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Introduction to Fractional Calculus with Brief Historical Background

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Introduction to Fractional Calculus with Brief Historical Background

MIHAILO P. LAZAREVIû
Faculty of Mechanical Engineering,
University of Belgrade,
Kraljice Marije 16, 11120 Belgrade,
SERBIA
mlazarevic@mas.bg.ac.rs

MILAN R. RAPAIû
Faculty of Technical Sciences,
University of Novi Sad
Trg Dositeja Obradoviüa 6, 21000 Novi Sad,
SERBIA
rapaja@uns.ac.rs

TOMISLAV B. ŠEKARA,
Faculty of Electrical Engineering,
University of Belgrade,
Bulevar Kralja Aleksandra 73, 11000 Belgrade,
SERBIA
tomi@etf.rs

Abstract: - The Fractional Calculus (FC) is a generalization of classical calculus concerned with operations of
integration and differentiation of non-integer (fractional) order. The concept of fractional operators has been
introduced almost simultaneously with the development of the classical ones. The first known reference can be
found in the correspondence of G. W. Leibniz and Marquis de l’Hospital in 1695 where the question of
meaning of the semi-derivative has been raised. This question consequently attracted the interest of many well-
known mathematicians, including Euler, Liouville, Laplace, Riemann, Grünwald, Letnikov and many others.
Since the 19th century, the theory of fractional calculus developed rapidly, mostly as a foundation for a number
of applied disciplines, including fractional geometry, fractional differential equations (FDE) and fractional
dynamics. The applications of FC are very wide nowadays. It is safe to say that almost no discipline of modern
engineering and science in general, remains untouched by the tools and techniques of fractional calculus. For
example, wide and fruitful applications can be found in rheology, viscoelasticity, acoustics, optics, chemical
and statistical physics, robotics, control theory, electrical and mechanical engineering, bioengineering, etc..In
fact, one could argue that real world processes are fractional order systems in general. The main reason for the
success of FC applications is that these new fractional-order models are often more accurate than integer-order
ones, i.e. there are more degrees of freedom in the fractional order model than in the corresponding classical
one. One of the intriguing beauties of the subject is that fractional derivatives (and integrals) are not a local (or
point) quantities. All fractional operators consider the entire history of the process being considered, thus being
able to model the non-local and distributed effects often encountered in natural and technical phenomena.
Fractional calculus is therefore an excellent set of tools for describing the memory and hereditary properties of
various materials and processes.

Key-Words: fractional calculus, historical background, Riemann-Liouville definition, Grunwald-Letnikov

definition, Caputo definition


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1. 1 Brief History of Fractional Calculus

Fractional calculus (FC) is an extension of ordinary calculus with more than 300 years of history. FC was
initiated by Leibniz and L`Hospital as a result of a correspondence which lasted several months in 1695. In that
year, Leibniz wrote a letter to L'Hospital raising the following question [1]:
“Can the meaning of derivatives with integer order be generalized to derivatives with non-integer
orders?" L'Hopital was somewhat curious about the above question and replied by another
simple one to Leibniz: “What if the order will be 1/2?". Leibniz in a letter dated September 30,
1695, replied: “It will lead to a paradox, from which one day useful consequences will be
drawn."
That date is regarded as the exact birthday of the fractional calculus. The issue raised by Leibniz for a
fractional derivative (semi-derivative, to be more precise) was an ongoing topic in decades to come [1,2].
Following L’Hopital’s and Liebniz’s first inquisition, fractional calculus was primarily a study reserved for
the best mathematical minds in Europe. Euler [2],wrote in 1730:
“When n is a positive integer and p is a function of x, p p  x  , the ratio of d n p to dx n can
always be expressed algebraically. But what kind of ratio can then be made if n be a fraction?“
Subsequent references to fractional derivatives were made by Lagrange in 1772, Laplace in 1812, Lacroix in
1819, Fourier in 1822, Riemann in 1847, Green in 1859, Holmgren in 1865, Grunwald in 1867, Letnikov in
1868, Sonini in 1869, Laurent in 1884, Nekrassov in 1888, Krug in 1890, Weyl in 1919, and others [3-5].
During the 19th century, the theory of fractional calculus was developed primarily in this way, trough insight
and genius of great mathematicians. Namely, in 1819 Lacroix [6], gave the correct answer to the problem raised
by Leibnitz and L’Hospital for the first time, claiming that d 1/ 2 x / dx1/ 2 2 x / S . In his 700 pages long book
on Calculus published in 1819, Lacroix developed the formula for n-th derivative of y x m , with m being a
positive integer

Dxn y
dx
dn
n  xm  m!
 m  n !
xm n , m t n (1)

