Introduction to fractional integrability and differentiability

Abstract

In this paper, we mainly consider fractional integral and derivatives including the Riemann-Liouville derivative, Caputo derivative,
Grünwald-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.

Full text

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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. Liaand 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 deriva-
tive, Gr¨ unwald-Letnikov derivative, Marchaud derivative, Riesz deriva-
tive, 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 frac-
tional 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
ae-mail: lcp@shu.edu.cn
be-mail: zgzhao999@yahoo.com.cn

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6The European Physical Journal Special Topics
the fractional differential equation, and so on [12–20]. In this paper, we firstly intro-
duce 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
?
ward 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 differ-
entiation, 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
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].
0
(y − z)−1/2u(z)dz =
?2gf(y),y ∈ [0,H], (1)
where u(z) =
1 + φ?2(z), φ(z) is an increasing function, g is the constant down-
1
Γ(α)
a
(x − t)α−1u(t)dt, a < x < b, (2)
Lemma 1. Consider, for α ∈ (0,1),−∞ ≤ a < b ≤ ∞, the classical Abel integral
equation
1
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
If a and f(a) are finite, then
?
?x
(x − t)α−1u(t)dt = f(x), a < x < b.(3)
u(x) =
1
Γ(1 − α)
d
dx
a
(x − t)−αf(t)dt,a < x < b. (4)
u(x) =
1
Γ(1 − α)
f(a)(x − a)−α+
?x
a
(x − t)−αf?(t)dt
?
,a < x < b. (5)

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Perspective on Fractional Dynamics and Control7
If a is finite and f is extended by 0 to the left of a, then
u(x) =
1
Γ(1 − α)
?x
a−0
(x − t)−αdf(t),a < x < b. (6)
If a = −∞ and limx→−∞|x|1−αf(x) = 0, then
u(x) =
1
Γ(1 − α)
1
Γ(1 − α)
?x
?x
−∞
(x − t)−αdf(t),
−∞ < x < b,(7)
u(x) =
−∞
(x − t)−αf?(t)dt,
−∞ < x < b.(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
where the integral is in the Lebsgue-Stieltjes sense. Also if a = −∞ and |x|1−αf(x)→0
as x → −∞, then equation (3) has a solution in L1(a,b), given by (10)
1
Γ(1 − α)
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
⎧
⎪
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,
(x − x0)λ,
is the solution of
1
Γ(α)
0
That is to sayRLDα
u(x) =
1
Γ(1 − α)
a
(x − t)−αdf(t), a < x < b,(9)
u(x) =
?x
a
(x − t)−αf?(t)dt =
1
Γ(1 − α)
?x
a
(x − t)−αdf(t). (10)
f(x) =
⎪
⎪
⎪
⎩
⎨
0,0 ≤ x ≤ x0,
Γ(1 + λ)
Γ(1 + λ + α)(x − x0)λ+α,x0< x ≤ 1,
u(x) =
0 ≤ x ≤ x0,
x0< x ≤ 1,
?x
(x − t)−αu(t)dt = f(x).
a,xf(x) exists and equals to u(x), which will be defined later.

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8The European Physical Journal Special Topics
Here, we found a lemma in [24] and [21] (but we guess it was firstly revealed by
Tamarkin [26] in 1930) about a sufficient and necessary condition forRLDα
L1(a,b) when α ∈ (0,1):
Lemma 3. For α ∈ (0,1),−∞ < a < b < ∞, there exists a function u ∈ L1(a,b)
such that
1
Γ(α)
a
if and only if f ∈ L1(a,b) and the function
1
Γ(1 − α)
is absolutely continuous withRLD−(1−α)
a,x
f(a) = 0.
Remark 4. That is to say,RLDα
and only if f ∈ L1(a,b) and (12) holds withRLD−(1−α)
a,xf(x) ∈
?x
(x − t)α−1u(t)dt = f(x),a ≤ x ≤ b,(11)
RLD−(1−α)
a,x
f(x) =
?x
a
(x − t)−αf(t)dt,a ≤ x ≤ b(12)
a,xf exists and belongs to L1(a,b) for 0 < α < 1 if
f(a) = 0.
a,x
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
?∞
That is now called the Liouville form of fractional integral with the factor (−1)pbeing
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
for fractional integration.
Furthermore, we have the most useful forms of left-hand and right-hand Riemann-
Liouville derivatives defined as follows
dm
dxm
D−pφ(x) =
1
(−1)pΓ(p)
0
φ(x + t)tp−1dt,−∞ < x < ∞, p > 0. (13)
1
Γ(α)
0
φ(t)dt
(x − t)1−α,x > 0, (14)
RLDα
a,xf(x) =
1
Γ(m − α)
(−1)m
Γ(m − α)
?x
?b
a
(x − t)m−α−1f(t)dt,(15)
RLDα
x,bf(x) =
dm
dxm
x
(t − x)m−α−1f(t)dt, (16)
where m − 1 ≤ α < m, a,b are the terminal points of the interval [a,b], which can
also be −∞,∞.
Lemma 4. ([21]) Iff(x)
∈
RLDα
forms
ACm([a,b]), thenthefractionalderivatives
a,xf,RLDα
x,bf exist almost everywhere on [a,b] and can be represented in the
RLDα
a,xf(x) =
m−1
?
k=0
f(k)(a)(x − a)k−α
Γ(k − α + 1)
+
1
Γ(m − α)
?x
a
(x − τ)m−α−1f(m)(τ)dτ,
(17)

