Pure and Applied Mathematics Journal
2013; 2(1) : 10-19
Published online February 20, 2013 (http://www.sciencepublishinggroup.com/j/pamj)
doi: 10.11648/j.pamj.20130201.12
Two definitions of fractional derivatives of powers
functions
Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary
Theoretical Physics Dept., Antananarivo, Madagascar
Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar), Antananarivo, Madagascar
Email address:
[email protected] (R. Andriambololona), [email protected] (R. Hanitriarivo), [email protected] (T. Ranaivoson)
To cite this article:
Raoelina Andriambololona, Rakotoson Hanitriarivo, Tokiniaina Ranaivoson, Roland Raboanary. Two Definitions of Fractional Derivative
of Powers Functions. Pure and Applied Mathematics Journal. Vol. 2, No. 1, 2013, pp. 10-19. doi: 10.11648/j.pamj.20130201.12
Abstract: We consider the set of powers functions defined on
and their linear combinations. After recalling some
properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive
and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality,
semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives
and integrals and their generalization.
Keywords: Gamma Function; Fractional Derivatives; Fractional Integrals; Power Functions
1. Introduction
The concept of fractional derivative can be considered as
rising from the idea to make sense to the expression
1.1
for a function
of a variable
and for a fractional
number . Historically, the origin of the concept of fractional derivative was attributed to Leibnitz and l’Hospital
(1695) but significant developments on the subject were
given later on by Riemann, Liouville, Grunwald-Letnikov
and others. Several approaches are considered by various
authors in the definitions of this concept [1], [2], [3], [4], [5]
[6], [7]. In our case, the study is based on the definition of
fractional derivative for power functions defined on the set
of positive number and their linear combination.We
utilize the properties of the gamma functions [8],[9] [10],
[11].
One of the major results that we obtain is the unification
of the definition for both positive and negative order of the
fractional derivative.
2. Recall on Some Properties of the
Gamma Function
The gamma function, noted Γ, is defined as the integral
Γ
2.1
0
for
and
are easily deduced
"
∞! . The following properties
Γ z
Γ 1
Γ z
1
$
1 %
2.2
&1 !
2.3
The definition of Γ may be generalized starting from the
relations (2.1) and (2.2) to negative integer, fractional,
real and complex. It may be shown that its extension in the
complex plan,
), is a meromorphic function. The poles
of the gamma function are any negative integer values of .
Then for any * +
lim $
∞
/01
2.4
Some properties of the gamma function may be demonstrated for any
) & 3*/* + 5. We have the multiplication theorem
Γ z Γ6
2:
;
<
1
8Γ6
7
7<
=
2
8…Γ6
7
Γ 7
7&1
8
7
2.5
Pure and Applied Mathematics Journal 2013, 2(1) : 10-19
particularly
For A
1
Γ6
2
Γ
1
2
Γ
2
, we have
2A !
√:
4B A!
A8
</
√:Γ 2
2.6
&4 B A!
√:
2A !
1
Γ 6 & A8
2
2.7
We give some useful numerical values of the gamma
functions
5
8√ :
G
Γ 6& 8 &
15
2
E
E
3
4√ :
1
EΓ 6& 8
Γ 6& 8 &2√:
2
3
2
1
3
F
√:
Γ6 8
E Γ 62 8 √ :
2
2
E
3√ :
5
E
Γ6 8
D
2
4
3.1.2 is equivalent to the ordinary derivative
Proof
For
A
, we have from the relation (3.1.2)
N
Γ J 1
Γ J&A 1
N
N Q
0
Q
P
O SB T
N
Q
O
is defined by
N
N
Γ J
Γ J&
1
1
O
O
0 NA J 0
0 NA J R 0 %
U 0 NA J R 0
N
V WX
XY
3.1.1
O
X
$ *&
3.1.2
in which 3WX 5 is a set of arbitrary constants with finite
values. We will see that the second part of the relation (3.1.2)
is the terms which allow the unified expression of fractional
derivative to be applicable for both negative and positive
orders. In the case of positive integer orders, this second
term of (3.1.2) is equal to zero. In the case of negative
integer orders, this term becomes a finite summation. So
there is no problem about the convergence in the case of
integer order. But in the case of fractional orders, this term is
an infinite summation and we have to discuss about the
convergence. We will see also that because of the existence
of the arbitrary constants WX , the fractional derivative is not
unique when the order is not a positive integer.
