Introduction to fractional integrability and differentiability - Springer Link

Eur. Phys. J. Special Topics 193, 5–26 (2011) c EDP Sciences, Springer-Verlag 2011  DOI: 10.1140/epjst/e2011-01378-2

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Review

Introduction to fractional integrability and differentiability Dedicated to Professor Yushu Chen on the Occasion of his 80th Birthday C.P. Lia and Z.G. Zhaob Department of Mathematics, Shanghai University, Shanghai 200444, China Received 01 December 2010 / Received in final form 27 January 2011 Published online 4 April 2011 Abstract. In this paper, we mainly consider fractional integral and derivatives including the Riemann-Liouville derivative, Caputo derivative, Gr¨ unwald-Letnikov derivative, Marchaud derivative, Riesz derivative, local fractional derivative, Canavati derivative. Then we introduce their existence conditions. Important issues on these fractional integral and derivatives are also included.

1 Introduction Fractional calculus is considered as the generalization of the classical (or integer-order) calculus with a history of at least three hundred years. It can be dated back to the Leibniz’s letter to L’Hospital and Wallis, see Leibiniz [1], dated 30 September 1695, in which the meaning of the one-half order derivative was first discussed and made some remarks about its possibility. “Subsequent mention of fractional derivatives was made, in some context or the other by (for example) Euler in 1730, Lagrange in 1772, Laplace in 1812, Lacroix in 1819, Fourier in 1822, Riemann in 1847, Green in 1859, Holmgren in 1865, Gr¨ unwald in 1867, Letnikov in 1868, Sonini in 1869, Laurent in 1884, Nekrassov in 1888, Krug in 1890, and Weyl in 1919, etc” [2]. Till now, it has been found that fractional calculus has many applications in various fields, such as in viscoelastic mechanics, chaos, chaos synchronization, anomalous diffusion phenomenon and power law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and so on [3–11]. Although the history of fractional calculus is three hundred years old, there are still a number of basic problems to be addressed. For example, what is the fractional differentiability of given functions, i.e., what is the condition of existence of the fractional derivatives. This problem is important and meaningful from the viewpoint of the theoretical analysis, such as the existence of solution and the stability analysis of a b

e-mail: [email protected] e-mail: [email protected]

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the fractional differential equation, and so on [12–20]. In this paper, we firstly introduce several kinds of fractional derivatives and then discuss them. The main reference is the encyclopedic book [21]. Therefore, this paper can be considered as a survey of fractional integrability and differentiability. In fact, the proper history of fractional calculus with its applications began with the papers by Abel in [22] (in 1823) and [23] (in 1826). The Abel integral equation can be regarded as the inverse of the fractional differential equation, which plays an important role in the fractional calculus. Therefore we will introduce the fractional differentiability from the viewpoint of the Abel integral equation in Sec. 2. Then we give some definitions of the fractional derivatives, such as the Riemann-Liouville derivative, Caputo derivative, Gr¨ unwald-Letnikov derivative, Marchaud derivative, Riesz derivative, local fractional derivative, Canavati derivative, and so on, and then further discuss their fractional differentiability in the following Secs. 3–11.

2 Abel integral equation In 1823, Abel considered a mechanical problem, namely Abel’s mechanical problem [24]. In the absence of friction, the problem is reduced to an integral equation  y  (y − z)−1/2 u(z)dz = 2gf (y), y ∈ [0, H], (1) 0

 where u(z) = 1 + φ 2 (z), φ(z) is an increasing function, g is the constant downward acceleration, f (y) is a prescribed function. Then Abel solved (1) in [22] and [23]. “Although it was not performed in the spirt of the idea of how to generalize differentiation, they played an enormous role in the development of these ideas” [21]. Also an Abel transform of a sufficiently well behaved function u was generalized to  x 1 (x − t)α−1 u(t)dt, a < x < b, (2) Γ(α) a where −∞ ≤ a < b ≤ ∞, α ∈ (0, 1). Here we discuss under what hypothesis the solution of classical Abel integral equation exists. That is also under what hypothesis the fractional derivative with order α ∈ (0, 1) exists in L1 (a, b). An answer to this question can be found in Tonelli [25]. Lemma 1. Consider, for α ∈ (0, 1), −∞ ≤ a < b ≤ ∞, the classical Abel integral equation  x 1 (x − t)α−1 u(t)dt = f (x), a < x < b. (3) Γ(α) a Then there exists at most one solution of equation (3) in L1 (a, b). Moreover, if the function f is absolutely continuous on [a, b], then equation (3) has a solution in L1 (a, b), given by (4)  x 1 d u(x) = (x − t)−α f (t)dt, a < x < b. (4) Γ(1 − α) dx a If a and f (a) are finite, then    x 1 f (a)(x − a)−α + u(x) = (x − t)−α f  (t)dt , Γ(1 − α) a

a < x < b.

(5)

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If a is finite and f is extended by 0 to the left of a, then  x 1 u(x) = (x − t)−α df (t), a < x < b. Γ(1 − α) a−0 If a = −∞ and limx→−∞ |x|1−α f (x) = 0, then  x 1 u(x) = (x − t)−α df (t), Γ(1 − α) −∞ u(x) =

1 Γ(1 − α)



x

−∞

(x − t)−α f  (t)dt,

−∞ < x < b, −∞ < x < b.

