Higher-order fractional Green and Gauss formulas

J. Math. Anal. Appl. 462 (2018) 157–171

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Higher-order fractional Green and Gauss formulas Jinfa Cheng a , Weizhong Dai b,∗ a

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, PR China Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA b

a r t i c l e

i n f o

Article history: Received 22 November 2017 Available online xxxx Submitted by J.J. Nieto Keywords: Green’s formula Gauss’s formula Caputo’s fractional derivative

a b s t r a c t Green’s formula and Gauss’s formula are two important formulas in vector calculus. In 2008, Tarasov [12] developed the fractional Green and Gauss formulas and also suggested two possible extensions of his fractional vector formulas (see V.E. Tarasov (2008) [12]). The first possible extension is to prove his fractional integral theorems for a general form of domains and boundaries, such as elementary regions. The second one is to generalize the formulations of fractional integral theorems for α > 1. The purpose of this article is to follow the above two interesting suggestions and present the higher-order fractional Green and Gauss formulas that are the extensions of fractional integral theorems obtained by Tarasov. In particular, the obtained formulas can be reduced to the classical Green and Gauss formulas when α = 1 and the fractional Green and Gauss formulas in [12] when 0 < α ≤ 1, respectively. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Fractional calculus (derivatives and integrals of non-integer order) and fractional differential equations [4–6,8–11] have wide applications in mechanics and physics (see [12,14,15] and references therein). In particular, fractional vector calculus is a very important tool for describing processes in complex media, non-local materials and distributed systems in three-dimensional space (see [14,15] and references therein). For instance, the fractional vector differential operators can be used for non-local continua and distributed systems with long-range power-law interactions [13]. Recently, Tarasov [14] has employed fractional vector differential operators and proposed a three-dimensional lattice approach for describing the fractional non-local continuum in three-dimensional space. Based on this new approach, he obtained a lattice model with long-range interaction for the fractional Maxwell equations of non-local continuous media. Furthermore, using a new fractional variational principle for Lagrangians with Riesz fractional derivatives, Tarasov and Aifantis [15] obtained the governing equations for the new fractional generalizations of the gradient elasticity theory for * Corresponding author. E-mail address: [email protected] (W. Dai). https://doi.org/10.1016/j.jmaa.2018.01.074 0022-247X/© 2018 Elsevier Inc. All rights reserved.

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

158

complex materials with power-law non-locality, long-term memory, and fractality. The fractional generalizations of the gradient elasticity theory may be important in describing unusual properties of nano-materials [1]. It should be pointed out that the main problem in fractional vector calculus appears when one tries to generalize not only differential vector operators (gradient, divergence, curl), but also the related integral theorems [12]. For example, the main problem in the formulation of fractional integral vector operators is connected with the complex form analogue of the Newton–Leibniz formula [15]. The non-commutativity between fractional integral vector operators and differential vector operators makes very difficult to obtain a convenient Riemann–Liouville fractional counterpart of the Newton–Leibniz formula. As pointed out in [15], a robust framework of fractional vector calculus must include generalizations of the differential operators (gradient, divergence, curl), the integral operations (flux, circulation), and the theorems of Gauss, Stokes and Green. In this study, we will focus on the generalization of the theorems of Green and Gauss. Recalling from ´ ˜ vector calculus, the well-known Green’s formula P dx + Qdy = [Qx − Py ]dxdy and Gauss’s formula ∂D D ˚ ¨ P dydz + Qdxdz + Rdxdy = ∂Ω

[Px + Qy + Rz ]dxdydz are two important formulas that describe the Ω

relationship between a line integral around a simple closed and positively oriented curve ∂D and a double integral over the plane D bounded by ∂D, and the relationship between a surface integral over a simple closed and positively oriented surface ∂Ω and a triple integral over the solid region Ω bounded by ∂Ω for  = (P, Q) in two dimensions and F  = (P, Q, R) in three dimensions, respectively. a given vector function F Here, P, Q, R are assumed to be continuously differentiable. In 2008, Tarasov [12] generalized the above classical Green and Gauss formulas to the fractional vector calculus case. For the two-dimensional case, he assumed that P (x, y) and Q(x, y) are absolutely continuous (i.e., continuously differentiable) real-value functions in a rectangular domain D = {(x, y)|a  x  b, c  y  d}, where the boundary of D is the closed curve ∂D. He introduced the following integral notations: α I∂D [x]P (x, y) ≡

1 Γ(α)

˛ (b − x)

α−1

P (x, y)dx,

(1)

α−1

P (x, y)dy,

(2)

∂D α I∂D [y]P (x, y) ≡

α [x, y]P (x, y) ≡ ID

˛

(d − y) ∂D

¨

1 [Γ(α)]

1 Γ(α)

α−1

(b − x)

2

α−1

(d − y)

P (x, y)dxdy,

(3)

D

and obtained a fractional Green formula as follows: Theorem 1.1. (Fractional Green’s formula) Let 0 < α ≤ 1, and P (x, y) and Q(x, y) be absolutely continuous (or continuously differentiable) real-value functions in a rectangular domain as D = {(x, y)|a  x  b, c  y  d}, where the boundary of D is the closed and positively oriented curve ∂D. Then it holds α α α I∂D [x]P (x, y) + I∂D [y]Q(x, y) = ID [x, y]

where

C

C

 α α D∂D [x]Q(x, y)−C D∂D [y]P (x, y) ,

(4)

α D∂D is the Caputo derivative on the boundary.