Replacing the factorial symbol by Gamma function (3), he developed the formula for the fractional derivative
of a power function
*  E  1
DDx x E x E D (2)
*  E  D  1
where D and E are fractional numbers and where the gamma function *  z  1 is defined for z ! 0 as:
f
 x z 1
*z ³e x dx (3)
0
In particular, Lacroix calculated
*  2  1/ 2 x
D1/
x x
2
x 2 (4)
*  3/ 2  S
Surprisingly, the previous definition gives a nonzero value for the fractional derivative of a constant function
 E 0  , since
DD D 0
x D z 0
1
x 1 Dx x (5)
* 1  D 
Using linearity of fractional derivatives, the method of Lacroix is applicable to any analytic function by term-
vise differentiation of its power series expansion. Unfortunately, this class of functions is too narrow in order
for the method to be considered general.
It is interesting to note that simultaneously with these initial theoretical developments, first practical
applications of fractional calculus can also be found. In a sense, the first of these was the discovery by Abel in

*
See the Appendix A.1.


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1823,[7-9]. Abel considered the solution of the integral equation related to the tautochrone problem . He found
that the solution could be accomplished via an integral transform, which could be written as a semi-derivative.
More precisely, the integral transform considered by Abel was
x
1/ 2
K ³ x  t f  t  dt , K const . (6)
0
Abel wrote the right hand side of (6) by means of a fractional derivative of order 1/ 2 ,
§ d 1/ 2 ·
1/ 2 
S¨ f  x  ¸ (7)
¨ ¸
© dx ¹
Abel’s solution had attracted the attention of Joseph Liouville, who made the first major study of fractional
calculus,[10-13]. The most critical advances in the subject came around 1832 when he began to study fractional
calculus in earnest and then managed to apply his results to problems in potential theory. Liouville began his
theoretical development using the well-known result for derivatives of integer order n
Dxn eax a n eax . (8)
Expression (8) can rather easily be formally generalized to the case of non-integer values of n, thus obtaining
DD
xe
ax
aD eax (9)
By means of Fourier expansion, a wide family of functions can be composed as a superposition of complex
exponentials.
f
f  x ¦ cn exp(an x ), Re an ! 0 (10)
n 0
Again, by invoking linearity of the fractional derivative, Liouville proposed the following expression for
evaluating the derivative of order D
f
DDx f ( x ) ¦ cn anD ea x . n
(11)
n 0
Formula (11) is known as the Liouville's first formula for a fractional derivative,[10,11]. However, this formula
cannot be seen as a general definition of fractional derivative for the same reason Lacroix formula could not:
because of its relatively narrow scope. In order to overcome this, Liouville labored to produce a second
definition. He started with a definite integral (closely related to the gamma function):
f
E 1  xu
I ³u e du, E ! 0, x ! 0. (12)
0
and derived what is now referred to as the second Liouville’s formula
* D  E  D  E
DD
xx
E
 1D x , E !0 (13)
*E 
None of previous definitions were found to be suitable for a general definition of a fractional derivative. In the
consequent years, a number of similar formulas emerged. Greer [14], for example, derived formulas for the
fractional derivatives of trigonometric functions using (9) in the form:
§ SD SD ·
DD
xe
iax
aD ¨ cos  i sin ¸  cos ax  i sin ax  (14)
© 2 2 ¹
Joseph Fourier [15] obtained the following integral representations for f  x  and its derivatives

The tautochrone problem consists of the determination of a curve in the (x, y) plane such that the time required for a
particle to slide down the curve to its lowest point under the influence of gravity is independent of its initial position (xo,
yo) on the curve.


&KDSWHU

f f
1
Dxn f  x ³ f [  d [ ³t
n
cos ª¬t  x  [   nS / 2 º¼dt , (15)
2S
f f
By formally replacing integer n by an arbitrary real quantity D he obtained
f f
DD D
1
x f  x ³ f [  d [ ³t cos ª¬t  x  [   DS / 2 º¼dt. (16)
2S
f f
Probably the most useful advance in the development of fractional calculus was due to a paper written by G. F.
Bernhard Riemann [16] during his student days. Unfortunately, the paper was published only posthumously in
1892. Seeking to generalize a Taylor series in 1853, Riemann derived different definition that involved a
definite integral and was applicable to power series with non-integer exponents
x
D 1
Dc,Dx f  x 
1
* D  ³ x  t f  t  dt  <  x  (17)
c