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Perspective on Fractional Dynamics and Control9
RLDα
x,bf(x) =
m−1
?
k=0
(−1)kf(k)(b)(b − x)k−α
Γ(k − α + 1)
+
(−1)m
Γ(m − α)
?b
x
(τ − x)m−α−1f(m)(τ)dτ,
(18)
where ACm(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)
(x − a)μ(b − x)ν, μ,ν ∈ [0,1 − α), (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 =
RLD−α
For the function (19), its fractional derivatives exist and can be written as follows
?x
?b
Remark 6. The above equations can be extended to value α > 1
a,xφ or f =RLD−α
x,bφ of a summable function φ ∈ L1(a,b).
RLDα
a,xf(x) =
1
Γ(1 − α)(x − a)
a
(1 − α)f(t) + (t − a)f?(t)
(x − t)α
dt,0 < α < 1,(20)
RLDα
x,bf(x) =
1
Γ(1 − α)(b − x)
x
(1 − α)f(t) − (b − t)f?(t)
(t − x)α
dt,0 < α < 1.(21)
RLDα
a,xf(x) =
1
Γ(m − α)(x −a)m
?x
a
(t −a)α[(t − a)m−αf(t)](m)
(x − t)α−m+1
dt,m−1 < α < m.
(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
dxk+1fβ(x),(β = k + 1 − α) (23)
where fβindicates the integral of f with order β which is adopted by Riesz’s definition
?∞
where Kγ(x) = |x|−γ.
The main idea of [36] is that f(α)exists at x0if 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 L2sense
at x0if fβhas a (k + 1)th derivative in the L2sense at x0.
There are some conditions needed to be given to characterize the existence of such
a αth order derivative.
fβ(x) =
−∞
f(t)|x − t|β−1dt = (f ∗ K1−β)(x),(0 < β < 1)(24)

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10The European Physical Journal Special Topics
(i) If α is fractional, k < α < k + 1, f satisfies a Lipschitz condition of order α at
x0, that is f belongs to Λαat x0if
R(x) = O(|x|α)asx → 0,(25)
where R(x) is the Peano remainder of f ∗ K1−βat x0.
(ii) Similarly, f belongs to Λ2
αat x0, if
?1/2
?1
x
?x
0
|R(t)|2dt= O(|x|α)asx → 0. (26)
(iii) f satisfies condition N2
αat x0if
?δ
−δ
[R(t)]2
|t|1+2αdt < ∞
(27)
for some δ > 0.
If R(x) is the Peano remainder of f ∗K∗
call (25), (26) and (27) as Λ?
1−βat x0, where K∗
αand N?2
γ(x) = sign x|x|−γ, we
α, Λ?2
αcorrespondingly.
Lemma 5. ([37]) If f belongs to Λαfor each point x0of 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 Λαat 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 N2
αalmost everywhere in E.
Lemma 7. ([36]) The necessary and sufficient condition that f satisfies the condition
Nαalmost everywhere in a set E is that fsatisfies the condition Λ2
in the L2sense, almost everywhere in E.
α, and f(α)exists
The following assumptions were stated in [38] which generalized the cases in [36]
from L2to Lp(2 ≤ p < ∞), and also from R1to Rn:
(i) f satisfies Λp
αat x0if
?
ρn
|x|<ρ
1
?
|R(x)|pdx
?1/p
= O(ρα).
(ii) f satisfies Np
αif for some ρ > 0
?
|x|≤ρ
|R(x)|p
|x|n+pαdx < ∞.
Lemma 8. ([38]) f(α)exists in the sense of Lp(2 ≤ p < ∞) and f satisfies the
condition Λp
almost everywhere in the set E.
αalmost everywhere in a set E if and only if f satisfies both N2
αand Np
α
Theorem 1. Suppose that f satisfies the condition Λαat every point of a set E with
positive measure. ThenRLDα
which is omitted here) if and only if f satisfies conditions N2
where in E.
a,xf exists almost everywhere in E (so doesRLDα
x,bf,
αand N?2
αalmost every-