For the three-dimensional case, he considered a vector function F = (P (x, y, z), Q(x, y, z), R(x, y, z)) and assumed that P, Q, R are continuously differentiable real-value functions in a parallelepiped domain as Ω = {(x, y, z) : a  x  b, c  y  d, g  z  h}, where the boundary of Ω is a closed surface ∂Ω. He introduced the following notations:

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

α α α α (I∂Ω , F ) = I∂Ω [y, z]P + I∂Ω [z, x]Q + I∂Ω [x, y]R ¨ 1 α−1 α−1 (d − y) (h − z) P (x, y, z)dydz = 2 [Γ(α)] ∂Ω ¨ 1 α−1 α−1 (b − x) (h − z) Q(x, y, z)dzdx + 2 [Γ(α)] ∂Ω ¨ 1 α−1 α−1 + (b − x) (d − y) R(x, y, z)dxdy, 2 [Γ(α)]

159

(5)

∂Ω

α F DivΩ α F = IΩα DivΩ

˚

1 [Γ(α)]

=

C α a Dx P

(b − x)

3

+

α−1

C α c Dy Q

(d − y)

+

α−1

C α g Dz R,

(h − z)

α−1

(6) α F dxdydz, DivΩ

(7)

Ω

and obtained a fractional Gauss formula as follows: Theorem 1.2. (Fractional Gauss’s formula) Let 0 < α ≤ 1 and F = (P (x, y, z), Q(x, y, z), R(x, y, z)). Assume that P (x, y, z), Q(x, y, z), R(x, y, z) are continuously differentiable real-value functions in a parallelepiped domain as Ω = {(x, y, z) : a  x  b, c  y  d, g  z  h}, where the boundary of Ω is a closed and positively oriented surface ∂Ω. Then it holds α α (I∂Ω , F ) = IΩα DivΩ F.

(8)

In the conclusion of his article, Tarasov also suggested two possible extensions regarding the above fractional Green and Gauss formulas. The first possible extension is to prove the suggested fractional integral theorems for a general form of domains and boundaries, such as elementary regions and boundaries, and the second one is to generalize the formulations of fractional integral theorems for the case of α > 1. These two extensions are important because they complement the fractional Green and Gauss formulas, which may be used for theoretical analysis for fractional order multi-dimensional partial differential equations and for applications in the theory of non-local media and processes aforementioned. The purpose of this article is to follow the above two interesting suggestions and present the higher-order fractional Green and Gauss formulas that are the extensions of fractional integral theorems obtained by Tarasov in [12]. The organization of the work is set out as follows: several definitions and lemmas are given in section 2; the higher-order fractional Green formula and the higher-order fractional Gauss formula are presented in sections 3 and 4, respectively; and finally, the conclusion is given in section 5. 2. Definitions and lemmas We first introduce the following three definitions and a lemma. Definition 2.1. [7] A function f (x)(x ≥ 0) is said to be in space Cα [0, ∞) where α > 0 if it can be written as (m) f (x) = xp f1 (x) for p > α and f1 (x) is continuous in interval [0, ∞), and it is said to be in space Cα [0, ∞) if the mth-order derivative f (m) (x) ∈ Cα [0, ∞) where m is an integer. Definition 2.2. [3], [4, p. 69], [9, p. 65] Assume f (x) ∈ Cα [a, ∞). The Riemann–Liouville integral operator of order α > 0 is defined as (Da−α f )(x)

1 = Γ(α)

ˆx α−1

(x − t) a

f (t)dt.

(9)

160

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

(m)

Definition 2.3. [3], [4, p. 92], [9, p. 79] Assume f (x) ∈ Cα [a, ∞). The Caputo fractional αth-order derivative of f (x) is defined as C α a Dx f (x)

=

Daα−m f (m) (x)

1 = Γ(m − α)

ˆx

f (m) (t)

α+1−m dt,

a

(x − t)

(10)

where m − 1 < α ≤ m, m is an integer, and x ≥ a. α Lemma 2.1. [3], [4, p. 96] Let Da−α be the Riemann–Liouville integral operator of order α > 0, and C a Dx be the Caputo fractional derivative operator of order α > 0, where 0 < α ≤ 1. Assume aC Dxα f (x) ∈ Cα [a, b]. Then it holds

[Da−α ·

C α a Dx f ](x)

= f (x) − f (a).

(11)

We now consider the case only in the sense of the Caputo fractional derivative for simplicity. For other definitions of fractional derivatives including their theories and applications, we refer the readers to these references [2,5–11,13–17] in the literature. Let D be an elementary region as D = {a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x)}, where y1 (x), y2 (x) are continuously real-value functions. We assume that P (x, y) is a continuously differentiable real-value function in domain D. Defining a function F (t) as F (t) = P (x, y1 (x) + t(y2 (x) − y1 (x))), 0  t  1,

(12)

one may see that the nth-order derivative of F (t) can be written as Dn F (t) = [y2 (x) − y1 (x)]n Pyn (x, y1 (x) + t(y2 (x) − y1 (x))),

(13)

where Pyn is the nth-order partial derivative of P with respect to the second component y, and n is an integer. We define the negative first-order integral of function F (t) as −1 0 Dt F (t)

ˆt =

ˆt P (x, y1 (x) + s(y2 (x) − y1 (x))) ds.

F (s)ds = 0

(14)

0

Thus, the negative nth-order integral of function F (t) can be written as −n 0 Dt F (t) =

1 (n − 1)!

1 = (n − 1)!

ˆt (t − s)

n−1

F (s)ds

(t − s)

n−1

P (x, y1 (x) + s(y2 (x) − y1 (x)))ds.