7

(6)

(7)

(8)

Remark 1. From the viewpoint of fractional calculus, we can see that (4)–(8) are just some other forms of fractional derivatives, with order α ∈ (0, 1), under some different hypotheses on f . If the absolutely continuous condition of f (x) is replaced by the bounded variation and continuous conditions, we have the following lemma. Lemma 2. ([24]) If the function f is of bounded variation and continuous from the right, then equation (3) has a solution in L1 (a, b), given by (9)  x 1 u(x) = (x − t)−α df (t), a < x < b, (9) Γ(1 − α) a where the integral is in the Lebsgue-Stieltjes sense. Also if a = −∞ and |x|1−α f (x) → 0 as x → −∞, then equation (3) has a solution in L1 (a, b), given by (10)  x  x 1 1 −α  u(x) = (x − t) f (t)dt = (x − t)−α df (t). (10) Γ(1 − α) a Γ(1 − α) a Remark 2. In fact, (9) and (10) are some other forms of the Caputo derivative, with order α ∈ (0, 1), here we just give the necessary conditions of their existences. Remark 3. The formula (4) is more general than formulas (9) and (10). In [22], the conclusion was tested by an example from the viewpoint of the Abel integral equation, where the function was chosen as ⎧ 0, 0 ≤ x ≤ x0 , ⎪ ⎪ ⎨ f (x) = Γ(1 + λ) ⎪ ⎪ (x − x0 )λ+α , x0 < x ≤ 1, ⎩ Γ(1 + λ + α) in which −1 < λ < −α. Obviously this function is not of bounded variation in [0, 1], hence formula (10) can not be used, but the function  0, 0 ≤ x ≤ x0 , u(x) = (x − x0 )λ , x0 < x ≤ 1, is the solution of

That is to say

1 Γ(α)

 0

x

(x − t)−α u(t)dt = f (x).

α RL Da,x f (x) exists and equals to u(x), which will be defined later.

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Here, we found a lemma in [24] and [21] (but we guess it was firstly revealed by α f (x) ∈ Tamarkin [26] in 1930) about a sufficient and necessary condition for RL Da,x 1 L (a, b) when α ∈ (0, 1): Lemma 3. For α ∈ (0, 1), −∞ < a < b < ∞, there exists a function u ∈ L1 (a, b) such that  x 1 (x − t)α−1 u(t)dt = f (x), a ≤ x ≤ b, (11) Γ(α) a if and only if f ∈ L1 (a, b) and the function  x 1 −(1−α) f (x) = (x − t)−α f (t)dt, a ≤ x ≤ b RL Da,x Γ(1 − α) a is absolutely continuous with Remark 4. That is to say,

−(1−α) f (a) RL Da,x

α RL Da,x f

(12)

= 0.

exists and belongs to L1 (a, b) for 0 < α < 1 if

and only if f ∈ L1 (a, b) and (12) holds with

−(1−α) f (a) RL Da,x

= 0.

3 Riemann-Liouville derivative In 1832-1837 a series of papers by Liouville [27–34] reported the earliest form of the fractional integral, though not quite rigorously from the mathematical point of view. The formula was taken as follows  ∞ 1 φ(x + t)tp−1 dt, −∞ < x < ∞, p > 0. (13) D−p φ(x) = (−1)p Γ(p) 0 That is now called the Liouville form of fractional integral with the factor (−1)p being omitted. Next the significant work was done by Riemann [35], who wrote that paper in 1847 when he was just a student. But it was published until 1876, ten years after his death. Riemann had arrived at the expression  x φ(t)dt 1 , x > 0, (14) Γ(α) 0 (x − t)1−α for fractional integration. Furthermore, we have the most useful forms of left-hand and right-hand RiemannLiouville derivatives defined as follows  x 1 dm α D f (x) = (x − t)m−α−1 f (t)dt, (15) RL a,x Γ(m − α) dxm a  b (−1)m dm α (t − x)m−α−1 f (t)dt, (16) RL Dx,b f (x) = Γ(m − α) dxm x where m − 1 ≤ α < m, a, b are the terminal points of the interval [a, b], which can also be −∞, ∞. Lemma 4. ([21]) If f (x) ∈ AC m ([a, b]), then the fractional derivatives α α RL Da,x f, RL Dx,b f exist almost everywhere on [a, b] and can be represented in the forms  x m−1 f (k) (a)(x − a)k−α 1 α + (x − τ )m−α−1 f (m) (τ )dτ, RL Da,x f (x) = Γ(k − α + 1) Γ(m − α) a k=0 (17)

Perspective on Fractional Dynamics and Control

α RL Dx,b f (x)

=

m−1 k=0

(−1)m (−1)k f (k) (b)(b − x)k−α + Γ(k − α + 1) Γ(m − α)



b

x

9

(τ − x)m−α−1 f (m) (τ )dτ,

(18) where AC m (Ω) represents the space of functions which have continuous derivatives up to order m − 1 on Ω with f (m−1) (x) ∈ AC(Ω). For the necessary condition of the lemma, we can even weaken the condition to the following form [21] (take 0 < α < 1 for example) f (x) =

f ∗ (x) , μ, ν ∈ [0, 1 − α), (x − a)μ (b − x)ν

(19)

where f ∗ (x) ∈ AC([a, b]). That is f  (x) isn’t necessarily integrable at the point x = a or x = b respectively. Remark 5. Function of the form (19) is representable by a fractional integral f = −α −α 1 RL Da,x φ or f = RL Dx,b φ of a summable function φ ∈ L (a, b). For the function (19), its fractional derivatives exist and can be written as follows  x 1 (1 − α)f (t) + (t − a)f  (t) α dt, 0 < α < 1, (20) RL Da,x f (x) = Γ(1 − α)(x − a) a (x − t)α α RL Dx,b f (x)

1 = Γ(1 − α)(b − x)



b x

(1 − α)f (t) − (b − t)f  (t) dt, 0 < α < 1. (t − x)α

(21)

Remark 6. The above equations can be extended to value α > 1 α RL Da,x f (x) =

1 Γ(m − α)(x −a)m

 a

x

(t −a)α [(t − a)m−α f (t)](m) dt, m−1 < α < m. (x − t)α−m+1 (22)

3.1 The sufficient and necessary conditions of existence of Riemann-Liouville derivative Stein and Zygmund [36] discussed the fractional differentiability in the following sense f (α) (x) =

dk+1 fβ (x), (β = k + 1 − α) dxk+1

(23)

where fβ indicates the integral of f with order β which is adopted by Riesz’s definition  ∞ fβ (x) = f (t)|x − t|β−1 dt = (f ∗ K1−β )(x), (0 < β < 1) (24) −∞

where Kγ (x) = |x|−γ . The main idea of [36] is that f (α) exists at x0 if fβ = f ∗ K1−β has a (k + 1)th derivative in the sense of (23), with β = k + 1 − α, and also f (α) exists in the L2 sense at x0 if fβ has a (k + 1)th derivative in the L2 sense at x0 . There are some conditions needed to be given to characterize the existence of such a αth order derivative.