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Perspective on Fractional Dynamics and Control11
Theorem 2. RLDα
conditions Λp
N?2
a,xf exists in the sense of Lp(2 ≤ p < ∞) and f satisfies the
αalmost everywhere in a set E if and only if f satisfies N2
αalmost everywhere in the set E.
α, Λ?p
α, Np
α,
αand N?p
Proof. (Proofs of Theorems 1 and 2) They are obviously true just because Riemann-
Liouville derivativesRLDα
defined by
?x
?∞
where K∗
a,xf andRLDα
x,bf are the special cases of (23), where fβis
−∞
f(t)(x − t)β−1dt =
?1
?1
2f ∗ (K1−β+ K∗
1−β)
?
?
(x), (28)
x
f(t)(t − x)β−1dt =
2f ∗ (K1−β− K∗
1−β)(x), (29)
γ(x) = sign x|x|−γ.
Theorem 3. f(α)exists in the sense of Lp(1 ≤ p < 2) and f satisfies the condition
Λp∗
Np∗
K such that for all such p (1 ≤ p < ∞)
α(2 ≤ p∗ < ∞) almost everywhere in a set E if and only if f satisfies both N2
α almost everywhere in the set E. If we further assume that there exists a constant
αand
?f(α)?p≤ K, (30)
then f(α)exists in the sense of L∞and
?f(α)?∞≤ K. (31)
Proof. From the condition Λp∗
of Lp∗(2 ≤ p∗ < ∞), that is ? f(α)?p∗< ∞. Therefore, by using the imbedding
relationship Lp∗→ Lpfor 1 ≤ p ≤ p∗ < ∞, we have that ?f(α)?p≤?f(α)?p∗< ∞.
Therefore, if f satisfies both N2
derive that f has an αth order derivative in the sense of Lp(1 ≤ p < ∞). If f / ∈ L∞
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,
?
It follows that ?f(α)?p≥ (μ(A))1/pK1, whence
α, we know that f has αth order derivative in the sense
αand Np∗
α
almost everywhere in the set E, we
E
|f(α)(t)|pdt ≥
?
A
|f(α)(t)|pdt ≥ μ(A)Kp
1.
liminfp→∞?f(α)?p≥ K1,
which contradicts (30).
Corollary 1. RLDα
satisfies the conditions Λp∗
satisfies N2
that there exists a constant K such that ?RLDα
such p, thenRLDα
?RLDα
a,xf andRLDα
α, Λ?p∗
α, N?2
x,bf exist in the sense of Lp(1 ≤ p < ∞) and f
α(2 ≤ p∗ < ∞) almost everywhere in a set E, if f
αand N?p∗
a,xf ?p≤ K, ?RLDα
a,xf andRLDα
x,bf ?∞≤ K.
α, Np∗
αalmost everywhere in the set E. If we further assume
x,bf ?p≤ K for all
a,xf ?∞≤ K,
x,bf exist in the sense of L∞, and ?RLDα