Theorem 3.1
For
A
, the derivative of order M as defined in
V WX
O B
N
B
O
XY
N
X B
$ *&A
3.1.3
1
Γ J 1
Γ J&A 1
O B
3.1.4
O B
The relation (3.14) is the expression of ordinary derivative
of order A.
Example
\
3
Z
\
Z
]
Γ 7 1
Γ 7&6 1
3
&8
]
Γ 6 1
Γ 6&3 1
&8
Theorem 3.2
11!
6!
[ Z
Z \
Z
7!
1!
3
5040
6!
3!
\
&443520
360
\
Z
If the order is a negative integer
&A with A
,
the fractional derivative as defined in (3.1.2) is not unique.
It depends on A arbitrary constants.
B
1
O
N
NJ J & 1 … J & A
Z [
For any doublet , J
K L & 3 &1, &1 5 and for
. a derivative of order M of the function
such
B
O
Because of the property 2.4 , the second part of the expression (3.1.3) is equal to zero for positive integer A. Then
3. First Definition
3.1. First Definition
N
B
B
2.8 %
11
N
Γ J 1
Γ J&A 1
N
B
O
N
O B
B O
B
V WX
XY B
*
X B
A !
3.1.5
Proof
If
&A with A
and according to the properties
(2.3) and (2.4) of the gamma function. We have
Γ *&
^
∞
*
1
Γ *
A
Q * _ &A %
A ! Q * ` &A
1
3.1.6
From the relation (3.1.2) and the relation (3.1.6) we may
deduce easily the relation (3.1.5).
B
N O
Γ J 1
O B
N
B
Γ J&A 1
V WX
XY B
*
X B
A !
3.1.7
The right hand side is the A-times indefinite integral
(primitive) of N O . If we derivate A times the relation
12
Raoelina Andriambololona et al.: Two definitions of fractional derivatives of powers functions
3.1.7 all the terms containing the constants WX disappear.
Examples
Γ 3 1
Γ 3 1 1
\
\
5
\
a
Γ 1 1
5
Γ 1 3 1
5
Theorem 3.3
a
W
4
V WX
\
XY
W
4
\
<
W
2
Γ *
V WX
XY \
W
<
Γ *
X
1
1
X \
3
1
\
If the order M is a negative integer,
&A with
A
, the fractional derivative as defined by the relation
(3.1.2) is not unique. It depends on A arbitrary constants.
Proof
It is evident by looking at the second part of the relation
(3.1.7). If b
is a derivative of order of
N O,
then the function
c
b
V dX
XY B
*
X B
3.1.8
A !
is also a derivative of order of
in which 3dX 5 is a
set of arbitrary finite constant numbers. We may introduce
the notion of semi-equality because of the relation (3.1.8)
between c
and b . This relation is an equivalence
relation.
Theorem 3.4
K, a sufficient conIf the order is a fraction
f
dition for the fractional derivative, as defined in (3.1.2) to be
well defined for any
is
e
lim h*
Xg
WX
h
WX
Proof
The relation (3.1.2) gives for
N
N
Γ J
Γ α&
p
q
1
1
O
e
f
O
e
f
V WX
XY
0
3.1.9
e
N
f
e
f
o ,
V WX
XY
X
Γ *&
e
f
m
n
1
3.1.10
The last members of this relation is well defined if the
infinite summation is convergent. Let be
$ *&
VW
1
pB ,
W
B
Then
o ,
B
$ &A &
B
BY
and
B
$ &A &
1
1
V pB ,
BY
According to the Abel’s criterium, the series o ,
convergent if and only if
lim h
B0
pB
,
pB ,
h _ 1 q lim h& A
B0
W
q | | U lim h& A
B0
W
W
B
B
W
h
B
B
So the definition (3.1.10) is meaningful for all
the coefficients W B WX fulfill the condition
W1
h
W1
lim h * &
X0
WX
h
WX
0 q lim h*
Xg
is
h_1
U 0 if
0
In all following calculations, expressions and use of the
above defined fractional derivative, we assume that the
condition (3.1.9) is fulfilled. This condition gives a restriction on the possible values of constants WX .