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(i) If α is fractional, k < α < k + 1, f satisfies a Lipschitz condition of order α at x0 , that is f belongs to Λα at x0 if R(x) = O(|x|α )

as

x → 0,

(25)

where R(x) is the Peano remainder of f ∗ K1−β at x0 . (ii) Similarly, f belongs to Λ2α at x0 , if   x 1/2 1 2 |R(t)| dt = O(|x|α ) x 0

as

x → 0.

(iii) f satisfies condition Nα2 at x0 if  δ [R(t)]2 dt < ∞ 1+2α −δ |t|

(26)

(27)

for some δ > 0. ∗ If R(x) is the Peano remainder of f ∗ K1−β at x0 , where Kγ∗ (x) = sign x|x|−γ , we 2 2 call (25), (26) and (27) as Λα , Λ α and N  α correspondingly. Lemma 5. ([37]) If f belongs to Λα for each point x0 of a set E (here we denote it as the set of (a, b), where a, b can be −∞ and ∞ correspondingly), then f (ν) exists a.e. if 0 < ν < α. From the discussion of the paper [36], we know that Lemma 6. ([36]) Suppose that f satisfies the condition Λα at every point of a set E with positive measure. Then f (α) (x) exists almost everywhere in E if and only if f satisfies condition Nα2 almost everywhere in E. Lemma 7. ([36]) The necessary and sufficient condition that f satisfies the condition Nα almost everywhere in a set E is that f satisfies the condition Λ2α , and f (α) exists in the L2 sense, almost everywhere in E. The following assumptions were stated in [38] which generalized the cases in [36] from L2 to Lp (2 ≤ p < ∞), and also from R1 to Rn : (i) f satisfies Λpα at x0 if 

1 ρn

 |x| 0  |R(x)|p dx < ∞. n+pα |x|≤ρ |x| Lemma 8. ([38]) f (α) exists in the sense of Lp (2 ≤ p < ∞) and f satisfies the condition Λpα almost everywhere in a set E if and only if f satisfies both Nα2 and Nαp almost everywhere in the set E. Theorem 1. Suppose that f satisfies the condition Λα at every point of a set E with α α f exists almost everywhere in E (so does RL Dx,b f, positive measure. Then RL Da,x 2 2 which is omitted here) if and only if f satisfies conditions Nα and Nα almost everywhere in E.

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11

α Theorem 2. RL Da,x f exists in the sense of Lp (2 ≤ p < ∞) and f satisfies the p p conditions Λα , Λ α almost everywhere in a set E if and only if f satisfies Nα2 , Nαp , 2 p N  α and N  α almost everywhere in the set E.

Proof. (Proofs of Theorems 1 and 2) They are obviously true just because Riemannα α f and RL Dx,b f are the special cases of (23), where fβ is Liouville derivatives RL Da,x defined by    x 1 ∗ f ∗ (K1−β + K1−β f (t)(x − t)β−1 dt = ) (x), (28) 2 −∞ 

∞

x

β−1

f (t)(t − x)

 dt =

 1 ∗ f ∗ (K1−β − K1−β ) (x), 2

(29)

where Kγ∗ (x) = sign x|x|−γ . Theorem 3. f (α) exists in the sense of Lp (1 ≤ p < 2) and f satisfies the condition 2 Λp∗ α (2 ≤ p∗ < ∞) almost everywhere in a set E if and only if f satisfies both Nα and p∗ Nα almost everywhere in the set E. If we further assume that there exists a constant K such that for all such p (1 ≤ p < ∞)  f (α) p ≤ K,

(30)

then f (α) exists in the sense of L∞ and  f (α) ∞ ≤ K.

(31)

Proof. From the condition Λp∗ α , we know that f has αth order derivative in the sense of Lp∗ (2 ≤ p∗ < ∞), that is  f (α) p∗ < ∞. Therefore, by using the imbedding relationship Lp∗ → Lp for 1 ≤ p ≤ p∗ < ∞, we have that  f (α) p ≤ f (α) p∗ < ∞. Therefore, if f satisfies both Nα2 and Nαp∗ almost everywhere in the set E, we / L∞ derive that f has an αth order derivative in the sense of Lp (1 ≤ p < ∞). If f ∈ ∞ or else if f ∈ L , but (31) does not hold, then we can find a constant K1 > K and a set A ⊂ E with measure μ(A) > 0 such that for x ∈ A, |f (α) (x)| ≥ K1 . Therefore,   (α) p |f (t)| dt ≥ |f (α) (t)|p dt ≥ μ(A)K1p . E

A

It follows that  f (α) p ≥ (μ(A))1/p K1 , whence lim inf p→∞  f (α) p ≥ K1 , which contradicts (30). α α f and RL Dx,b f exist in the sense of Lp (1 ≤ p < ∞) and f Corollary 1. RL Da,x p∗  p∗ satisfies the conditions Λα , Λ α (2 ≤ p∗ < ∞) almost everywhere in a set E, if f 2 p∗ satisfies Nα2 , Nαp∗ , N  α and N  α almost everywhere in the set E. If we further assume α α f p ≤ K,  RL Dx,b f p ≤ K for all that there exists a constant K such that  RL Da,x α α ∞ α such p, then RL Da,x f and RL Dx,b f exist in the sense of L , and  RL Da,x f ∞ ≤ K, α  RL Dx,b f ∞ ≤ K.

