s Theorem for Generalized Fractional Derivatives Definition 3.1 (Dual p-set). Let Pti = hai , ti , bi , pi , qi i, i ∈ N. We denote by Pt∗i the p-set Pt∗i = hai , ti , bi , qi , pi i and call it the dual of Pti . Theorem 3.1 (Generalized 2D Integration by Parts). Let α ∈ (0, 1), Pti = hai , ti , bi , pi , qi i be a parameter set, and kα be a difference kernel, i.e., kα (ti , τ ) = kα (ti − τ ) such that kα ∈ L1 ([0, bi − ai ]), i = 1, 2. If f, g, η1 , η2 ∈ C (∆2 ), then the generalized partial fractional integrals satisfy the following identity:

Zb1 Zb2 h

a1 a2

i g(t) KPαt1 η1 (t) + f (t) KPαt2 η2 (t) dt2 dt1 =

Zb1 Zb2

a1 a2

η1 (t)

i i h h KPαt∗ g (t) + η2 (t) KPαt∗ f (t) dt2 dt1 , 2

1

where Pt∗i is the dual of Pti , i = 1, 2. P r o o f. Define

F1 (t, τ ) :=

|p1 kα (t1 − τ )| · |g(t)| · |η1 (τ, t2 )| if τ ≤ t1 |q1 kα (τ − t1 )| · |g(t)| · |η1 (τ, t2 )| if τ > t1

for all (t, τ ) ∈ [a1 , b1 ] × [a2 , b2 ] × [a1 , b1 ] and

F2 (t, τ ) :=

|p2 kα (t2 − τ )| · |f (t)| · |η2 (t1 , τ )| if τ ≤ t2 |q2 kα (τ − t2 )| · |f (t)| · |η2 (t1 , τ )| if τ > t2

for all (t, τ ) ∈ [a1 , b1 ] × [a2 , b2 ] × [a2 , b2 ]. Since f, g and ηi , i = 1, 2, are continuous functions on ∆2 , they are bounded on ∆2 . Hence, there exist real numbers C1 , C2 , C3 , C4 > 0 such that |f (t)| ≤ C1 ,

|g(t)| ≤ C2 ,

|η1 (t)| ≤ C3 ,

|η2 (t)| ≤ C4 ,

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for all t ∈ ∆2 . Therefore, Z Z b1 Z b2 Z b1 F1 (t, τ )dt1 dt2 dτ + a1

a2

=

Z

a1 b1 Z b2

b1

a1

Z

b2

Z

Z

b2

F2 (t, τ )dt2 dτ dt1

a2

a2

b1

|p1 kα (t1 − τ )| · |g(t)| · |η1 (τ, t2 )| dt1 Z τ + |q1 kα (τ − t1 )| · |g(t)| · |η1 (τ, t2 )| dt1 dt2 dτ a2

a1

+ +

Z

a1 b1

Z

b2

τ

Z

b2

|p2 kα (t2 − τ )| · |f (t)| · |η2 (t1 , τ )| dt2 |q2 kα (τ − t2 )| · |f (t)| · |η2 (t1 , τ )| dt2 dτ dt1

a Z 1τ a2

Z

a2

b1

Z

τ

b2

b1

Z

|p1 kα (t1 − τ )| dt1 ≤ C2 C3 τ a2 a1 Z τ |q1 kα (τ − t1 )| dt1 dt2 dτ + a1

+ C1 C4 +

Z

+

+

a1

Z

b2 a2

Z

b2

|p2 kα (t2 − τ )| dt2 τ

|q2 kα (τ − t2 )| dt2 dτ τ

Z

b1

a1 b1 −a1

Z

b2 a2

Z

dt1

b1 −a1

|p1 kα (u1 )| du1 0

|q1 kα (u1 )| du1 dt2 dτ 0

+ C1 C4 Z

b1

b2

≤ C2 C3 Z

Z

Z

b2 −a2

b1 a1

Z

b2 a2

Z

b2 −a2

|p2 kα (u2 )| du2 0

|q2 kα (u2 )| du2 dτ 0

dt1

= C2 C3 (|p1 | + |q1 |) kkα k (b2 − a2 )(b1 − a1 ) + C1 C4 (|p2 | + |q2 |) kkα k (b2 − a2 )(b1 − a1 ) < ∞. Hence, we can use Fubini’s theorem to change the order of integration in the iterated integrals: Z b1 Z b2 h i g(t) KPαt1 η1 (t) + f (t) KPαt2 η2 (t) dt2 dt1 a1