0

ˆt (15)

0

Introducing y = y1 (x) + s(y2 (x) − y1 (x)), 0 ≤ s ≤ t, and y(t) = y1 (x) + t(y2 (x) − y1 (x)), we substitute them into Eq. (15). This gives −n 0 Dt F (t)

−n

[y (t) − y1 (x)] = 2 (n − 1)!

y(t) ˆ

[y(t) − y]n−1 P (x, y)dy y1 (x)

−n = [y2 (x) − y1 (x)]−n [y1 (x) Dy(t) P ](x, y).

(16)

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

161

We now extend the above negative nth-order integral to the negative fractional order integral. In particular, we have the following three lemmas. Lemma 2.2. Let n − 1 < α ≤ n. Assume that D is an elementary region as D = {a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x)}, where y1 (x), y2 (x) are continuously real-value functions, and P (x, y) is a continuously differentiable real-value function in region D. Then it holds −α 0 Dt F (t)

−α = [y2 (x) − y1 (x)]−α [y1 (x) Dy(t) ]P (x, y),

C α 0 Dt F (t)

C

α = [y2 (x) − y1 (x)]α [y1 (x) Dy(t) ]P (x, y),

−α where 0 Dt−α and y1 (x) Dy(t) are the Riemann–Liouville integral operators of order α > 0, and C α D are the Caputo fractional derivative operators of order α > 0. y1 (x) y(t)

Proof. From Eq. (9) and Eq. (13), we obtain 1 = Γ(α)

−α 0 Dt F (t)

ˆt (t − s)

α−1

F (s)ds

(t − s)

α−1

P (x, y1 (x) + s(y2 (x) − y1 (x)))ds

0

1 = Γ(α)

ˆt 0 y(t) ˆ

−α

[y2 (x) − y1 (x)] = Γ(α)

[y(t) − y]α−1 P (x, y)dy y1 (x)

−α = [y2 (x) − y1 (x)]−α [y1 (x) Dy(t) ]P (x, y),

and C α 0 Dt F (t)

= [0 Dt−n+α Dn ]F (t) 1 = Γ(n − α) 1 = Γ(n − α)

ˆt n−α−1

(Dn F )(s)ds

n−α−1

[y2 (x) − y1 (x)]n Pyn (x, y1 (x) + s(y2 (x) − y1 (x)))ds

(t − s) 0

ˆt (t − s) 0

1 = [y2 (x) − y1 (x)] Γ(n − α)

ˆt n−α−1

(t − s)

n

= [y2 (x) − y1 (x)]

Pyn (x, y1 (x) + s(y2 (x) − y1 (x)))ds

0

− y1 (x)]α−n Γ(n − α)

n [y2 (x)

(17)

y(t) ˆ

[y(t) − y]n−α−1 Pyn (x, y)dy y1 (x)

α = [y2 (x) − y1 (x)]α [C y1 (x) Dy(t) P ](x, y). 2

(18) C α 0 Dt

and

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

162

Lemma 2.3. Let n − 1 < α ≤ n. Assume that D is an elementary region as D = {a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x)}, where y1 (x), y2 (x) are continuously real-value functions, and P (x, y) is a continuously differentiable real-value function in region D. Then it holds ˆ1

yˆ 2 (x)

(1 −

α−1 C α t) [0 Dt F ](t)dt

0

where

C α 0 Dt

α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dy,

=

(19)

y1 (x)

is the Caputo fractional derivative operator of order α > 0 and similarly for

C α y1 (x) Dy .

Proof. By Lemma 2.2, we obtain ˆ1 α−1 C α [0 Ds F ](s)ds

(1 − s) 0

ˆ1 =

(1 − s)

α−1

α [y2 (x) − y1 (x)] [C y1 (x) Dy(s) P ](x, y)ds

α

(1 − s)

α−1

α [y2 (x) − y1 (x)] [C y1 (x) Dy(s) P ](x, y1 (x) + s(y2 (x) − y1 (x)))ds

0

ˆ1 =

α

0

ˆ1 = [y2 (x) − y1 (x)]

(1 − s)

α

α−1 C α [y1 (x) Dy(s) P ](x, y1 (x)

+ s(y2 (x) − y1 (x)))ds

0 yˆ 2 (x)

[y2 (x) − y]

=

α−1 C [y1 (x) Dyα P ](x, y)dy,

y1 (x)

where y = y1 (x) + s(y2 (x) − y1 (x)), 0 ≤ s ≤ 1. 2 Lemma 2.4. (Fractional Taylor’s formula) Let F (t) be a nth-order continuously differentiable real-value α function in interval [a, b] and C a Db be the Caputo fractional derivative operator of order α > 0. Then, the following fractional Taylor formula holds F (b) =

n  F (j−1) (a) j=1

(j − 1)!

(b − a)

j−1

α α + I[a,b] (C a Db F (b)),

n − 1 < α ≤ n,

(20)

α where I[a,b] is defined as that in Eq. (1).

´b α−1 1 α α Proof. Since (I[a,b] F )(b) = Γ(α) (b − s) F (s)ds and (C a Db F )(b) = a one may obtain based on the integration by parts that

α α I[a,b] (C a Db F (b))

=

α I[a,b] [

1 Γ(n − α)

ˆb (b − s) a

n−α (n) α I[a,b] F )(b) = (I[a,b] n = (I[a,b] F (n) )(b)

n−α−1

F

(n)

1 Γ(n−α)

(s)ds]

´b a

n−α−1

(b − s)

F (n) (s)ds,

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

1 = Γ(n)

163

ˆb n−1

(b − s)

F (n) (s)ds

a

F (n−1) (a) 1 n−1 =− (b − a) + (n − 1)! (n − 2)!