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3.2 Remarks about the nonexistence of Riemann-Liouville derivative −α −α u and RL Dx,b u exist almost everywhere for the integrable function, Remark 7. RL Da,x even if function u has finite discontinuity points. −α −α u and RL Dx,∞ u exist almost everywhere for function u ∈ Remark 8. ([21]) RL D−∞,x p L (−∞, ∞) if 0 < α < 1 and 1 ≤ p < 1/α.

From Sec. 3.1, we always talk the fractional differentiability in the sense of almost everywhere. But at any point in the integral interval, we observe that ([39]) Remark 9.

α RL Da,x u

possibly does not exist at the jump discontinuity points of u.

Consider a discontinuous function which has a jump discontinuity point for example  1 − x, if 0 < x ≤ 1, u(x) = (32) 2 − x, if 1 < x < 2. Then u(x) exists the classical derivative at x = 0. However, if 0 < α < 1, ⎧ x1−α x−α ⎪ ⎪ − , if 0 < x ≤ 1, ⎪ ⎨ Γ(1 − α) Γ(2 − α) α D u(x) = RL 0,x ⎪ (x − 1)−α x1−α x−α ⎪ ⎪ + − , if 1 < x < 2. ⎩ Γ(1 − α) Γ(1 − α) Γ(2 − α) Therefore

α RL D0,x u(x)

Remark 10.

α RL Da,x u

(33)

does not exist at x = 0 and x = 1. possibly exists at the removable discontinuity points of u.

Consider a discontinuous function which has a removable discontinuity point for example ⎧ if 0 < x < 1, ⎪ ⎨x, u(x) =

⎪ ⎩

0,

if x = 1,

(34)

2 − x, if 1 < x < 2.

Then, if 0 < α < 1, we get

α RL D0,x u(x)

=

⎧ 1−α x ⎪ ⎪ ⎪ ⎨ Γ(2 − α) ,

if 0 < x < 1,

⎪ x1−α − 2(x − 1)1−α ⎪ ⎪ , if 1 < x < 2. ⎩ Γ(2 − α)

α α Therefore limx→1− RL D0,x u(x) = limx→1+ RL D0,x u(x) = α RL D0,x u(x) exists at x = 1.

Remark 11. kind of u.

α RL Da,x u

1 Γ(2−α) .

(35)

Then we know that

possibly does not exist at the discontinuity points of the second 

Consider u(x) =

0 ≤ x ≤ x0 ,

0, λ

(x − x0 ) , x0 < x ≤ 1,

(36)

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where −1 < λ < 0. Then, if 0 < α < 1, we have ⎧ 0, ⎪ ⎪ ⎨ α RL D0,x u(x) = Γ(1 + λ) ⎪ ⎪ (x − x0 )λ−α , ⎩ Γ(1 + λ − α)

13

0 ≤ x ≤ x0 , x0 < x ≤ 1.

(37)

α u(x) does not exist at x0 . Therefore RL D0,x From (33), (35) and (37), we get the following remark. α u(x) possibly has no connection with Dm u(x), Remark 12. The existence of RL Da,x α i.e., RL Da,x u(x) cannot be said a generalization of Dm u(x), where m − 1 < α < m.

3.3 Riemann-Liouville derivative of the monotone function Lemma 9. Suppose that f (x) is a monotone function on [a, b], then (1) there exists f  (x) almost everywhere over [a, b]; (2) f  (x) is integrable in [a, b]; b (3) if f (x) is an increasing function, then a f  (x)dx ≤ f (b) − f (a). By using Lemma 9, we can quickly get the following theorem. Theorem 4. Suppose that f (x) is a monotone function over [a, b], then α −α −α α RL Da,x f, RL Dx,b f exist and are differentiable, so do RL Da,x f, RL D x,b f almost everywhere over [a, b].

3.4 Riemann-Liouville derivative of the Weierstrass function In this section, we discuss the Weierstrass function which is continuous everywhere but differentiable nowhere. One form is as follows W (x) = λ−μj sin(λj x), (0 < μ < 1, λ > 1). (38) j≥1

Then we get −α RL D0,x f (x)

=



λ−μj Sx (α, λj ), 0 < α + μ < 1,

(39)

λ(1−μ)j Cx (1 − β, λj ), 0 < β < μ < 1,

(40)

j≥1 β RL D0,x f (x)

=

j≥1

where −α RL D0,x

1 sin ax = Γ(α)

−α RL D0,x

1 cos ax = Γ(α)

 

x

0

0

x

tα−1 sin a(x − t)dt := Sx (α, a), tα−1 cos a(x − t)dt := Cx (α, a).

These results show that the αth order Riemann-Liouville integral and the βth order Riemann-Liouville derivative of the Weierstrass function exist for 0 < α + μ < 1, 0 < β < μ < 1 correspondingly [40].

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4 Gr¨ unwald-Letnikov derivative Gr¨ unwald [41] (in 1867) and Letnikov [42] (in 1868) developed an approach to fractional differentiation based on the definition GL D

where

α



α h f (x) =

( α h f )(x) , h→0 hα

f (x) = lim

(41)

  α f (x − jh), (h > 0). j

(−1)|j|

0≤|j| 0 [21].  ∞ (m−1) dm−1 {α} {α} f (x) − f (m−1) (x − t) α D f = dt, (49) M D+ f = + dxm−1 Γ(1 − {α}) 0 t1+{α} α M D− f

dm−1 {α} {α} = m−1 D− f = dx Γ(1 − {α})

 0

∞

f (m−1) (x) − f (m−1) (x + t) dt, t1+{α}

(50)

where {α} = α − m + 1, m = α . (49) and (50) will be called Marchaud derivatives (in 1927), or called Weyl-Marchaud derivatives. Remark 17. ([21]) Evidently, (49) and (50) exist for bounded function f (m−1) satisfying the local H¨ older condition of order λ > α. This may be weakened to λ = α if one takes function f (m−1) belonging locally to the space H α,−a (a > 1) and bounded at infinity.