a2


=

Z

b1 a1

b2

Z

a2 Z b1

+ q1

Z g(t) p1

t1

kα (t1 − τ )η1 (τ, t2 )dτ

a1

kα (τ − t1 )η1 (τ, t2 )dτ

t1

Z + f (t) p2 + q2 =

Z

b2 a2

+

Z

a1

Z

+

kα (t2 − τ )η2 (t1 , τ )dτ a2

kα (τ − t2 )η2 (t1 , τ )dτ b1

a1

b2

Z

Z

τ

η2 (t1 , τ ) p2

Z

τ

kα (τ − t2 )f (t)dt2 a2 b2

b2 τ

kα (t2 − τ )f (t)dt2 dτ dt1

η1 (τ, t2 ) KPαt∗ g (τ, t2 )dt2 dτ 1

a2

b1 Z

a1

dt2 dt1

b1

a2

Z

Z

kα (t1 − τ )g(t)dt1 kα (τ − t1 )g(t)dt1 dτ dt2 η1 (τ, t2 ) p1

a1 τ

a1

b1

t2

t2

b1

+ q2 =

Z

b2

Z

+ q1 Z

Z

9

b2 a2

η2 (t1 , τ ) KPαt∗ f (t1 , τ )dτ dt1 . 2

✷ We are now in conditions to state and prove the main result of the paper: the Green theorem for generalized fractional derivatives. Theorem 3.2 (Generalized Green’s Theorem). Let 0 < α < 1 and f, g, η ∈ C 1 (∆2 ). Let kα be a difference kernel, i.e., kα (ti , τ ) = kα (ti − τ ) 1−α 1 such that kα ∈ L1 ([0, bi − ai ]), i = 1, 2, and KP1−α ∗ g, KP ∗ f ∈ C (∆2 ). t2 t1 Then, the following formula holds: Z

b1

a1

Z

b2

a2

h

i g(t) BPαt η (t) + f (t) BPαt η (t) dt2 dt1

=−

1

Z

b1

a1

Z

2

b2 a2

+

η(t) I

h

∂∆2

i AαPt∗ g (t) + AαPt∗ f (t) dt2 dt1 2

1

i h 1−α (t)dt1 . η(t) KP1−α ∗ g (t)dt2 − KP ∗ f t1

t2

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P r o o f. By the definition of generalized partial Caputo fractional derivative, Theorem 3.1, and the standard Green’s theorem, one has Z b1 Z b2 h i g(t) BPαt1 η (t) + f (t) BPαt2 η (t) dt2 dt1 a1

a2

=

Z

b1

a1 Z b1

b2

Z

a2 Z b2

g(t)

∂ KP1−α t1 ∂t

η (t) + f (t) 1

∂ KP1−α t2 ∂t

2

η (t) dt2 dt1

∂ ∂ 1−α 1−α = η(t) KP ∗ g (t) + η(t) KP ∗ f (t) dt2 dt1 t1 t2 ∂t1 ∂t2 a2 a1 Z b1 Z b2 ∂ ∂ 1−α 1−α =− η(t) KP ∗ g (t) + KP ∗ f (t) dt2 dt1 t t ∂t1 ∂t2 1 2 a1 a2 I i h 1−α (t)dt1 . η(t) KP1−α + ∗ g (t)dt2 − KP ∗ f

t2

t1

∂∆2

✷ Corollary 3.1. Let 0 < α < 1 and f, g, η ∈ C 1 (∆2 ). If t1 I1−α g (t) b1 and t2 Ib1−α f (t) are continuously differentiable on the rectangle ∆2 , then 2 Z

b1

a1

Z

b2

a2

g(t) =

Z

C α a1 Dt1 η b1

a1

Z

b2

(t) + f (t)

C α a2 Dt2 η

(t) dt2 dt1

η(t) t1 Dαb1 g (t) + t2 Dαb2 f (t) dt2 dt1 a2 I h i 1−α η(t) t1 I1−α + g (t)dt − I f (t)dt 2 t2 b2 1 . b1 ∂∆2

Acknowledgements This work received The Grunwald–Letnikov Award: Best Student Paper (theory), at the 2012 Symposium on Fractional Differentiation and Its Applications (FDA’2012), May 16, 2012, Hohai University, Nanjing. It was supported by FEDER funds through COMPETE (Operational Program Factors of Competitiveness) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-010124-FEDER-022690. Odzijewicz was also supported by FCT under Ph.D. fellowship SFRH/BD/33865/2009; Malinowska by Bialystok University of Technology grant S/WI/02/2011; and Torres by FCT through the project PTDC/MAT/113470/2009.


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[15] S. Russenschuck, Field Computation for accelerator magnets: analytical and numerical methods for electromagnetic design and optimization, Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim (2010). [16] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon (1993). [17] C.H. Sherman, J.L. Butler, Transducers and arrays for underwater sound, Springer-Verlag (2007). [18] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations, Ann. Physics 323, No 11 (2008), 2756–2778. 1

Center for Research and Development in Mathematics and Applications Department of Mathematics University of Aveiro 3810-193 Aveiro, PORTUGAL e-mail: [email protected], delfi[email protected] 2

Faculty of Computer Science Bialystok University of Technology 15-351 Bialystok, POLAND e-mail: [email protected]

Received: July 09, 2012