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12The European Physical Journal Special Topics
3.2 Remarks about the nonexistence of Riemann-Liouville derivative
Remark 7. RLD−α
even if function u has finite discontinuity points.
a,xu andRLD−α
x,bu exist almost everywhere for the integrable function,
Remark 8. ([21])RLD−α
Lp(−∞,∞) 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])
−∞,xu andRLD−α
x,∞u exist almost everywhere for function u ∈
Remark 9. RLDα
a,xu 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,
Then u(x) exists the classical derivative at x = 0. However, if 0 < α < 1,
⎧
⎪
ThereforeRLDα
u(x) =
if 0 < x ≤ 1,
if 1 < x < 2.2 − x,
(32)
RLDα
0,xu(x) =
⎪
⎪
⎪
⎪
⎪
⎩
⎨
x−α
Γ(1 − α)−
x−α
Γ(1 − α)+(x − 1)−α
x1−α
Γ(2 − α), if 0 < x ≤ 1,
Γ(1 − α)
−
x1−α
Γ(2 − α),if 1 < x < 2.
(33)
0,xu(x) does not exist at x = 0 and x = 1.
Remark 10. RLDα
a,xu possibly exists at the removable discontinuity points of u.
Consider a discontinuous function which has a removable discontinuity point for
example
⎧
⎪
Then, if 0 < α < 1, we get
⎧
⎪
Therefore limx→1−RLDα
RLDα
u(x) =
⎪
⎩
⎨
x,if 0 < x < 1,
0,
2 − x,
if x = 1,
if 1 < x < 2.
(34)
RLDα
0,xu(x) =
⎪
⎪
⎪
⎪
⎪
⎩
⎨
x1−α
Γ(2 − α),
x1−α− 2(x − 1)1−α
Γ(2 − α)
if 0 < x < 1,
,if 1 < x < 2.
(35)
0,xu(x) = limx→1+RLDα
0,xu(x) =
1
Γ(2−α). Then we know that
0,xu(x) exists at x = 1.
Remark 11. RLDα
kind of u.
a,xu possibly does not exist at the discontinuity points of the second
Consider
u(x) =
?0,0 ≤ x ≤ x0,
x0< x ≤ 1,
(x − x0)λ,
(36)

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Perspective on Fractional Dynamics and Control13
where −1 < λ < 0. Then, if 0 < α < 1, we have
RLDα
0,xu(x) =
⎧
⎪
⎪
⎪
⎪
⎩
⎨
0,0 ≤ x ≤ x0,
Γ(1 + λ)
Γ(1 + λ − α)(x − x0)λ−α,x0< x ≤ 1.
(37)
ThereforeRLDα
From (33), (35) and (37), we get the following remark.
0,xu(x) does not exist at x0.
Remark 12. The existence ofRLDα
i.e.,RLDα
a,xu(x) possibly has no connection with Dmu(x),
a,xu(x) cannot be said a generalization of Dmu(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];
(3) if f(x) is an increasing function, then?b
By using Lemma 9, we can quickly get the following theorem.
af?(x)dx ≤ f(b) − f(a).
Theorem 4. Suppose
RLD−α
where over [a,b].
thatf(x) isa monotonefunction
a,xf,RLDα
over
x,bf almost every-
[a,b], then
a,xf,RLD−α
x,bf exist and are differentiable, so doRLDα
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) =
j≥1
λ−μjsin(λjx), (0 < μ < 1,λ > 1).(38)
Then we get
RLD−α
0,xf(x) =
?
λ(1−μ)jCx(1 − β,λj), 0 < β < μ < 1,
j≥1
λ−μjSx(α,λj), 0 < α + μ < 1,(39)
RLDβ
0,xf(x) =
?
j≥1
(40)
where
RLD−α
0,xsinax =
1
Γ(α)
?x
?x
0
tα−1sina(x − t)dt := Sx(α,a),
RLD−α
0,xcosax =
1
Γ(α)
0
tα−1cosa(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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14The European Physical Journal Special Topics
4 Gr¨ unwald-Letnikov derivative
Gr¨ unwald [41] (in 1867) and Letnikov [42] (in 1868) developed an approach to frac-
tional differentiation based on the definition
GLDαf(x) = lim
h→0
?α
(?α
hf)(x)
hα
,(41)
where
?α
hf(x) =
?
0≤|j|<∞
(−1)|j|
j
?
f(x − jh),(h > 0).
This is the left Gr¨ unwald-Letnilov derivative as a limit of a fractional order backward
difference. Similarly, we have the right one as
?α
hf(x) =
?
0≤|j|<∞
(−1)|j|
?α
j
?
f(x + jh),(h < 0).
Therefore, we can define the new form of Gr¨ unwald-Letnikov derivative as follows
GLDαf(x) =
1
2cos(πα
2)lim
h→0
(?α
hf + ?α
|h|α
−hf)(x)
, (42)
which is called the Gr¨ unwald-Letnikov-Riesz derivative.
Remark 13. When α < 0, (41) may be divergent when f is a periodic function case.
In the non-periodic case, the series may convergent for α < 0, if f(x) has a “good”
decay at infinity, for example, |f(x)| ≤ c(1 + |x|−μ),μ > |α|.
Remark 14. If the function f(x) ∈ Cm([a,b]), then the Gr¨ unwald-Letnilov derivative
with order α can be defined by
GLDα
a,xf(x) =
m−1
?
k=0
f(k)(a)(x − a)−α+k
Γ(−α + k + 1)
+
1
Γ(m − α)
?x
a
(x − τ)m−α−1f(m)(τ)dτ,
(43)
where m − 1 ≤ α < m ∈ Z+.
5 Caputo derivative
The definition (15) of the fractional differentiation of Riemann-Liouville type leads to
a conflict between the well-established and polished mathematical theory and proper
needs, such as the initial problem of the fractional differential equation, and the
nonzero problem related to the Riemann-Liouville derivative of a constant, and so
on.
A certain solution to this conflict was proposed by Caputo first in his paper [43] (in
1967) and two years later in his book [44], and El-Sayed [45,46]. Caputo’s definitions
can be written as
?x
(−1)m
Γ(m − α)
where m − 1 ≤ α < m ∈ Z+.
CDα
a,xf(x) =
1
Γ(m − α)
a
(x − τ)m−α−1f(m)(τ)dτ, (44)
CDα
x,bf(x) =
?b
x
(τ − x)m−α−1f(m)(τ)dτ,(45)