Theorem 3.5
K, the fractional deIf the order M is a fraction
f
rivative, as defined in (3.1.2) is not unique.
Proof
e
The relation (3.1.2) gives for
the relation (3.1.10).
f
According to this relation, we remark that if b
is a
e
derivative of order f of the function
N O , then the
function
c
O
X
e
b
V dX
XY
e
X
Γ *&
e
f
m
n
1
3.1.11
is also a derivative of order f of
. The constants dX
are arbitrary constants but they have to fulfill the conditions
(3.1.9) for the convergence of the infinite summation. For
the relation (3.1.11), we recall and extend the notion of
“semi-equality” as defined in the relation (3.1.8).
3.2. Properties
3.2.1 Linearity and Semi-Linearity
Theorem 3.6
Pure and Applied Mathematics Journal 2013, 2(1) : 10-19
If the order M is positive integer, the fractional derivative
of order M verifies the property of linearity
N
O
d
s
O
N
s
d
Theorem 3.7
If the order is a negative integer, or a negative or positive fraction, the fractional derivative of order M does not
have the property of linearity. But it has a property that we
may define as “semi-linearity”
tN
O
N
s
d
d
O
s
V WX
XY
u
X
$ *&
3.2.2
1
Proof
If is not a positive integer, we have
Γ J
N
Γ J&
Γ v
d
Γ v&
1
1
Γ J
N
Γ J&
1
s
O
N
1
1
V WX
O
XY
V WX<
XY
tN
1
O
O
d
XY
$ *&
X
u
Γ v
d
Γ v&
V WX\
X
1
X
$ *&
3.2.3
1
1
Because of the arbitrariness on the values of the constants WX , WX< , and WX , we may identify WX WX\ & WX &
WX< and then deduce from the relation (3.2.3) and (3.2.4)
the relation (3.2.2). We may qualify the property (3.2.2) as
“semi-linearity” on the same footing as semi-equality defined in the relations (3.1.8) and (3.1.11).
3.2.2. Commutativity and Semi-Commutativity
Theorem 3.8
If the order is a positive integer, the fractional derivative
x
w
N
x
w
x
O
N
w
O
3.2.5
If the order is a negative integer, or a negative or positive fraction, the fractional derivative of order does not
have the property of commutativity. But it has a property
that we may define as “semi-commutativity” (semi-group
property)
Proof
using the definition (3.1.2) we obtain
x
w
x
w
V WX <
x
w
N
N
w
O
X
x
w
x
N
O
w
x
N
<
N
w
w
X
$ *&
N
x
V WX<
Γ J
Γ J&
x
XY
x
1
XY
V WX
w
V WX<
x
x
O
$ *&
O
Γ J
Γ J&
w
X
Then we have
XY
N
$ *&
x
w
w
3.2.4
w
x
Theorem 3.9
XY
s
x
O
Proof
This property is evident because of the Theorem 3.1, in
the case of positive integer order, the fractional derivative is
exactly the ordinary derivative which is well known to have
the property of commutativity.
1
1
N
x
w
XY
$ *&
s
w
V WX<
s
d
has the property of commutativity
3.2.1
Proof
This property is evident according to the theorem 3.1. In
the case of positive integer order, the fractional derivative is
equivalent to the ordinary derivative which is well known
to be linear at any order.
13
x
O
1
1
<
1
<
V WX<
XY
1
1
<
V WX
x
x
& V WX
O
w
X
1
O
X
$ *&
N
w
1
3.2.6
w
1
w
x
x
$ *&
x
$ *&
w
$ *&
XY
X
XY
X
&
1
x
w
1
3.2.7
O
X
$ *&
w
1
3.2.8
According to the relation (3.2.8) we don’t have in general the property of commutativity. But on equal footing as
the definitions of “semi-equality” and ‘semi-linearity”
defined in the relations (3.1.8), (3.1.11) and (3.2.2),we may
define the property of “semi-commutativity” for the relation (3.2.8).