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Remark 18. Marchaud derivative M Dα f is more convenient on R1 than Liouville derivative RL Dα f as the former allows more freedom for f (x) at infinity. Marchaud derivative for a “not very good” function f (x) will be understood to be conditionally convergent [21]. Namely, let  ∞ (m−1) {α} f (x) − f (m−1) (x ∓ t) α dt (51) M D±, f (x) = Γ(1 − {α})

t1+{α} α f = be the truncated Marchaud derivative. Then by definition, we know that M D± α lim →0 M D±, f , where the character of the convergence will be defined by the problems under consideration.

6.2 The Marchaud derivative on an interval In order to get the Marchaud derivative on an interval, we first consider the function f (x) to be mth differentiable. Then integrating in (17) by parts, we have α RL Da,x f (x)

m−1

f (k) (a) (x − a)k−α Γ(1 + k − α) k=0  x 1 + (x − t)m−α−1 d[f (m−1) (t) − f (m−1) (x)] Γ(m − α) a  m−1 1 f (k) (a) f (m−1) (t) − f (m−1) (x) (x − a)k−α + lim = Γ(1 + k − α) Γ(m − α) t→x (x − t)α−m+1 k=0   x (m−1) f (x) − f (m−1) (t) + {α} dt . (52) (x − t)1+{α} a

=

Because the middle term here vanishes for f (t) ∈ C m , and 1 − {α} = m − α, we denote α M Da,x f (x) =

m−1

f (k) (a) (x − a)k−α Γ(1 + k − α) k=0  x (m−1) {α} f (x) − f (m−1) (t) + dt Γ(1 − {α}) a (x − t)1+{α}

(53)

as an analogue of the Marchaud derivative in interval [a, b], −∞ < a < b < ∞. Similarly, the right hand side Marchaud derivative is introduced α M Dx,b f (x)

=

m−1

f (k) (b) (b − x)k−α Γ(1 + k − α) k=0  (−1)m {α} b f (m−1) (t) − f (m−1) (x) + dt. Γ(1 − {α}) x (t − x)1+{α}

(54)

Remark 19. We can also get (52) and (54) by the continuous function f (m−1) (x), which is zero beyond the interval [a, b] and apply the usual Marchaud derivative on the whole real line. Remark 20. For the case of the finite interval, the difference between the RiemannLiouville and Marchaud derivatives discussed in Remark 18 and connection with their behavior at infinity will not exist.

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17

−α Remark 21. In [21], we know that for function f = RL Da,x φ, φ ∈ L1 (a, b), the Riemann-Liouville and Marchaud derivatives coincide almost everywhere: α α (RL Da,x f )(x) ≡ (M Da,x )f (x) = φ(x).

(55)

7 Riesz fractional integro-differentiation We now study Riesz fractional integro-differentiation of functions which is a fractional power (−Δ)α/2 of the Laplace operator. The idea of how to define such a power is obviously in Fourier transforms: (−Δ)α/2 f = F −1 |x|α Ff in the case of sufficiently good functions f . The negative powers (−Δ)α/2 , α > 0 (the case α > 0 can be referred to [55]), will be Riesz potentials  1 φ(y)dy , α = n, n + 2, n + 4, · · · , (56) I α φ(x) = γn (α) Rn |x − y|n−α where the normalizing constant γ(α) is defined as  2α π n/2 Γ(α/2)[Γ(n − α)/2]−1 , γn (α) = (−1)(n−α)/2 2α−1 π n/2 Γ(1 + (α − n)/2),

α − n = 0, 2, 4, · · · , α − n = 0, 2, 4, · · · .

The operation (−Δ)α/2 f (x) = F −1 |x|−α Ff =



I α f, RZ D

α > 0, −α

f, α < 0,

(57)

is called the Riesz fractional integro-differentiation, where the positive powers of the Laplace operator will be realized as the so called hypersingular integrals RZ Dα f (the order of singularity is higher than the dimension of the space Rn ), defined below by  (Δly f )(x) 1 α dy, (l > α), (58) RZ D f (x) = dn,l (α) Rn |y|n+α α where the normalizing constant dα n,l will be chosen so that RZ D f would not depend on l, only if l > α, 2−α π 1+n/2 Al (α) , dn,l (α) = α n+α Γ(1 + 2 )Γ( 2 ) sin( απ 2 )

where Al (α) =

l

k−1

(−1)

k=0

  l kα . k

And (Δlh f )(x) is a finite difference of order l of a function f (x) of many variables with a vector step h and with center at the point x ∈ Rn : (Δlh f )(x)

l

= (τ−h/2 − τh/2 ) f =

l

  l f [x + (l/2 − k)h] (−1) k k

k=0

or with non-centered differences (Δlh f )(x) = (E − τh )l f =

l

(−1)k

k=0

  l f (x − kh). k

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Lemma 10. Let f (x) ∈ C m (Rn ) and let l ≥ m. Then (Δlh f )(x) =

hm (m − 1)!



1

0

(1 − τ )m−1

l

(−1)m−k k m

k=0

  l f (m) (x − khτ )dτ. k

(59)

Corollary 2. Let f (x) ∈ C m (Rn ) and f (m) (x). Then |(Δlh f )(x)| ≤ c|h|m sup |f (m) (x)|, x

where c =

1 m!

l

k=0

k m ( kl ) ≤

l ≥ m,

(60)

l m 2l m! .