In fact, the obtained expression is the most-widely utilized modern definition of fractional integral. Due to the
ambiguity in the lower limit of integration c, Riemann added to his definition a “complementary” function
<  x  where the present-day definition of fractional integration is without the troublesome complementary
function. Since neither Riemann nor Liouville solved the problem of the complementary function, it is of
historical interest how today's Riemann-Liouville definition was finally deduced.
The earliest work that ultimately led to what is now called the Riemann-Liouville definition appears to be
the paper by N. Ya. Sonin in 1869, [17] where he used Cauchy`s integral formula as a starting point to reach
differentiation with arbitrary index. A. V. Letnikov [18] extended the idea of Sonin a short time later in 1872,
[19]. Both tried to define fractional derivatives by utilizing a closed contour. Starting with Cauchy's integral
formula for integer order derivatives, given by
f t 
f   z
n!
³
n
dt , (18)
2S i C  t  z n 1

the generalization to the fractional case can be obtained by replacing the factorial with Euler's Gamma function
D ! * 1  D  . However, the direct extension to non-integer values D results in the problem that the integrand
in (18) contains a branching point, where an appropriate contour would then require a branch cut which was
not included in the work of Sonin and Letnikov. Finally, Laurent [20], used a contour given as an open circuit
(known as Laurent loop) instead of a closed circuit used by Sonin and Letnikov and thus produced today's
definition of the Riemann-Liouville fractional integral
x
D 1
Dc,Dx f  x 
1
* D  ³x  t f  t  dt , Re D  ! 0 . (19)
c
In expression (19) one immediately recognizes Riemann’s formula (17), but without the problematic
complementary function. In nowadays terminology, expression (19) with lower terminal c = í ’ is referred as
Liouville fractional integral; by taking c = 0 the expression reduces to the so called Riemann fractional integral,
whereas the expression (19) with arbitrary lower terminal c is called Riemann-Liouville fractional integral.
Expression (19) is the most widely utilized definition of the fractional integration operator in use today. By
choosing c = 0 in (19) one obtains the Riemann's formula (17) without the problematic complementary function
<  x  and by choosing c f , formula (19) is equivalent to Liouville's first definition (10). These two facts
explain why equation (19) is called Riemann-Liouville fractional integral. While the notation of fractional
integration and differentiation only differ in the sign of the parameter D in (19), the change from fractional
integration to differentiation cannot be achieved directly by inserting negative D at the right-hand side of (19).
The problem originates from the integral at the right side of (19) which is divergent for negative integration
orders. However, by analytic continuation it can be shown that


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dn § 1 ·
x
D E 1
Dc, x f  x  Dcn,x E f  x Dcn, x f  x  Dc,Ex f  x ¨
³  x  t  f  t  dt ¸ , (20)
dx n ¨© *  E  c ¸
¹

holds, which is known today as the definition of the Riemann-Liouville fractional derivative. In (20) n >D @ is
the smallest integer greater than D with 0  E n  D  1 . For either c 0 or c f the integral in (20) is the
Beta-integral (see Appendix A.2) for a wide class of functions and thus easily evaluated.
Nearly simultaneously, Grunwald and Letnikov provided the basis for another definition of fractional
derivative [21] which is also frequently used today. Disturbed by the restrictions of the Liouville`s approach
Grunwald (1867) adopted the definition of a derivative as the limit of a difference quotient as its starting point.
He arrived at definite-integral formulas for ordinary derivatives, showed that Riemann’s definite integral had to
be interpreted as having a finite lower limit, and also that the Liouville’s definition, in which no distinguishable
lower limit appeared, correspond to a lower limit f . Formally,

§D ·
'h f   x 
D ¦  1k ¨© k ¸¹ f  x  kh 
GL D
Dx f  x lim lim k 0 , D !0 (21)
h o0 hD h o0 hD
§D ·
which is today called the Grunwald-Letnikov fractional derivative. In definition (21), ¨ ¸ is the generalized
©k ¹
binomial coefficient, wherein the factorials are replaced by Euler's Gamma function. Letnikov [18] also showed
that definition (21) coincides, under certain relatively mild conditions, with the definitions given by Riemann
and Liouville. Today, the Grunwald-Letnikov definition is mainly used for derivation of various numerical
methods, which use formula (21) with finite sum to approximate fractional derivatives. Together with the
advances in fractional calculus at the end of the nineteenth century the work of O. Heaviside [22] has to be
mentioned. The operational calculus of Heaviside, developed to solve certain problems of electromagnetic
theory, was an important next step in the application of generalized derivatives. The connection to fractional
calculus has been established by the fact that Heaviside used arbitrary powers of p, mostly p , to obtain
solutions of various engineering problems.
Weyl [23] and Hardy,[24,25], also examined some rather special, but natural, properties of differintegrals of
functions belonging to Lebesgue and Lipschitz classes in 1917. Moreover, Weyl showed that the following
fractional integrals could be written for 0  D  1 assuming that the integrals in (22) are convergent over an
infinite interval
x f
I D M  x   x  t D 1M  t  dt, I D M  x 
D 1
1 1
* D  ³ * D  ³ t  x  M  t  dt , (22)
f x