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Perspective on Fractional Dynamics and Control15
Remark 15. From the discussion in Sec. 3, we can further know that one of the exis-
tence conditions of the Cpauto’s definitions (44) and (45) is f(x) ∈ ACm.
Remark 16. When f =
ous, then we can extend the scope from the absolutely continuous to a wilder space,
that is f ∈ ACm(Ω), but f?(x) is not necessarily integrable at the point x = a or
x = b.
f∗
(x−a)μ(b−x)ν, μ,ν ∈ (0,1−α) where f∗is absolutely continu-
Theorem 5. If f(m−1)(x) is of bounded variations and continuous from the right,
thenCDα
?x
Besides, we have the following relationship between Caputo derivative and Riemann-
Liouville derivative if f(x) ∈ ACm(a,b), that is
a,xf(x) is in L1(a,b), and also has the following form
CDα
a,xf(x) =
1
Γ(m − α)
a
(x − τ)m−α−1df(m−1)(τ). (46)
CDα
a,xf(x) =RLDα
a,xf(x) −
m−1
?
k=0
f(k)(a)(x − a)k−α
Γ(1 + k − α)
, (47)
CDα
x,bf(x) =RLDα
x,bf(x) −
m−1
?
k=0
(−1)kf(k)(b)(b − x)k−α
Γ(1 + k − α)
.(48)
Obviously, the Caputo derivative is more strict than Riemann-Liouville derivative,
one reason is that the mth order derivative is required to exist.
“We concluded that the Riemann-Liouville definition serves as a generalization of
the notion of the generalized (in the sense of generalized functions) derivative, while
the Caputo derivative is a generalization of differentiation in the classical sense [47].”
More relationships between the Riemann-Liouville derivative and the Caputo
derivative are discussed in details [48–55].
6 Marchaud derivative
6.1 The Marchaud derivative on the axis R1
Riemann-Liouvlle derivative (15) on the axis R1may be reduced in general to a more
convenient form than (15). Let us suppose temporarily that f(x) is a sufficiently
“good” function, for example, f(x) is continuously differentiable and its derivative,
f(i)(x),i = 1,··· ,m − 2, vanishes at infinity as |x|α−1−?,? > 0 [21].
dm−1
dxm−1D{α}
Γ(1 − {α})
dm−1
dxm−1D{α}
Γ(1 − {α})
where {α} = α − m + 1,m = ?α?. (49) and (50) will be called Marchaud derivatives
(in 1927), or called Weyl-Marchaud derivatives.
MDα
+f =
+ f =
{α}
?∞
?∞
0
f(m−1)(x) − f(m−1)(x − t)
t1+{α}
dt, (49)
MDα
−f =
− f =
{α}
0
f(m−1)(x) − f(m−1)(x + t)
t1+{α}
dt,(50)
Remark 17. ([21]) Evidently, (49) and (50) exist for bounded function f(m−1)satis-
fying 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 Hα,−a(a > 1) and bounded
at infinity.