Corollary 3. Let f (x) ∈ C m (Rn ) and |f (m) (x + h) − f (m) (x)| ≤ A|h|λ , where A = A(x) does not depend on h. Then |(Δlh f )(x)| ≤ Ac|h|m+h ,

l > m.

(61)

From Lemma 10, Corollary 2 and Corollary 3, we can easily get the next theorem. Theorem 6. Let α > 0. The Riesz derivative (58) is defined for function f (x) ∈ C [α] (Rn ) such that supx∈Rn |f (x)| < ∞, and |f ([α]) (x + h) − f ([α]) (x)| ≤ A(x)hλ , where λ > α − [α]. Note that the hypersingular integral (RZ Dα y)(x) does not depend on the choice of l (l > α). Such a construction is also called the Riesz derivative of order α > 0 in the sense that (F(RZ Dα y))(x) = |x|α Fy(x), (α > 0) for a “sufficiently good” function y(x). α RZ D f (x) =

1 dn,l (α)

 |y|≥

(Δly f )(x) dy |y|n+α

(62)

can be called a truncated hypersingular integral, and RZ D

α

f (x) = lim

→0+

α RZ D f (x),

(63)

can be understood to be conditionally convergent. Lemma 11. ([2]) Let α > 0, and [α] be the integer part of α. Also let a function y(x) be the bounded together with its derivative (Dk y)(x), (|k| = [α] + 1). Then the hypersingular integral RZ Dα f (x) in (58) is absolutely convergent. If l > 2[α/2], then this integral is only conditionally convergent.

8 Partial and mixed fractional integral and fractional derivatives Starting from the one-dimensional definition, we can naturally define the partial Riemann-Liouville fractional derivative of the order αk with respect to the kth variable by the relation  xk 1 u(x1 , · · · , xk−1 , ξ, xk+1 , · · · , xn ) −αk dξ, (64) RL Dak ,xk u(x) = Γ(αk ) ak (xk − ξ)1−αk

Perspective on Fractional Dynamics and Control

19

where αk > 0. This definition assumes function u(x1 , · · · , xn ) to be given for xk > αk . Introducing the notation ek = (0, · · · , 0, 1, 0, · · · , 0) for the kth unit xk > αk , we can rewrite the fractional integral (64) in the following shorter form  xk −ak 1 u(x − τ ek ) −αk dτ. (65) RL Dak ,xk u(x) = Γ(αk ) 0 τ 1−αk Further, the expression −α RL Da,x u(x)

1 n = (RL Da−α , · · · , RL Da−α u)(x) 1 ,x1 n ,xn  x1  xn ϕ(τ ) 1 ··· dτ, α > 0, = n Πk=1 Γ(αk ) a1 (x − τ )1−α an

(66)

with α = (α1 , · · · , αn ) ∈ Rn+ , (x−τ )1−α = Πnk=1 (xk −τk )1−αk , 1−α = (1−α1 , · · · , 1− αn ), is called a left hand sided mixed Riemann-Liouville integral of order α. The right hand sided mixed fractional integral can be defined in a similar way, which is omitted here. By the same idea of the partial and mixed Riemann-Liouville integrals, we can also give the partial and mixed Riemann-Liouville derivatives in the n dimensional case. αk RL Dak ,xk u(x)

αk

k −αk ) · RL Da−( α u(x) k ,xk αk  xk 1 ∂ (xk − τ ) αk −αk −1 = Γ(αk − αk ) ∂x αk ak k ×u(x1 , · · · , xk−1 , xk − τ, xk , · · · , xn )dτ,

= Dk

(67)

is called the left hand sided partial derivative, and the right hand sided partial derivative can be defined similarly. 1 ∂m α m −(m−α) u(x) = n RL Da,x u(x) = D · RL Da,x Πk=1 Γ([αk ] − αk ) ∂x1 α1 · · · ∂xn αn

 x1  xn × ··· (x − τ )m−α−1 u(τ )dτ, (68) a1

an

α −α−1

= (x1 − τ1 ) α1 −α1 −1 · · · (xn − where m = α1 + · · · + αn , (x − τ ) αn −αn −1 . τn ) The right hand sided mixed derivative can be defined similarly. Remark 22. (66) and (68) are the α ∈ Rn+ order integral and derivative in n dimension separately, where αi for i = 1, · · · , n are fixed. From another angle, if αi is not fixed, we give another definitions as follows −α RL Da,x u(x) α RL Da,x u(x)

1 n = (RL Da−α , · · · , RL Da−α u)(x), 1 ,x1 n ,xn

−(m−α) = Dm · RL Da,x u(x) 1 )−α1 n −αn ) , · · · , RL Da−( α u)(x), = Dm · (RL Da−( α 1 ,x1 n ,xn

or where

(69)

α RL Da,x u(x)

= (RL Daα11,x1 , · · · , RL Daαnn,xn u)(x),

(70) (71)

|α| = α1 + · · · + αn ∈ R1+ , αi ∈ R1+ , i = 1, · · · , n; m = α . (72) The problem is that the number of the pair (α1 , · · · , αn ) satisfying (72) is infinite. Exactly speaking, the number is aleph one, so uncountable. Therefore, how to determine αi for i = 1, · · · , n is a problem. We hope that the reader can solve the problem and give more proper definitions.

