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Comparison of Definition of Several Fractional Derivatives
Yun Ouyang1, a and Wusheng Wang1, b* 1School of Mathematics and
Statistics, Hechi University, Guangxi, Yizhou 546300, P. R.
China
[email protected],
[email protected]
*The corresponding author
Keywords: Riemann-liouville fractional derivatives; Caputo's
fractional derivative; Grunwald-letnikov fractional derivatives;
Fractional integral.
Abstract. The idea of fractional derivatives was raised first by
L'Hospital in 1695. The fractional
calculus and its mathematical consequences attracted many
mathematicians such as Fourier, Euler,
Laplace. Various definitions of non-integer order integral or
derivative was given by many
mathematicians. In this paper, firstly, we discuss the positive
integer higher order derivative of a
function, and obtain general formula of high order derivative.
Secondly, we introduce the definitions of
Gr\"unwald-Letnikov, Riemann-Liouville and Caputo. Finally, we
point out the relationship between
these definitions. Caputo's integral definition and
Gr\"unwald-Letnikov integral definition are
consistent with the Riemann-Liouville integral definition. When
f has 1m order continuous
derivative and 1 mpm , Gr\"unwald-Letnikov positive non-integer
order derivative definition is
consistent with the Riemann-Liouville positive non-integer order
derivative definition.
Introduction
The idea of fractional derivatives was raised first by
L'Hospital in 1695. The fractional calculus and its
mathematical consequences attracted many mathematicians such as
Fourier, Euler, Laplace. Various
definitions of non-integer order integral or derivative was
given by many mathematicians (see [1]). For
fractional derivative, the most popular definitions are Riemann-
Liouville and Caputo definitions as
follows, respectively. Recently, the fractional calculus methods
have become increasingly more
popular and successful in a large number of physical
applications (see for example, [2, 3]). The
theoretical and experimental scientists observed that certain
materials admit anomalous, fractional
behavior, as [4]. There are many systems which evolution is
governed by some fractional differential
equations (for example, [5]). Fractional calculus is nowadays a
broad and interesting field of very
active research (see [6-12]).
In this paper, firstly, we discuss the positive integer high
order derivative of a function, and obtain
general formula of high order derivative. Secondly, we introduce
the definitions of Gr\"
unwald-Letnikov, Riemann-Liouville and Caputo. Finally, we point
out the relationship between these
definitions.
General Formula of High Order Derivative
Suppose that function )(tf has n order continuous differentiable
derivative on ],[ ba . From the
definition of the derivative, we have
h
htftf
dt
dftf
h
)()(lim)('
0
, (1)
h
htftf
dt
fdtf
h
)(')('lim)(''
02
2
2016 6th International Conference on Education, Management and
Computer Science (ICEMC 2016)
© 2016. The authors - Published by Atlantis Press 553
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h
htfhtf
h
htftf
hh
)2()()()(1lim
0
20
)2()(2)(lim
h
htfhtftf
h
, (2)
By induction we can obtain
),()1(1
lim)(0
0
)( rhtfr
n
hdt
fdtf
n
r
r
nhn
nn
(3)
where n is a natural number,
!
)1()2)(1(
r
rnnnn
r
n
(4)
Let p be any arbitrary integer, define a new derivative (see
[9,10])
0
0
lim ( 1)n
p p r
a th
rnh t a
pD f t h f t rh
r
, (5)
where nath /)( ,
!
)1()2)(1(
r
rpppp
r
p
. (6)
If ,0p let ,pq
!
)1()2)(1(
r
rqqqq
r
q
, (7)
then
r
q
r
rqqqq
r
pr)1(
!
)1()2)(1( , (8)
or,
r
q
r
pr)1( . (9)
From (5) and (9), we have
0
0
lim , 0, 0.n
q q
a th
rnh t a
qD f t h f t rh p q p
r
(10)
In particular, we have
10
0
1lim
n
a th
rnh t a
D f t h f t rhr
00
0
lim ,n t a t
ahrnh t a
h f t rh f t s ds f s ds
(11)
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2 2 20 0
0 0
2lim lim 1
n n
a th h
r rnh t a nh t a
D f t h f t rh h r f t rhr
1
001
lim ,n t a t
ahrnh t a
h rh f t h rh sf t s ds t f d
(12)
3 3 30 0
0 0
3 1 2lim lim
2!
n n
a th h
r rnh t a nh t a
r rD f t h f t rh h f t rh
r
21 1
2
0 01 1
lim lim2! 2!
n n
h hr rnh t a nh t a
h hrh f t h rh rh f t h rh
22
0
1 1,
2! 3
t a t
as f t s ds t f d
(13)
Generally, for any positive integer p , define the p order
derivative of a function )(tf
0 0
0 0
1 2 1lim lim
!
n np p p
a th h
r rnh t a nh t a
p p p p p rD f t h f t rh h f t rh
r r
00
1 2 1lim
( 1)!
np
hrnh t a
p p p p rh f t rh
p
11
01
lim1 !
np
hrnh t a
hrh f t h rh
p
2 12
0 01 0
lim lim2 2 !
n np p
h hr rnh t a nh t a
phrh f t h rh h f t rh
p
11
0
1 1.
1 !
t a t pp
as f t s ds t f d
p p
(14)
Since
2 11 .