3.2.3. Important Relation Verified by the Fractional De-
14
Raoelina Andriambololona et al.: Two definitions of fractional derivatives of powers functions
rivatives.
e
For any f K, we may always assume, without loss of
y . We have the theorem:
generality, that m + and n
Theorem 3.9
For A
and n z n
e
f
Be
f
N
Be
f
e
f
e
f
e.
e…
e N
f |||~
f
{|f|||}|
B
O
=e
f
X
7m
Γ *&
n
=Y XY
For n
e
<
e
<
N
e
<
O
N
2 and A
!
e
<
N
e
<
2
O
Γ J
Γ •J &
V WX
XY
e
<
e
<
!N
m
2
1
1‚
X
$ *&
Γ J
Γ •J &
V WX
1
e
<
m
2
e
For n
N
V WX
e
O
XY
3 and A
e
\
N
e
\
O
$ *&
V WX
XY
1
N
e
<
m
2
X
Γ *&
m
2
1‚
Γ J
Γ •J &
m
3
1
1‚
O
e
<
e
<
X
1
e
<
m
2
!
<e
\
1
1
O
e
\
<e
\
N
<e
\
For n
e
\
e e
< <
X
e e
< <
3.2.10
1
e
\
N
e
\
O
2
e
\
e
\
V WX
m
3
!N
3.2.11
1
Γ J
m
Γ •J &
3
X
$ *&
XY
Γ J
e
O
<
m
$ *&
1
XY
2
m
1‚
Γ •J &
Γ J 1
2
O
N
.
m
m m
Γ •J &
1‚ Γ •J & &
1‚
2
2 2
m
Γ •* &
1‚
1
2
V WX
m
m m
$ *&
1 Γ •* & &
1‚
XY
2
2 2
X
3 and A
e
\
$ *&
e
\
e
\
m
3
1
e
\
O
1‚
1
<e
\
XY
O
V V
3 and A
e
\
e
\
e
\
N
e
\
O
X
$ *&
3
e
\
2m
3
1
!N
e
\
m
3
=e
\
3.2.12
1
Γ J
1
2m
Γ •J &
1‚
3
V WX
XY
X
Γ *&
V
XY
X
Γ *&
e
\
m
3
1
e
\
e
\
<e
\
1
<e e
\ \
O
<e
2m
X
1‚
\
3
V WX
2m m
2m
&
1‚ Γ * &
Γ •* &
XY
3
3 3
e
m
X
Γ •J &
1‚
\
3
V WX
m m
m
1‚ $ •* &
Γ •J & &
XY
3 3
3
O
m
3
Γ •* &
WX<
e e
< <
X
1
2m
Γ •J &
1‚
1
3
N
.
2m
2m m
Γ •J &
1‚ Γ •J &
&
1‚
3
3 3
Γ J
e e
\ \
1
7m
3
Γ *&
e
\
X
Γ *&
X
WX=
<e
\
m
3
$ *&
V WX
XY
e
\
X
V WX
=Y XY
V WX
XY
N
1
e
\
O
m
Γ •J &
1‚
3
O
.
N
m m
m
1‚ Γ •J & &
1‚
Γ •J &
3
3 3
m
Γ •* &
1‚
1
3
V WX
m
m m
$ *&
1 Γ •* & &
1‚
XY
3
3 3
3.2.9
Proof
We use a mathematical induction.
For n 1, the equality is evident.