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16The European Physical Journal Special Topics
Remark 18. Marchaud derivative MDαf is more convenient on R1than Liouville
derivativeRLDαf 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
?∞
be the truncated Marchaud derivative. Then by definition, we know thatMDα
lim?→0 MDα
lems under consideration.
MDα
±,?f(x) =
{α}
Γ(1 − {α})
?
f(m−1)(x) − f(m−1)(x ∓ t)
t1+{α}
dt(51)
±f =
±,?f, where the character of the convergence will be defined by the prob-
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
RLDα
a,xf(x) =
m−1
?
+
k=0
f(k)(a)
Γ(1 + k − α)(x − a)k−α
1
Γ(m − α)
m−1
?
+{α}
a
?x
a
(x − t)m−α−1d[f(m−1)(t) − f(m−1)(x)]
=
k=0
f(k)(a)
Γ(1 + k − α)(x − a)k−α+
?x
1
Γ(m − α)
?
?
lim
t→x
f(m−1)(t) − f(m−1)(x)
(x − t)α−m+1
f(m−1)(x) − f(m−1)(t)
(x − t)1+{α}
dt.(52)
Because the middle term here vanishes for f(t) ∈ Cm, and 1 − {α} = m − α, we
denote
MDα
a,xf(x) =
m−1
?
k=0
f(k)(a)
Γ(1 + k − α)(x − a)k−α
{α}
Γ(1 − {α})
+
?x
a
f(m−1)(x) − f(m−1)(t)
(x − t)1+{α}
dt(53)
as an analogue of the Marchaud derivative in interval [a,b],−∞ < a < b < ∞.
Similarly, the right hand side Marchaud derivative is introduced
MDα
x,bf(x) =
m−1
?
+(−1)m{α}
Γ(1 − {α})
k=0
f(k)(b)
Γ(1 + k − α)(b − x)k−α
?b
x
f(m−1)(t) − f(m−1)(x)
(t − x)1+{α}
dt.(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 Riemann-
Liouville and Marchaud derivatives discussed in Remark 18 and connection with their
behavior at infinity will not exist.

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Perspective on Fractional Dynamics and Control17
Remark 21. In [21], we know that for function f = RLD−α
Riemann-Liouville and Marchaud derivatives coincide almost everywhere:
a,xφ,φ ∈ L1(a,b), the
(RLDα
a,xf)(x) ≡ (MDα
a,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/2of the Laplace operator. The idea of how to define such a power is
obviously in Fourier transforms: (−Δ)α/2f = F−1|x|αFf 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
?
where the normalizing constant γ(α) is defined as
?
(−1)(n−α)/22α−1πn/2Γ(1 + (α − n)/2),
The operation
Iαφ(x) =
1
γn(α)
Rn
φ(y)dy
|x − y|n−α,α ?= n,n + 2,n + 4,··· ,(56)
γn(α) =
2απn/2Γ(α/2)[Γ(n − α)/2]−1,α − n ?= 0,2,4,···,
α − n = 0,2,4,···.
(−Δ)α/2f(x) = F−1|x|−αFf =
?Iαf,α > 0,
RZD−αf,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 integralsRZDαf (the
order of singularity is higher than the dimension of the space Rn), defined below by
?
where the normalizing constant dα
on l, only if l > α,
2−απ1+n/2
Γ(1 +α
RZDαf(x) =
1
dn,l(α)
Rn
(Δl
|y|n+α
yf)(x)
dy,(l > α),(58)
n,lwill be chosen so thatRZDαf would not depend
dn,l(α) =
2)Γ(n+α
2)
Al(α)
sin(απ
2),
where
Al(α) =
l?
k=0
(−1)k−1
?l
k
?
kα.
And (Δl
with a vector step h and with center at the point x ∈ Rn:
hf)(x) is a finite difference of order l of a function f(x) of many variables
(Δl
hf)(x) = (τ−h/2− τh/2)lf =
l?
k=0
(−1)k
?l
k
?
f[x + (l/2 − k)h]
or with non-centered differences
(Δl
hf)(x) = (E − τh)lf =
l?
k=0
(−1)k
?l
k
?
f(x − kh).