ˆb (b − s)

n−2

F (n−1) (s)ds

a

= ··· = F (b) −

n  F (j−1) (a) j=1

(j − 1)!

(b − a)j−1 .

It should be pointed out that Lemma 2.4 is the same as the formula given in Eq. (2.4.42) of lemma 2.22 on page 96 in [4]. Also, Trujillo et al. [16] gave a similar but Riemann–Liouville generalized Taylor formula. 2 3. Higher order fractional Green formulas In this section, we consider two-dimensional cases and derive the higher-order fractional Green formulas. Theorem 3.1. (Higher order fractional Green’s formulas) (i) Let n − 1 < α ≤ n, and D be an elementary region as D = {a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x)} where y1 (x) and y2 (x) are continuous real-value functions. Assume that P (x, y) is the nth-order continuously differentiable with respect to y. Then it holds ˆ P (x, y)dx =

n  j=1

l2

+

1 (j − 1)!

1 Γ(α)

¨

ˆ [y2 (x) − y1 (x)]j−1 Pyj−1 (x, y1 (x))dx l1

α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dxdy,

(21)

D α where l1 is the curve of y = y1 (x), l2 is the curve of y = y2 (x), and a ≤ x ≤ b. Here, C y1 (x) Dy is the Caputo fractional derivative operator of order α > 0. (ii) Let n − 1 < α ≤ n, and D be an elementary region as D = {c ≤ y ≤ d, x1 (y) ≤ x ≤ x2 (y)} where x1 (y) and x2 (y) are continuous real-value functions. Assume that Q(x, y) is the nth-order continuously differentiable with respect to x. Then it holds

ˆ Q(x, y)dy =

n  j=1

l2

+

1 (j − 1)!

1 Γ(α)

¨

ˆ [x2 (y) − x1 (y)]j−1 Qxj−1 (x1 (y), y)dy l1

α [x2 (y) − x]α−1 [C x1 (y) Dx Q](x, y)dxdy,

(22)

D

where l1 is the curve of x = x1 (y), l2 is the curve of x = x2 (y), and c ≤ y ≤ d. Here, fractional derivative operator of order α > 0.

C α x1 (y) Dx

is the Caputo

Proof. Letting F (t) = P (x, y1 (x) + t(y2 (x) − y1 (x))), 0 ≤ t ≤ 1, we have F (1) = P (x, y2 (x)) and F (0) = P (x, y1 (x)). By Lemma 2.4, one may obtain

J. Cheng, W. Dai / J. Math. Anal. Appl. 462 (2018) 157–171

164

n−1 

F (1) − F (0) =

j=1 n−1 

=

j=1

F (j) (0) α α +I[0,1] [C 0 Dt F (t)] j! 1 F (j) (0) + j! Γ(α)

ˆ1 α−1 C α [0 Dt F ](t)dt,

(1 − t)

(23)

0

where n − 1 < α  n. Thus, we obtain from Eq. (13) that ˆ ˆ P (x, y)dx − P (x, y)dx l2

l1

ˆb [P (x, y2 (x))−P (x, y1 (x))]dx

= a

ˆb [F (1) − F (0)]dx

= a

ˆb =

1 F (0)dx + 2! 

ˆb

a

+

a

1 Γ(α)

ˆb

=

F (n−1) (0)dx a

α−1 C α [0 Dt F ](t)dt

(1 − t)

dx 0

1 Py (x, y1 (x))[y2 (x) − y1 (x)]dx + 2!

a

ˆb Py2 (x, y1 (x))[y2 (x) − y1 (x)]2 dx a

+ ... +

+

ˆb

ˆ1

a

ˆb

1 F (0)dx + ... + (n − 1)! 

1 Γ(α)

1 (n − 1)! ˆb

ˆb Pyn−1 (x, y1 (x))[y2 (x) − y1 (x)]n−1 dx a

ˆ1 α−1 C α [0 Dt F ](t)dt.

(1 − t)

dx a

(24)

0

Note that from Lemma 2.3 ˆ1

yˆ 2 (x)

(1 −

α−1 C α t) [0 Dt F ](t)dt

α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dy.

=

0

(25)

y1 (x)

Substituting it into the last term on the right-hand-side of Eq. (24), we obtain 1 Γ(α)

ˆb

ˆ1 α−1 C α [0 Dt F ](t)dt

(1 − t)

dx a

1 = Γ(α)

0

ˆb dx a

=

1 Γ(α)

yˆ 2 (x)

¨

α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dy

y1 (x) α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dxdy.

D

(26)