For n 2 and A 1 according to (3.1.2) we have
e
<
XY
e
\
O
B •=€
V V WX=
For n
V WX
X
e
\
1
1‚
Pure and Applied Mathematics Journal 2013, 2(1) : 10-19
N
e
e
<
O
V V
=Y XY
For any A z n, n
Be
f
N
B
O
Be
f
y
e
f
=e
\
X
WX=
Γ *&
7m
3
3.2.13
1
V
N
, we assume the hypothesis (3.2.9)
e
f
e
f
e.
e…
e N
f |||~
f
{|f|||}|
B •=€
V V WX=
=Y XY
O
=e
f
X
Γ *&
7m
n
B
e
f
B
f
e
B
N
f
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
e
B
O
B
•=€
V V
=Y XY
WX=
Let us perform the calculation
e
f
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B
e
f
e
f
e
f
e
f
!N
•=€
!
Be
f
N
Be
f
Γ α
N
e
f
O
e
f
B
O
Am
Γ J&
n
N
V V
O
X
7m
Γ *&
n
e
f
! e. e… e N
f |||~
f
{|f|||}|
1
B
=Y XY
1
V V WX=
=Y XY
Γ •* &
V WX
XY
X
Γ •* &
X
=e
f
7m
n
=e
f
7m
n
Am
Γ J&
1
Γ α 1
n
N
Am
Am m
Γ J&
1 $ J&
&
1
n
n
n
X
Γ *&
1‚
1‚
Be
f
Am
n
B
N
f
O
e
O
$ 6* &
WX=
X
B
Γ *&
V V
=Y XY
O
Be
f
A
$ •* &
WXB
1
1
n
X
1 m
=e
f
B
18
7m m
&
n
n
X
e
f
Am
n
WX=
e
f
1‚
1
X
Γ *&
=e
f
7m
n
1
4.1. Second Definition
1
For any doublet , J
K L & 3 &1, &1 5 and for
, a derivative of order of the function
such
N
O
is defined by
Be
f
Be
Am
X
Γ *&
1
f
n
V WX
Am m
Am
$ *&
&
1 Γ *&
1
XY
n
n
n
Γ *&
e
n
1
1 m
Am
n
Replacing the order by its entire part in the summation
which appears in the relation (3.1.2), we may give another
definition of the fractional derivative which looks simpler
because it eliminates the complexity introduced by the
existence of the infinite summation. So we do not have to
study the convergence of this term.
O
e
f
=e
7m
X
1
f
n
V V WX=
7m m
7m
$ *&
&
1 Γ •* &
1‚
=Y XY
n
n
n
B
f
V
XY
Γ *&
e
f
X
4. Second Definition
e
f
B •=€
B
3.2.15
1
e
f
V V WX=
Be
O
f
=e
f
XY
=Y XY
We have now to prove that this formula remains true for
A 1 _ n i.e
WXB
Γ α
A
$ J&
V WX
3.2.14
1
XY
15
N
Γ α 1
Γ J&s 1
N Q
P 0 Q
N O SB T Q
N
O
O
0 NA J 0
0 NA J R 0 %
U 0 NA J R 0
N
V WX
XY |! |
4.1.1
O
X
$ *&!
1
4.1.2
!M is the entire part of M. This second definition gives
exactly the same results as the definition (3.1.2) in the case
of integer order. But some differences appear between the
two definitions in the case of fractional order.
Theorem 4.1
For
A with A
, the derivative of order as defined in (4.1.2) is the ordinary derivative of order A.
16
Raoelina Andriambololona et al.: Two definitions of fractional derivatives of powers functions
Proof
From the definition (4.1.2) and using the property (2.4) of
the gamma function, we obtain
B
N
B
Γ α 1
N
Γ J&A 1
O
O B
NJ J & 1 … J & A
The relation (4.1.3) is the ordinary expression of derivative of order A for A
.
Theorem 4.2
For
K , the derivative of order
…
(4.1.2) is given by the expression
„
e
f
N
e
f
O
e
f
N
O
N
e
f
Γ J
Γ J&
m
n
1
e
f
4.1.4
Proof
The relation (4.1.4) is deduced easily from the relation
e
(4.1.2) by replacing
and noting that the second part of
f
this expression is equal to 0 because of the relation (2.4).
According to (4.1.3) and (4.1.4), we remark that for positive
value of the order (integer or fractional), the fractional
derivative as defined in (4.1.2) is unique. We will see that it
is not always the case when is negative.