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18 The European Physical Journal Special Topics
Lemma 10. Let f(x) ∈ Cm(Rn) and let l ≥ m. Then
?1
(Δl
hf)(x) =
hm
(m − 1)!
0
(1 − τ)m−1
l?
k=0
(−1)m−kkm
?l
k
?
f(m)(x − khτ)dτ.(59)
Corollary 2. Let f(x) ∈ Cm(Rn) and f(m)(x). Then
|(Δl
hf)(x)| ≤ c|h|msup
x
|f(m)(x)|,l ≥ m,(60)
where c =
1
m!
?l
k=0km(l
k) ≤lm2l
m!.
Corollary 3. Let f(x) ∈ Cm(Rn) and |f(m)(x + h) − f(m)(x)| ≤ A|h|λ, where A =
A(x) does not depend on h. Then
|(Δl
hf)(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 (RZDαy)(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(RZDαy))(x) = |x|αFy(x),
for a “sufficiently good” function y(x).
(α > 0)
RZDα
?f(x) =
1
dn,l(α)
?
|y|≥?
(Δl
|y|n+α
yf)(x)
dy(62)
can be called a truncated hypersingular integral, and
RZDαf(x) = lim
?→0+RZDα
?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 (Dky)(x),(|k| = [α] + 1). Then the
hypersingular integralRZDαf(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 αkwith respect to the kth vari-
able by the relation
?xk
RLD−αk
ak,xku(x) =
1
Γ(αk)
ak
u(x1,··· ,xk−1,ξ,xk+1,··· ,xn)
(xk− ξ)1−αk
dξ,(64)

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Perspective on Fractional Dynamics and Control19
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
RLD−αk
ak,xku(x) =
1
Γ(αk)
?xk−ak
0
u(x − τek)
τ1−αk
dτ.(65)
Further, the expression
RLD−α
a,xu(x) = (RLD−α1
a1,x1,··· ,RLD−αn
1
an,xnu)(x)
?xn
k=1(xk−τk)1−αk,1−α = (1−α1,··· ,1−
=
Πn
k=1Γ(αk)
?x1
a1
···
an
ϕ(τ)
(x − τ)1−αdτ,α > 0, (66)
with α = (α1,··· ,αn) ∈ Rn
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.
ak,xku(x) = D?αk?
k
·RLD−(?αk?−αk)
1
Γ(?αk? − αk)
×u(x1,··· ,xk−1,xk− τ,xk,··· ,xn)dτ,
is called the left hand sided partial derivative, and the right hand sided partial deriv-
ative can be defined similarly.
+,(x−τ)1−α= Πn
RLDαk
ak,xk
u(x)
?xk
=
∂?αk?
∂x?αk?
k
ak
(xk− τ)?αk?−αk−1
(67)
RLDα
a,xu(x) = Dm·RLD−(m−α)
a,x
u(x) =
1
Πn
k=1Γ([αk] − αk)
?x1
∂m
∂x1?α1?···∂xn?αn?
(x − τ)m−α−1u(τ)dτ,
×
a1
···
?xn
an
(68)
where m = ?α1? + ··· + ?αn?,(x − τ)?α?−α−1= (x1− τ1)?α1?−α1−1···(xn−
τn)?αn?−αn−1.
The right hand sided mixed derivative can be defined similarly.
Remark 22. (66) and (68) are the α ∈ Rn
separately, where αifor i = 1,··· ,n are fixed. From another angle, if αiis not fixed,
we give another definitions as follows
+order integral and derivative in n dimension
RLD−α
a,xu(x) = (RLD−α1
a1,x1,··· ,RLD−αn
an,xnu)(x), (69)
RLDα
a,xu(x) = Dm·RLD−(m−α)
= Dm· (RLD−(?α1?)−α1
a,x
u(x)
a1,x1
,··· ,RLD−(?αn?−αn)
an,xn
u)(x),(70)
or
RLDα
a,xu(x) = (RLDα1
a1,x1,··· ,RLDαn
an,xnu)(x),(71)
where
|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 deter-
mine αifor i = 1,··· ,n is a problem. We hope that the reader can solve the problem
and give more proper definitions.