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9 Directional integral and directional derivatives In this section we will introduce directional integral and directional derivative operators in Rn case. Firstly, we will discuss the operators in R2 . To this end, we associate with each unit vector v = [v1 , v2 ]t ∈ R2 a unique angle θ ∈ [0, 2π) such that v = [cos θ, sin θ]t . 9.1 Fractional directional integral [56, 57] Let α > 0, θ ∈ [0, 2π) be given. The αth order fractional integral in the directional of θ is given by  ∞ α−1 ξ −α u(x − ξ cos θ, y − ξ sin θ)dξ. (73) Dθ u(x, y) = Γ(α) 0 Remark 23. We note that for special directions the directional integral operator is equivalent to the left-hand and right-hand Riemann-Liouville integral operators, i.e., −α D0−α u(x, y) = D−∞,x u(x, y), −α −α u(x, y) = D−∞,y u(x, y), Dπ/2 −α Dπ−α u(x, y) = Dx,∞ u(x, y), −α −α u(x, y) = Dy,∞ u(x, y). D3π/2

9.2 Fractional directional derivative Let α > 0, θ ∈ [0, 2π) be given. The mth order derivative in the directional of θ is given by Dθm u(x, y) = (cos θ

∂ ∂ + sin θ )m u(x, y) = ([cos θ, sin θ]t · ∇)m u(x, y). ∂x ∂y

(74)

Let α > 0, m = α , θ ∈ [0, 2π) be given. Then the αth order fractional derivative in the directional of θ [56,57] is given by −(m−α)

Dθα u(x, y) = Dθm Dθ

u(x, y).

(75)

And the αth order fractional derivative with respect to M is defined as α DM u(x, y)

 = 0

2π

Dθα u(x, y)M (dθ),

(76)

where M (dθ) is a probability measure on [0, 2π). Remark 24. For n dimensional case, we define the direction θ = (θ1 , · · · , θn ) ∈ [0, 2π)n , then we can generalize the fractional directional integral of αth order from the form of (73) to the following way  ∞ α−1 ξ u(x1 − ξ cos θ1 , · · · , xn − ξ cos θn )dξ. (77) Dθ−α u(x, y) = Γ(α) 0

Perspective on Fractional Dynamics and Control

21

The directional derivative of mth and αth order in n dimensional case can be also defined separately m  ∂ ∂ m Dθ u(x) = cos θ1 + · · · + cos θn u(x) = ([cos θ1 , · · · , cos θn ]t · ∇)m u(x), ∂x1 ∂xn (78) −(m−α)

Dθα u(x) = Dθm Dθ

u(x),

(79)

where x = (x1 , · · · , xn ) ∈ Rn , similar to the forms (74) and (75). But taking the special direction θ = (θ1 , · · · , θn ) = (0, · · · , 0) for example, we find that  ∞ α−1 ξ −α u(x1 − ξ, · · · , xn − ξ)dξ, D0 u(x, y) = Γ(α) 0 m  ∂ ∂ m D0 u(x) = + ··· + u(x) = ∇m u(x), ∂x1 ∂xn −(m−α)

D0α u(x) = D0m D0

u(x).

Obviously the above definitions are not consistent with the αth order fractional integrals and derivatives defined in Sec. 8 either from (64) to (68) when α ∈ Rn or from (69) to (71) when α ∈ R1 . It leads to a paradox, which we will study in further.

10 The local fractional derivative The local fractional derivative was defined by Kolwankar and Gangal(KG-FD) in 1996 [40, 58, 59]. It is defined as follows: for a function f : (a, b) → R, if the right (left) limit dα (f (x) − f (y)) α (80) Lo D± f (y) := lim x→y± d(±(x − y))α exists and is finite, where dα f (x) α := RL Da,x f (x), d(x − a)α then f has the right (left) local fractional derivative of order α, 0 < α < 1. Remark 25. The local fractional derivative is not equivalent to the Riemann-Liouville derivative. The necessary and sufficient condition for both KG-LFD and Riemannx Liouville derivative to exist and to be equal is that y (x − t)−α (f (t) − f (y))dt should be continuously differentiable. Remark 26. For a function f: (a, b) → R, if the limit α Lo D± f (y)

:= lim

x→y

m−1

f (k) (y) k=0 Γ(k+1) (x d(±(x − y))α

dα (f (x) −

− y)k )

(81)

exists and is finite, then f has the left local fractional derivative of order α, m − 1 < α < m. But here we only discuss the local fractional derivative of order 0 < α < 1 for simplicity. And one can get similar results about α > 1 case.

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In [60], another new definitions of the right and left local fractional derivatives were introduced by α L D+ f (y)

f (y + h) − f (y) , h→0+ hα

:= Γ(1 + α) lim

(82)

f (y − h) − f (y) . (83) hα α α f (y) = L D− f (y), we say that function f is locally α−differentiable at y and If L D+ denote the common value by L Dα f (y). For α > 1, let α = n + β, where n ≥ 1 is the integer and 0 < β < 1. We call f α−differentiable at y if the nth derivative of f near y exists and f (n) is β−differentiable at y. α L D− f (y)

:= −Γ(1 + α) lim

h→0+

Theorem 7. ([60]) For 0 < α < 1, f is α−differentiable at y if and only if f (x + h) = f (x) + sign(h)A|h|α , h → 0.

(84)

For α = n + β, f is α−differentiable if and only if f (x + h) =

n−1

f (k) (x)hk + sign(hn+1 )A|h|α , h → 0.

(85)

k=0

Let f : (a, b) → R and y ∈ (a, b), we say that f is locally right (respectively left) C α (0 < α < 1) at y if there is a δ > 0 and a constant Cy > 0 such that |f (x) − f (y)| ≤ Cy |x − y|α

(86) α

for x ∈ (y, y+δ) (respectively for x ∈ (y−δ, y)). We say that f is locally C (0 < α < 1) at y if f is both locally left and right C α at y. Then we have some lemmas and theorems about the function satisfying C α . Lemma 12. ([61]) Suppose f is locally right (respectively left) C α at y ∈ (a, b) such that  1  1 f (y + th) − f (y) f (y − th) − f (y) dt (respectively lim − dt) exists. lim h→0+ 0 h→0+ hα hα 0 (87) Then  1 f (y + th) − f (y) α dt, (88) Lo D+ f (y) = (1 + α)Γ(1 − α) lim h→0+ 0 hα respectively  1 f (y) − f (y − th) α dt. (89) Lo D− f (y) = −(1 + α)Γ(1 − α) lim h→0+ 0 hα α f (y) exists and Lemma 13. ([61]) Let f : (a, b) → R be continuous such that Lo D+ α ∞ RL Dy,x (f (x) − f (y)) belongs to L (y, y + δ) for some δ > 0, then α Lo D+ f (y)

= (1 + α)Γ(1 − α) lim

h→0+

f (y + th) − f (y) . hα

(90)

Consequently α f (y)(x − y)α + o(x − y)α f (x) = f (y) + Γ(1 + α) · Lo D+

as x → y+ .