Theorem 4.3
K , with m, A
„
…
For
equality
e
f
e
f
e
f
e.
e…
e N
f |||~
f
{|f|||}|
e
f
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
N
f •=€
y
Be
f
O
B •=€
and particularly
,n
e
O
N
Be
f
N
e
4.1.5
O
4.1.6
Proof
We utilize mathematical induction.
For n 1 the relation is evident
For n 2 and A 1, according to (4.1.4), we have
<
N
For n
<
O
<
<
N
2 and A
<
<
O
<
O
<
N
Γ J
m
Γ J&
n
1
1
O
4.1.7
<
2, according to (4.14), we have
<
<
†N
Γ J
1
Γ •J &
2
1
1‚
O
<‡
,n
e
f
e
f
y
NJ
A
N
O
O
4.1.8
, we assume that
e.
e…
e N
f |||~
f
{|f|||}|
Be
f
O
B •=€
N
Be
f
O
4.1.9
We have now to prove that this formula remains true for
1
e
f
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B
•=€
N
B
O
Let us perform the calculation
e
f
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B
e
f
e
f
we have the
O
Γ α 1
Γ J&1 1
e
f
as defined in
O
1
For any A
4.1.3
O B
N
N
•=€
†
Be
f
N
N
Be
f
O
e
f
O
e
f
‡
e
f
e
f
!N
!
e
f
B
e
f
N
f
e
O
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B •=€
Γ J
1
Am
Γ •J &
1‚
n
Am
Γ •J &
1‚
1
n
N
Am
Am m
1‚ Γ •J &
&
1‚
Γ •J &
n
n
n
Γ J
B
f
B
e
f
B
O
e
e
f
B
N
f
4.1.10
e
O
N
O
O
Be
f
Be
f
O
Theorem 4.4
If the order is negative and the absolute value of is
such | | ` 1, the fractional derivative as defined by the
relation (4.1.2) is not unique and depends on |! | arbitrary
constants WX .
Proof
If _ 0 and | | ` 1 ˆ * & !
1 U 0 in general.
then the second term of the relation (4.1.2) is not equal to
zero and the expression of a derivative b
of order of
N O is
N
N
b
Γ α 1
Γ J&s 1
O
V WX
O
XY |! |
N
O
X
$ *&!
1
4.1.11
As the values of the constants WX are arbitrary, any other
function c
such
c
b
&1
V W*
* &|! |
*&
$ *&!
1
4.1.12
Pure and Applied Mathematics Journal 2013, 2(1) : 10-19
is a derivative of order
of
5!
38
Γ
7
too.
Theorem 4.5
If
&A + , the derivative of order as defined by
the relation (4.1.2) is the indefinite integral of order A.
Proof
If we replace
&A in the relation (4.1.2), we obtain
B
N
Γ α 1
Γ J A 1
N
O
B
V WX
O B
XYB
B
X
$ *
A
4.1.13
1
The last member of the relation (4.1.13) is equal to the
result obtained by integrating A times. The constants WX
may be identified to the integration constants.
Theorem 4.5
K , the expression of the derivative of orFor
f
der , according to (4.1.2) is
e
e
f
N
Γ α
Γ J&
m
n
1
O
1
N
e
f
e
f
O
V WX
XY !
e
f
e
f
e
X
f
$ *&!
m
n
4.1.14
1
Proof
This relation is obtained directly from (4.1.2) by replacing
e
. We recall, according to the theorem 4.4, this fracf
tional derivative is not unique for ‰f ‰ U 1
e
Example
Applying the relation (4.1.14) , we obtain
Γ 5
a ]
Γ •5
a
]
\ \
Γ 3
]
\
[
[ ]
[
[
1
4
Γ •3
N
Γ 5
Γ •5
1
5
3
1‚
1
]
a
5!
25
Γ
4
1‚
3
3!