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´ Hence, after moving the term l1 P (x, y)dx to the right-hand-side, Eq. (24) can be expressed as Eq. (21). Using a similar argument, we may obtain Eq. (22). Thus, we have completed the proof. 2 To verify and illustrate the above higher order fractional Green formulas, we give the following three examples. Example 3.1. Let D be a simple region as D = {0  x  1, 0  y  x2 }, 1 < α ≤ 2, and P (x, y) = x + y α . (x,y) α C α α For this case, n = 2, y1 (x) = 0, y2 (x) = x2 , ∂P∂y = αy α−1 , and [C y1 (x) Dy P ] = 0 Dy (x + y ) = Γ(α + 1). ´1 ´ 2α 1 Thus, the left-hand-side of Eq. (21) can be expressed as l2 P (x, y)dx = 0 (x + x )dx = 12 + 1+2α . On the ´1 2 ´1 other hand, the first term on the right-hand-side of Eq. (21) can be written as 0 (x + 0)dx+ 0 x α0dx = 12 , ˜ ´1 ´ x2 2 2 α−1 α−1 1 while the second term can be simplified as Γ(α) (x −y) Γ(α + 1)dxdy = α 0 dx 0 (x −y) dy = D 1 1+2α . Hence, the left-hand-side of Eq. (21) equals to the right-hand-side of Eq. (21). Example 3.2. Let D = {0  x  1, 0  y  x2 }, 1 < α ≤ 2, and P (x, y) = xy α . For this case, n = 2, (x,y) α C α α y1 (x) = 0, y2 (x) = x2 , ∂P∂y = αxy α−1 , and [C y1 (x) Dy P ] = 0 Dy (xy ) = Γ(α+1)x. Thus, the left-hand-side ´ ´1 2α 1 of Eq. (21) can be expressed as l2 P (x, y)dx = 0 (xx )dx = 2+2α . On the other hand, the first term on the ´1 ´1 3 right-hand-side of Eq. (21) can be written as 0 0dx+ 0 αx 0dx =0, while the second term can be simplified ´1 ´ x2 2 ˜ 2 α−1 α−1 1 1 as Γ(α) (x −y) Γ(α + 1) xdxdy = α 0 xdx 0 (x −y) dy = 2+2α . Hence, the left-hand-side of D Eq. (21) equals to the right-hand-side of Eq. (21). Example 3.3. Let D be a simple region as D = {0  x  1, x  y  1}, 1 < α ≤ 2, and P (x, y) = (y − x)α . (x,y) α C α α For this case, n = 2, y1 (x) = x, y2 (x) = 1, ∂P∂y = α(y −x)α−1 , and [C y1 (x) Dy P ] = x Dy (y −x) = Γ(α+1). ´ ´1 α 1 Thus, the left-hand-side of Eq. (21) can be expressed as l2 P (x, y)dx = 0 (1 − x) dx = 1+α . On the other ´1 ´1 hand, the first term on the right-hand-side of Eq. (21) can be written as 0 0dx+ 0 (1 − x)α0dx =0, while ´1 ´1 ˜ α−1 α−1 1 1 the second term can be simplified as Γ(α) (1 − y) Γ(α + 1)dxdy = α 0 dx x (1 − y) dy = 1+α . D Hence, the left-hand-side of Eq. (21) equals to the right-hand-side of Eq. (21). Corollary 3.1. (Fractional Green’s formula) If 0 < α ≤ 1 and D is a simple region as D = {a  x  b, y1 (x)  y  y2 (x)} = {c  y  d, x1 (y)  x  x2 (y)}, where y1 (x), y2 (x), x1 (y), and x2 (y) are continuous real-value functions. Assume that P (x, y) and Q(x, y) are continuously differentiable in D. Then it holds ˛ P (x, y)dx + Q(x, y)dy ∂D

=

1 Γ(α)

¨ 

 α α−1 C α D Q](x, y)−[y (x) − y] [ D P ](x, y) dxdy, [x2 (y) − x]α−1 [C 2 x1 (y) x y1 (x) y

(27)

D α C α where C x1 (y) Dx and y1 (x) Dy are the Caputo fractional derivative operators of order α > 0 with respect to x and y, respectively. Here, the line integration is performed counterclockwise around ∂D. In particular, when α = 1, Eq. (27) can be reduced to the classical Green formula.

Proof. It can be seen from Theorem 3.1(i) that if 0 < α ≤ 1 (i.e., n = 1), D = {a ≤ x ≤ b, y1 (x) ≤ y ≤ y2 (x)}, and y1 (x), y2 (x) are continuous real-value functions, then it holds ˆ

ˆ P (x, y)dx = l2

implying that

l1

1 P (x, y)dx + Γ(α)

¨ α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dxdy, D

(28)

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166

˛

1 P (x, y)dx = − Γ(α)

¨ α [y2 (x) − y]α−1 [C y1 (x) Dy P ](x, y)dxdy.

(29)

D

∂D

Similarly, if 0 < α ≤ 1, D = {c ≤ y ≤ d, x1 (y) ≤ x ≤ x2 (y)}, and x1 (y), x2 (y) are continuous real-value functions, then it holds from Theorem 3.1(ii) that ˛

1 Q(x, y)dy = Γ(α)

¨ α [x2 (y) − x]α−1 [C x1 (y) Dx Q](x, y)dxdy.

(30)

D

∂D

Adding Eqs. (29) and (30) together, we obtain Eq. (27) and hence complete the proof. 2 Corollary 3.2. Let P (x, y) and Q(x, y) be absolutely continuous (or continuously differentiable) real-value functions in a rectangular domain as D = {(x, y)|a  x  b, c  y  d}, where the boundary of D is the closed and positively oriented curve ∂D. Then it holds α α α α α I∂D [x]P (x, y) + I∂D [y]Q(x, y) = ID [x, y]{C D∂D [x]Q(x, y)−C D∂D [y]P (x, y)},

(31)

which is exactly the same as Eq. (4) that was obtained by Tarasov in [12]. Proof. It can be seen from Corollary 3.1 that if the domain D is a rectangular region as D = {(x, y)|a  x  b, c  y  d}, Eq. (27) can be reduced to ˛ P (x, y)dx + Q(x, y)dy ∂D

1 = Γ(α)

¨   α−1 C α α−1 C α [a Dx Q](x, y) − (d − y) [c Dy P ](x, y) dxdy. (b − x)

(32)

D

Replacing P (x, y) by expressed as 1 Γ(α)

(b−x)α−1 Γ(α) P (x, y)

˛ (b − x) ∂D

=

respectively, Eq. (32) can be further

(d − y)

α−1

Q(x, y)dy

∂D

(b − x)

2

(d−y)α−1 Γ(α) Q(x, y),

˛

1 P (x, y)dx + Γ(α)

¨

1 [Γ(α)]

α−1

and Q(x, y) by

α−1

(d − y)

α−1

C α  α [a Dx Q](x, y) − [C c Dy P ](x, y) dxdy.