(91)

Perspective on Fractional Dynamics and Control

23

Remark 27. Under Lemma 13, we can get that α Lo D+ f (y)

and

=

Γ(1 − α) α L D+ f Γ(α)

α f (y)(x − y)α + o(x − y)α f (x) = f (y) + (1 + α)Γ(1 − α) · L D+

(92)

(93)

as x → y+ . Lemma 14. ([61]) For 0 < α < 1 and δ > 0, let  f (x) =

xα g(x),

0 < x < δ,

0,

x = 0,

(94)

where g ∈ C 1 (0, δ) is bounded and for some C > 0, |g  (x)| ≤ Cx−β for x ∈ (0, δ) with 0 < β < 2 + δ. Let F (x) = x1+α g  (x)(0 < x < δ) and define F (0) = 0. Then α α α α Lo D+ f (0) exists if and only if both D+ f (0) and D+ F (0) exist with D+ F (0) = 0, where  1 ± f (th ± y) − f (y) α lim f (y) := (1 − t)−α dt D± Γ(1 − α) h→0+ 0 hα are called the right (left) singular integral difference-quotent local fractional derivatives of order α (0 < α < 1). α f (y) exists Theorem 8. ([61]) Suppose f ∈ C α (a, b) for some 0 < α < 1 and Lo D± α α for a.e., y ∈ (a, b), then Lo D− f (y) = Lo D+ f (y) = 0 for a.e., y ∈ (a, b). Furthermore,

 lim

h→0+

 lim −

h→0+

Consequently,

1 0

α Lo D± f (y)

|

0

1

|

f (y + th) − f (y) |dt = 0, hα

f (y − th) − f (y) |dt = 0 a.e. y ∈ (a, b). hα

(95)

(96)

= 0 for a.e. y ∈ (a, b).

11 Canavati derivative Let g ∈ C α ([a, b]), m = α , ν = α − (m − 1) = 1 − (m − α) (0 < ν < 1). We define the Canavati αth order derivative of g as [62] α Ca Da,x g(x)

where

−(1−ν) m−1 = DDa,x D g(x),

−(1−ν) g ∈ C 1 ([a, b])}. C α ([a, b]) := {g ∈ C m ([a, b]) : Da,x

Obviously, C α ([a, b]) is a Banach space with norm: [[f ]]α = max{ f ∞ ,  f  ∞ , · · · ,  f (m−1) ∞ ,  Dα f ∞ } if α ≥ 1, and if 0 < α < 1.

[[f ]]α = Dα f ∞

(97)

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Remark 28. Here, we can give another necessary condition of the existence of the Canavati derivative defined above, that is −(1−ν) g ∈ AC([a, b])}. AC α ([a, b]) := {g ∈ AC m−1 ([a, b]) : Da,x

(98)

Obviously, it is weaker than g ∈ AC m which is the necessary condition of the existence of the Caputo derivative, and stronger than the Riemann-Liouville derivative’s condition. In fact, the Canavati derivative can be considered as one of the middle states between the Riemann-Liouville derivative and the Caputo derivative.

12 Conclusion In this paper, we have presented the introduction to the fractional integrability and differentiability of the given functions. The corresponding fractional integral and derivatives are often applied to physics, chemistry, biology and engineering and other related fields, such as the memory/hereditary property of various materials and processes, the anomalous diffusion phenomenon, the long-term and/or the long-range actions, the long-tail phenomenon, the power law phenomenon, the allometric scaling law phenomenon, and so on. This introduction almost covers the contributions in this area. If some important references happened not to be here, we do apologize for these omissions. We have mentioned that fractional means fractional integration and fractional differentiations. For fractional integration, only the Riemman-Liouville ingetral is mostly −α u used. Roughly speaking, for the fractional integrability we point out that RL D−∞,x −α p and RL Dx,∞ u exist almost everywhere for function u ∈ L (R) if 0 < α < 1 and 1 ≤ p < 1/α. For fractional differentiation, although there are more than seven definitions for the fractional derivatives, Riemann-Liouville derivative and Caputo one are often utilized. For fractional differentiability in the sense of Riemann-Liouville we α f exists and belongs to L1 (a, b) for 0 < α < 1 if and only if point out that RL Da,x −(1−α)

−(1−α)

f ∈ L1 (a, b) and RL Da,x f (x) is absolutely continuous with RL Da,x f (a) = 0. α α f and RL Dx,b f exist in the sense of Lp (1 ≤ p < ∞) if f We further get that RL Da,x p∗ satisfies conditions Nαp∗ and N  α (2 ≤ p∗ < ∞) almost everywhere in the considered set E. For fractional differentiability in the sense of Caputo, we conclude that the existence condition of Caputo derivative is the absolute continuity of the given function. It should be noted that the problem of fractional integrability and differentiability is the basic one in fractional calculus. We hope that more further contributions in this respect will be appeared elsewhere. This work was partially supported by the Natural Science Foundation of China under Grant No. 10872119, the Key Disciplines of Shanghai Municipality under Grant No. S30104, and the Innovation Foundation of Shanghai University under Grant No. SHUCX101070.

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