-1
t pp p
a t a ta
dD f t t f d D f t
dt p
(15)
Then
1 2t t s
p p p
a t a t a ta a a
D f t D f s ds ds D f d
1 2 1 13
1 2 3 3 1 2 .pt t t t t tp
a t p pa a a a a adt dt D f t dt dt dt f t dt
(16) It implies that the derivative of negative integer p- order
and the p-fold integral (2.21) of the
continuous function )(tf are equivalent.
Several Fractional Derivatives
Grunwald-Letnikov fractional derivative and integral definitions
(see [9,10]): Let p be any arbitrary
positive real number, define p order derivative and p order
integral of a continuous function )(tf :
0
0
lim -1)n
G p p ra t
hrnh t a
pD f t h f t rh
r
（ , (17)
and
555
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0
0
limn
G p pa t
hrnh t a
pD f t h f t rh
r
. (18)
Riemann-Liouville fractional derivative and integral definitions
(see [9]): Let m be positive integer,
1 mpm , define Riemann-Liouville p order derivative and p order
integral of a continuous
function )(tf : 1
( ) ( ) ( ) ,
mt
R p m p
a ta
dD f t t f d
dt
(19)
and
11( ) ( ) ( ) .( )
tR p p
a ta
D f t t f dp
(20)
Define Caputo's fractional derivative and integral
definitions(see [9]): Suppose that )(tf
has m order
continuous derivative, define Caputo's p order derivative and p
order integral: ( )
1
1 ( ),( 1 )
( ) ( )
mt
C p
a t p ma
fD d m p m
m p t
, (21)
and
11( ) ( ) ( ) .( )
tC p p
a ta
D f t t f dp
(22)
Comparison of the Definition of Riemann-Liouville, Caputo and
Grunwald-Letnikov
Caputo's integral definition and Grunwald-Letnikov integral
definition are consistent with the
Riemann-Liouville integral definition
11( )= ( )= ( ) ( ) ( ) .( )
tG p R p C p p
a a at t ta
D f t D f t D f t t f dp
(23)
When p is positive integer and ),,2,1(0)()( pkaf k , Caputo's
derivative definition and
Grunwald-Letnikov derivative definition are consistent with the
Riemann-Liouville derivative
definition. ( )= = ( )G p R p C p pa t a t a tD D D f t .
(24)
If )(tf
has 1m order continuous derivative and 1 mpm , then
1
( ) ( ) ( )
mt
R p m p
a ta
dD f t t f d
dt
( )( 1)
0
( )( ) 1( ) ( )
( 1) ( 1)
k p km tm p m
ak
f a t at f d
p k p m
( ), 1.G pa tD f t m p m (25)
It implies that Grunwald-Letnikov positive non-integer order
derivative definition is consistent with the
Riemann-Liouville positive non-integer order derivative
definition.
Summary
Caputo's integral definition and Grunwald-Letnikov integral
definition are consistent with the
Riemann-Liouville integral definition. When p is positive
integer and ),,2,1(0)()( pkaf k ,
Caputo's derivative definition and Grunwald-Letnikov derivative
definition are consistent with the
Riemann-Liouville derivative definition. It implies that
Grunwald-Letnikov positive non-integer order
derivative definition is consistent with the Riemann-Liouville
positive non-integer order derivative
definition.
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Acknowledgements
This research was supported by National Natural Science
Foundation of China (Project No. 11161018,
11561019) and Scientific Research Foundation of the Education
Department of Guangxi Autonomous
Region of China (No. 201106LX599).
References
[1] A. Loverro, Fractional calculus: history, definitions and
applications for the engineer, Rapport
technique, Univeristy of Notre Dame: Department of Aerospace and
Mechanical Engineering,
France, 2004.
[2] V. J. Ervin, N. Heuer and J. P. Roop, Numerical
approximation of a time dependent, nonlinear,
space-fractional diffusion equation, SIAM J. Numer. Anal. Vol.
45 (2007), No 2, 572-591.
[3] N. Heymans and I. Podlubny, Physical interpretation of
initial conditions for fractional differential
equations with Riemann-Liouville fractional derivatives,
Rheologica Acta vol. 45 (2006), No. 5,
765-771.
[4] R. Gorenflo and F. Mainardi, Random walk models for
space-fractional diffusion processes,
Fractional Calulus and Applied Analysis, vol. 1 (1998), No 2,
167-191.
[5] E. Gerolymatou, I. Vardoulakis and R. Hilfer, Modelling
infiltration by means of a nonlinear
fractional diffusion model, J. Phys. D: Appl. Phys. Vol. 39
(2006) 4104-4110.
[6] K. Oldham, J. Spanier, The Fractional Calculus, Theory and
Applications of Differentiation and
Integration of Arbitrary Order, Academic Press, USA, 1974.
[7] G. Samko, A. A. Kilbas, Marichev, Fractional Integrals and
Derivatives: Theory and Applications,
Gordon and Breach, Switzerland, 1993.
[8] K.S. Miller, An Introduction to Fractional Calculus and
Fractional Differential Equations, J.
Wiley and Sons, New York, 1993.
[9] I. Podlubny, Fractional Differential Equations, Academic
Press, New York, 1999.
[10] A. V. Letnikov, Theory of differentiation of arbitrary
order, Mat. Sb. vol. 3 (1868) 1-68 (in
Russian).
[11] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new
Denition of Fractional Derivative, J.
Comput. Appl. Math. Vol. 264(2014) 65-70.
[12] T. Abdeljawad, On conformable fractional calculus, J.
Computational Appl. Math. Vol. 279(2015)
57-66.
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