(33)

D

By applying those notations given in Eqs. (1)–(3) for Eq. (33), we obtain Eq. (31). 2 Remark 3.1. (i) In Corollary 3.1, D is considered to be a simple region as D = {a  x  b, y1 (x)  y  y2 (x)} = {c  y  d, x1 (y)  x  x2 (y)}. This indicates that we have given a proof for the fractional integral theorem for a more general form of domains and boundaries, which was suggested by Tarasov. (ii) In Theorem 3.1, the value of α is chosen to be in the range of n − 1 < α ≤ n, where n is an integer. Therefore, when n ≥ 2, it indicates α > 1, implying that we have proposed a generalized formulation of the fractional Green theorem for the case of α > 1, which was also suggested by Tarasov. 4. Higher order fractional Gauss formulas In this section, we consider three-dimensional cases and derive the higher-order fractional Gauss formulas.

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Theorem 4.1. (Higher order fractional Gauss’s formulas) (i) Let n − 1 < α ≤ n, and Ω be a simple region as Ω = {(x, y, z)|(y, z) ∈ Dyz , x1 (y, z)  x  x2 (y, z)} where Dyz is a domain in the yz-plane, x1 (y, z) and x2 (y, z) are continuous real-value functions in Dyz . Assume that P (x, y, z) is the nth-order continuously differentiable with respect to x. Then it holds ¨ P (x, y, z)dydz =

n  j=1

Γ2

1 (j − 1)!

1 Γ(α)

+

˚

¨ [x2 (y, z) − x]j−1 Pxj−1 (x1 (y, z), y, z)dydz Γ1

α [x2 (y, z) − x]α−1 [C x1 (y,z) Dx P ](x, y, z)dxdydz,

(34)

Ω α where Γ1 is the surface of x = x1 (y, z), Γ2 is the surface of x = x2 (y, z), and (y, z) ∈ Dyz . Here, C x1 (y,z) Dx is the Caputo fractional derivative operator of order α > 0. (ii) Let n − 1 < α ≤ n, and Ω be a simple region as Ω = {(x, y, z)|(z, x) ∈ Dzx , y1 (z, x)  y  y2 (z, x)} where Dzx is a domain in the zx-plane, y1 (z, x) and y2 (z, x) are continuous real-value functions in Dzx . Assume that Q(x, y, z) is the nth-order continuously differentiable with respect to y. Then it holds

¨ Q(x, y, z)dzdx =

n  j=1

Γ2

1 (j − 1)!

1 Γ(α)

+

˚

¨ [y 2 (z, x) − y]j−1 Qyj−1 (x, y1 (z, x), z)dzdx Γ1

α [y 2 (z, x) − y]α−1 [C y1 (z,x) Dy Q](x, y, z)dxdydz,

(35)

Ω α where Γ1 is the surface of y = y1 (z, x), Γ2 is the surface of y = y2 (z, x), and (z, x) ∈ Dzx . Here, C y1 (z,x) Dy is the Caputo fractional derivative operator of order α > 0. (iii) Let n − 1 < α ≤ n, and Ω be a simple region as Ω = {(x, y, z)|(x, y) ∈ Dxy , z1 (x, y)  z  z2 (x, y)} where Dxy is a domain in the xy-plane, z1 (x, y) and z2 (x, y) are continuous real-value functions in Dxy . Assume that R(x, y, z) is the nth-order continuously differentiable with respect to z. Then it holds

¨ Q(x, y, z)dxdy =

n  j=1

Γ2

+

1 (j − 1)!

1 Γ(α)

˚

¨ [z 2 (x, y) − z]j−1 Rzj−1 (x, y, z1 (x, y))dxdy Γ1

α [z 2 (x, y) − z]α−1 [C z1 (x,y) Dz R](x, y, z)dxdydz,

(36)

Ω

where Γ1 is the surface of z = z1 (x, y), Γ2 is the surface of z = z2 (x, y), and (x, y) ∈ Dxy . Here, is the Caputo fractional derivative operator of order α > 0.

C α z1 (x,y) Dz

Proof. Using a similar argument as that in the proof of Theorem 3.1, we first define F (t) = P (x1 (y, z) + t(x2 (y, z) − x1 (y, z)), y, z) and obtain F (1) = P (x2 (y, z), y, z), F (0) = P (x1 (y, z), y, z). We then obtain ¨

¨ P (x, y, z)dydz − Γ2

¨

P (x, y, z)dydz Γ1

[P (x2 (y, z), y, z) − P (x1 (y, z), y, z)]dydz

= Dyz

¨

[F (1) − F (0)]dydz.

= Dyz

(37)

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By Lemma 2.4, we have n−1 

F (1) − F (0) =

j=1

F (j) (0) α α +I[0,1] (C 0 Dt F (t)) j!

n  F (j−1) (0)

=

1 + (j − 1)! Γ(α)

j=2

ˆ1 α−1 C α [0 Dt F ](t)dt,

(1 − t)

(38)

0

where n − 1 < α  n. Substituting Eq. (38) into Eq. (37) gives ¨

¨ P (x, y, z)dydz − Γ2

P (x, y, z)dydz Γ1

=

n  j=2

=

n  j=2

1 (j − 1)! 1 (j − 1)!