18
Γ
3
1
17
1‚
7
V
XY ‰Š
<
a
‹‰
WX
[
[
X
a
1
Š ‹
4
$ •*
1‚
W
W
<
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B •=€
N
Be
f
O
N
Be
f
O
4.1.15
but it may be proved that we have an equality of the form
Be
f
N
Be
f
e
f
O
e
f
e
f
e.
e…
e
f |||~
f
{|f|||}|
B •=€
N
Œ
O
4.1.16
in which Œ
is a linear combination of power functions
with arbitrary coefficients. As example, let us take m &1
then we can show by mathematical induction that
f
f
f
.
…
N
f
f
f
{|||||}|||||~
N
O
B •=€
But we have
N
O
N
So
N
O
J
f
O
1
f
J
O
4.1.17
1
W
f
N
.
…
f
f
f
{|||||}|||||~
4.1.18
O
B •=€
Œ
with Œ
W in our example. As the relation (4.1.17)
differs from the relation (4.1.18) by the constant W we
have a same property of the “semi-equality” introduced in
paragraph 3.
4.2. Properties
4.2.1. Linearity and Semi-linearity
V W*
a
\
]
a
&1
5
3
e
f
e
f O
N
\
[
Remark
e
K , it may be shown that we do not have
If
f
exactly the relation
B O
N
17
* &1
W
$ •*
V WX
XY <
$ •*
5
3
*
Theorem 4.5
5
Š ‹
3
X
1‚
[
[
17
Š ‹
7
1‚
If the order M is a positive rational number (M K or a
negative rational number (M K with |M| _ 1, the fractional derivative of order M (as defined in 4.1.2) has the
property of linearity
N
O
Proof
If
K or
d
s
N
O
d
s
K with |•| _ 1, we have
4.2.1
18
Raoelina Andriambololona et al.: Two definitions of fractional derivatives of powers functions
V WX
So
N
X
XY |! |
$ *&!
O
O
N
Then
Γ α
N
Γ J&
1
N
O
N
1
N
O
1
Theorem 4.9
For
, <
K L K with |MŽ | _ 1, |M• | _ 1 , the fractional derivative as defined by the relation (4.1.2) is
commutative.
O
1
w
Γ α
d
Γ J&
d
s
1
N
w
If the order M is a negative rational number (M K
with |M| U 1, the fractional derivative, as defined in (4.1.2),
doesn’t have the property of linearity. In fact we have in this
case
O
N
d
Proof
We have for
N
s
V WX
XY |! |
d
N
Γ α 1
Γ J&s 1
Γ α 1
Γ J&s 1
Γ α 1
Γ J&s 1
s
u
s
O
X
$ *&!
V WX
O
XY |! |
O
N
O
d
V WX<
XY |! |
w
w
N
w
x
O
x
w
N
x
N
N
x
N
O
x
O
N
w
0 so
Γ α
Γ J&
1
Γ α 1
Γ J& <&s
w
x
w
w
O
4.2.4
O
1
N
O
1
x
O
x
X
$ *&!
1
1
4.2.3
As the values of the constants WX and WX< are arbitrary,
we may identify WX
WX< WX then the relation (4.2.2) is
deduced easily from (4.2.3) one. According to the definition
in the paragraph 3, the relation (4.2.2) is a “semi-linearity”
one.
w
O
XY
w
$ *&!
$ *&!
N
w
1
X
X
If
, <
K L K with | | ` 1, | < | ` 1 , the fractional derivative as defined by the relation (4.1.2) doesn’t
have the commutativity propriety.
Proof
From the definition 4.1.2 , we obtain
x
s
V WX
XY |! |
4.2.2
1
O
N
x
O
N
x
x
Theorem 4.10
K and |M| U Ž !s R 0
Then
N
d
O
w
O
Then
Theorem 4.8
tN
x
O
Proof
For | | _ 1 we have !
s
1
N
x
w
s
d
O
0
1
Γ α
Γ J&
4.2.2. Commutativity and Semi-commutativity
w
x
N
N
V
Γ J
Γ J&
V
XY |! x |
x
O
N
V
XY |! x |
<
X
WX<
|! w |
1
$ *&!
WX<
w
X
$ *&!
Γ J
Γ J&
WX <
V
XY |! w |
X
$ *&!
WX
<
X
$ *&!