1 + Γ(α)

¨ (j−1)

F

1 (0)dydz + Γ(α)

Dyz

¨

ˆ1 α−1 C α [0 Dt F ](t)dt

(1 − t)

dydz Dyz

0

¨ [x2 (y, z) − x1 (y, z)]j−1 Pxj−1 (x1 (y, z), y, z)dydz Dyz

¨

ˆ1 (1 − t)

dydz

α−1 C α [0 Dt F ](t)dt.

(39)

0

Dyz

By Lemma 2.3, the last term on the right-hand-side of Eq. (39) can be changed to 1 Γ(α)

¨

ˆ1 0

Dyz

1 = Γ(α) 1 Γ(α)

x2ˆ(y,z)

¨ dydz Dyz

=

α−1 C α [0 Dt F ](t)dt

(1 − t)

dydz

˚

α [x2 (y, z) − x]α−1 [C x1 (y,z) Dx P ](x, y, z)dx

x1 (y,z) α [x2 (y, z) − x]α−1 [C x1 (y,z) Dx P ](x, y, z)dxdydz.

(40)

Ω

˜ Substituting Eq. (40) into Eq. (39) and then moving the term Γ1 P (x, y, z)dydz to the right-hand-side, we obtain Eq. (34). Furthermore, Eqs. (35) and (36) can be obtained using a similar argument. Hence, we have completed the proof. 2 To verify and illustrate the higher order fractional Gauss formulas, we give the following example. Example 4.1. Let Ω be a simple region as Ω = {(x, y, z)|0 ≤ z ≤ 1, 0 ≤ y ≤ z, 0  x  yz}, 1 < α ≤ 2, and P (x, y) = yzxα . For this case, n = 2, Dyz ={(y, z)|0 ≤ z ≤ 1, 0 ≤ y ≤ z} and x1 (y, z) = 0, x2 (y, z) = yz. ∂P (x,y,z) α C α α = αyzxα−1 and [C x1 (y,z) Dx P ] = 0 Dx (yzx ) = yzΓ(α + 1). Thus, the left-hand-side of Eq. (34) can ∂x ´1 ´z ˜ α 1 be expressed as Dyz P (x, y)dydz = 0 dz 0 yz(yz) dy = 2(2+α) 2 . On the other hand, the first term on the ˜ ˜ right-hand-side of Eq. (34) can be written as Dyz yz0dydz+ Dyz (yz − 0)αyz0dydz =0, while the second

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´1 ´z ´ yz ˝ α−1 1 term can be simplified as Γ(α) [yz − x]α−1 yzΓ(α + 1)dxdydz = α 0 dz 0 dy 0 yz (yz − x) dx = Ω 1 . Hence, the left-hand-side of Eq. (34) equals to the right-hand-side of Eq. (34). 2(2+α)2 Corollary 4.1. (Fractional Gauss’s formula) Let 0 < α ≤ 1 and Ω be a three-dimensional simple region as Ω = {(x, y, z)|(x, y) ∈ Dxy , z1 (x, y)  z  z2 (x, y)} = {(x, y, z)| (y, z) ∈ Dyz , x1 (y, z)  x  x2 (y, z)} = {(x, y, z)|(z, x) ∈ Dzx , y1 (z, x)  y  y2 (z, x)}. Assume that P (x, y, z), Q(x, y, z) and R(x, y, z) are continuously differentiable. Then it holds ¨

¨ P (x, y, z)dydz + ∂Ω

1 = Γ(α)

˚

¨ Q(x, y, z)dzdx +

∂Ω

R(x, y, z)dxdy ∂Ω

α α−1 C {[x2 (y, z) − x]α−1 [C [y1 (z,x) Dyα Q](x, y, z) x1 (y,z) Dx P ](x, y, z) + [y2 (z, x) − y] Ω

α + [z2 (x, y) − z]α−1 [C z1 (x,y) Dz R](x, y, z)}dxdydz,

(41)

where the surface integration is performed based on the positive orientation. In particular, when α = 1, Eq. (41) can be reduced to the classical Gauss formula. Proof. It can be seen from Theorem 4.1(i) that if 0 < α ≤ 1 (i.e., n = 1), Dyz is a domain in the yz-plane and Ω = {(x, y, z)|(y, z) ∈ Dyz , x1 (y, z)  x  x2 (y, z)}, then ¨ P (x, y, z)dydz Γ2

¨ =

1 P (x, y, z)dydz + Γ(α)

˚ α [x2 (y, z) − x]α−1 [C x1 (y,z) Dx P ](x, y, z)dxdydz,

Γ1

(42)

Ω

implying that ¨ P (x, y, z)dydz =

1 Γ(α)

˚ α [x2 (y, z) − x]α−1 [C x1 (y,z) Dx P ](x, y, z)dxdydz.

(43)

Ω

∂Ω

Here, Γ1 is the surface of x = x1 (y, z), Γ2 is the surface x = x2 (y, z), and (y, z) ∈ Dyz . Similarly, if 0 < α ≤ 1, Dzx is a domain in the zx-plane, and Ω = {(x, y, z)| (z, x) ∈ Dzx , y1 (z, x)  y  y2 (z, x)}, then it holds by Theorem 4.1 (ii) that ¨ Q(x, y, z)dzdx =

1 Γ(α)

˚ α [y2 (z, x) − y]α−1 [C y1 (z,x) Dy Q](x, y, z)dxdydz,

(44)

Ω

∂Ω

and if 0 < α ≤ 1, Dxy is a domain in the xy-plane, Ω = {(x, y, z)|(x, y) ∈ Dxy , z1 (x, y)  z  z2 (x, y)}, then it holds by Theorem 4.1 (iii) that ¨ R(x, y, z)dxdy = ∂Ω

1 Γ(α)

˚ α [z2 (x, y) − z]α−1 [C z1 (x,y) Dz R](x, y, z)dxdydz.