&
x
1
w
<
&
x
w
N
w
O
R
w
w
1
x
w
x
<
O
w
x
O
w
x
1
1
<
So in general we have
x
x
1
1
1
N
x
O
4.2.5
Pure and Applied Mathematics Journal 2013, 2(1) : 10-19
On the equal footing as in paragraph 3, as the values of the
constants WX are arbitrary, we may say that we have the
semi-commutativity.
5. Conclusion and Comparison of the
Two Definitions
The approach that we have introduced to study the fractional derivatives opens a new way and new insights concerning the definition and properties of the derivation operation, particularly in the possibility of an unified definition
for both orders : positive and negative.
The two definitions that we have considered have their
own advantages and disadvantages. The first one corresponding to the relation (3.1.2) seems to be more general
but the existence of the infinite summation introduces some
complexity which implies the study of the convergence of
this series. The second one corresponding to the relation
(4.1.2) seems to be simpler because it eliminates the existence of the infinite summation but it introduces some ambiguity particularly in the case of fractional negative orders
as it can be seen in the relations (4.1.16).
The two definitions are exactly the same if the order is
integer numbers: positive or negative.
We have introduced the notions of semi-equality,
semi-linearity and semi-commutativity giving a broad sense
to the relation (3.1.8). In this framework, the two definitions
may be said to be “equivalent modulo the second term of this
relation (3.1.8) ”.
In our study, we have considered fractional order
K
and power functions defined on
and their linear combinations
In references [12],[13], we study another method to define
real and complex order derivatives and integrals by using
operator approach in the case of more general functions
than power functions.
References
[1]
S. Miller Kenneth, “An introduction to the fractional calculus
and the fractional differential equations”, Bertram Ross
(Editor). Publisher: John Wiley and Sons 1st edition, 1993
19
ISBN 0-471-58884-9.
[2]
K.B.Oldham, J.Spanier, “The fractional calculus. Theory and
Application of differentiation and integration to arbitrary
order” (Mathematics in Science and engineering). Publisher:
Academic Press, Nov 1974, ISBN 0-12-525550-0.
[3]
Zavada, “Operator of fractional derivative in the complex
plane”, Institute of Physics, Academy of Sciences of Czech
Republic, 1997.
[4]
F. Dubois, S. Mengué, “Mixed Collocation for Fractional
Differential Equations”, Numerical Algorithms, vol. 34, p.
303-311, 2003.
[5]
A.C. Galucio, J.-F.Deü, S. Mengué, F. Dubois,“An adaptation of the Gear scheme for fractional derivatives”, Comp.
Methods in Applied Mech. Engineering, vol. 195, p.
6073–6085, 2006.
[6]
A. Babakhani, V. Daftardar-Gejji, “On calculus of local
fractional derivatives”,J. Math. Anal. Appl. 270, 66-79
(2002).
[7]
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, “Theory and
Applications of Fractional Differential Equations”
(North-Holland Mathematical Studies 204, Ed.Jan van Mill,
Amsterdam (2006))
[8]
E. Artin, “The Gamma Function”. New York: Holt, Rinehart,
and Winston, 1964.
[9]
G.E. Andrews, R. Askey, and R. Roy, “Special functions”,
Cambridge, 1999.
[10] S.C. Krantz, “The Gamma and Beta functions”, §13.1 in
Handbook of Complex analysis, Birkhauser, Boston, MA,
1999, pp. 155-158.
[11] G. Arfken, “The Gamma Function”. Ch 10 in Mathematical
Methods for Physics FL: Academic Press, pp.339-341and
539-572, 1985.
[12] Raoelina Andriambololona,“Definition of real order integrals
and derivatives using operator approach,” preprint Institut
National des Sciences et Techniques Nucléaires,
(INSTN-Madagascar), May 2012, arXiv:1207.0409
[13] Raoelina Andriambololona, Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, “Definition of complex order integrals
and derivatives using operator approach”, INSTN preprint
120829, arXiv 1409.400. Published in IJLRST, Vol.1,Issue
4:Page
No.317-323,
November-December(2012),
http://www.mnkjournals.com/ijlrts.htm.