(45)

Ω

Adding Eqs. (43)–(45) together, we obtain Eq. (41) and hence complete the proof. 2 Corollary 4.2. Let F = (P (x, y, z), Q(x, y, z), R(x, y, z)) be a vector function and P (x, y, z), Q(x, y, z), R(x, y, z) be continuously differentiable real-value functions in a parallelepiped domain as Ω = {(x, y, z) :

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170

a  x  b, c  y  d, g  z  h}, where the boundary of Ω is a closed and positively oriented surface ∂Ω. If 0 < α ≤ 1, then it holds α α (I∂Ω , F ) = IΩα DivΩ F,

(46)

which is exactly the same as Eq. (8) that was given by Tarasov in [12]. Proof. It can be seen that if Ω is a parallelepiped region as Ω = {(x, y, z) : a  x  b, c  y  d, g  z  h}, then Eq. (41) can be simplified to ¨

¨ P (x, y, z)dydz + ∂Ω

=

1 Γ(α)

¨ Q(x, y, z)dzdx +

∂Ω

˚

R(x, y, z)dxdy ∂Ω

α−1 C α [a Dx P ](x, y, z)

{(b − x)

+ (d − y)

α−1 C α [c Dy Q](x, y, z)

Ω

+ (h − z) Replacing P (x, y, z) by α−1

[(b−x)(d−y)] Γ(α)

α−1 C α [g Dz R](x, y, z)}dxdydz.

[(d−y)(h−z)]α−1 P (x, y, z), Γ(α)

(47)

Q(x, y, z) by

[(b−x)(h−z)]α−1 Q(x, y, z), Γ(α)

and R(x, y, z) by

R(x, y, z), respectively, Eq. (47) can be further expressed as

¨ [(d − y)

α−1

(h − z)

α−1

α−1

P (x, y, z)dydz + (b − x)

(h − z)

α−1

Q(x, y, z)dzdx

∂Ω α−1

α−1

+ (b − x) (d − y) R(x, y, z)dxdy] ˚ 1 α−1 α−1 α−1 C α α C α = (b − x) (d − y) (h − z) [a Dx P +C c Dy Q +g Dz R]dxdydz. Γ(α)

(48)

Ω

By applying those notations given in Eqs. (5)–(7) for Eq. (48), we obtain Eq. (46). 2 Remark 4.1. (i) In Corollary 4.1, Ω is considered to be a three-dimensional simple region as Ω = {(x, y, z)|(x, y) ∈ Dxy , z1 (x, y)  z  z2 (x, y)} = {(x, y, z)|(y, z) ∈ Dyz , x1 (y, z)  x  x2 (y, z)} = {(x, y, z)|(z, x) ∈ Dzx , y1 (z, x)  y  y2 (z, x)}. This indicates that we have given a proof for the fractional integral theorem for a more general form of domains and boundaries, which was suggested by Tarasov. (ii) In Theorem 4.1, the value of α is chosen to be in the range of n − 1 < α ≤ n, where n is an integer. Therefore, when n ≥ 2, it indicates α > 1, implying that we have proposed a generalized formulation of the fractional Gauss theorem for the case of α > 1, which was also suggested by Tarasov. 5. Conclusion We have obtained the higher-order fractional Green and Gauss formulas that are the extensions of the fractional Green and Gauss formulas obtained by Tarasov in [12]. The higher-order fractional formulas can be reduced to the classical Green and Gauss formulas when α = 1 and the fractional Green and Gauss formulas in [12] when 0 < α ≤ 1, respectively. The obtained higher-order fractional Green and Gauss formulas may be used for theoretical analysis for fractional order multi-dimensional partial differential equations, such as well-posedness, and for applications in the theory of non-local media and processes.

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Acknowledgments We would like to express our gratitude to the anonymous reviewers for their valuable comments and constructive suggestions. The research is supported by the Fundamental Research Funds for the Central Universities, grant No. 20720150006, and Natural Science Foundation of Fujian Province, grant No. 2016J01032. References [1] E.C. Aifantis, Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals, Metall. Mater. Trans. A 42 (2011) 2985–2998. [2] J.F. Cheng, Theory of Fractional Difference Equations, Xiamen University Press, Xiamen, China, 2011 (in Chinese). [3] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [4] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, vol. 204, Elsevier, Amsterdam, 2006. [5] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Press, San Francisco, 1979. [6] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. [7] Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007) 286–293. [8] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [9] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [10] B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, Lecture Notes in Math. 457 (1975) 1–36. [11] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Philadelphia, 1993. [12] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations, Ann. Physics 323 (2008) 2756–2778. [13] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2011. [14] V.E. Tarasov, Toward lattice fractional vector calculus, J. Phys. A: Math. Theor. 47 (2014) 355204. [15] V.E. Tarasov, E.C. Aifantis, Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality, Commun. Nonlinear Sci. Numer. Simul. 22 (2015) 197–227. [16] J.J. Trujillo, M. Rivero, B. Bonilla, On a Riemann–Liouville generalized Taylor’s formula, J. Math. Anal. 231 (1999) 255–265. [17] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford, London, 2005.