Specially, the Riemann-Liouville definition of a fractional integral given in (19) with lower limit c f , the
form equivalent to the definition of fractional integral proposed by Liouville, is also often referred to as Weyl
fractional integral. In the modern terminology one recognizes two distinct variants of all fractional operators,
left sided and right sided ones. Weyl operators defined in (22) are sometimes also referred to as the left and
right Liouville fractional integrals, respectively.
Later, in 1927 Marchaud [27] developed an integral version of the Grunwald-Letnikov definition (21) of
fractional derivatives, using

D
Dx f  x 
D
f
 'lt f   x dt D
f
f ( x)  f ( x  t )
*(1  D ) ³ *(1  D ) ³
M
1D
dt , D !0 (23)
0
t 0
t1D

as fractional derivative of a given function f , today known as Marchaud fractional derivative. The term
'lt f   x  is a finite difference of order l ! D and c is a normalizing constant. Since this definition is related to
the Grunwald-Letnikov definition,it also coincides with the Riemann-Liouville definition under certain


&KDSWHU

conditions. M. Riesz published a number of papers starting from 1938 [28, 29] which are centered around the
integral
f
R D 1 M t 
I M
2 * D  cos DS / 2  ³ tx
1D
dt , Re D ! 0, D z 1,3,5,... (24)
f

today known as Riesz potential. This integral (and its generalization in the n-dimensional Euclidean space) is
tightly connected to Weyl fractional integrals (22) and therefore to the Riemann-Liouville fractional integrals
by
R D
I  I D  I D   2cos DS / 2 1 (25)
In 1949 Riesz [29] also developed a theory of fractional integration for functions of more than one variable.
A modification of the Riemann-Liouville definition of fractional integrals, given by

2x 
2 D K  x
2 x 2K
x

³  ³ x 
D 1 2K 1 D 1 1 2D  2K
x2  t2 t M  t  dt , 2
 t2 t M  t  dt , (26)
* D  * D 
0 0

were introduced by Erdelyi et al. in [30-32], which became useful in various applications. While these ideas are
tightly connected to fractional differentiation of the functions x 2 and x , already done by Liouville 1832, the
fact that Erdelyi and Kober used the Mellin’s transform for their results is noteworthy.
Among the most significant modern contributions to fractional calculus are those made by the results of M.
Caputo in 1967,[33]. One of the main drawbacks of Riemann-Liouville definition of fractional derivative is that
fractional differential equations with this kind of differential operator require a rather “strange” set of initial
conditions. In particular, values of certain fractional integrals and derivatives need to be specified at the initial
time instant in order for the solution of the fractional differential equation to be found. Caputo [33,34]
reformulated the more “classic” definition of the Riemann-Liouville fractional derivative in order to use
classical initial conditions, the same one needed by integer order differential equations [34]. Given a function f
with an  n  1 absolutely continuous integer order derivatives, Caputo defined a fractional derivative by the
following expression
t n
n D 1 § d ·
D*D f  x 
1
* n  D  ³ t  s  ¨
© ¹
ds
¸ f  s  ds, (27)
0
Derivative (27) is strongly connected to the Riemann-Liouville fractional derivative and is today frequently
used in applications. It is interesting to note that Rabotnov [35] introduced the same differential operator into
the Russian viscoelastic literature a year before Caputo’s paper was published. Regardless of this fact, the
proposed operator is in the present-day literature commonly named after Caputo.
By the second half of the twentieth century, the field of fractional calculus had grown to such extent that in
1974 the first conference “The First Conference on Fractional Calculus and its Applications” concerned solely
with the theory and applications of fractional calculus was held in New Haven. In the same year, the first book
on fractional calculus by Oldham and Spanier [3] was published after a joint collaboration started in 1968. A
number of additional books have appeared since then, for example McBride (1979) [36], Nishimoto (1991)
[37], Miller and Ross (1993), [4], Samko et al. (1993),[38], Kiryakova (1994) [39], Rubin (1996) [40],
Carpinteri and Mainardi (1997),[41], Davison and C. Essex (1998), [42],Podlubny (1999) [43], R. Hilfer (2000)
[44], Kilbas et.al (2006),[5], Das (2007)[45], J. Sabatier et. al (2007) [46], and others. In 1998 the first issue of
the mathematical journal “Fractional calculus & applied analysis” was printed. This journal is solely
concerned with topics on the theory of fractional calculus and its applications. Finally, in 2004 the first
conference “Fractional differentiation and its applications” was held in Bordeaux, and it is organized every
second year since 2004,[47].


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1. 2 Basic Definitions of Fractional Order Differintegrals

There are many different forms of fractional operators in use today. Riemann-Liouville, Grunwald-Letnikov,
Caputo, Weyl and Erdely-Kober derivatives and integrals are the ones mentioned in the previous historical
survey. In addition, most of these operators can be defined in two distinct forms, as the left and as right
fractional operators. The two most frequently used definitions for the general fractional differintegral are: the
Grunwald-Letnikov (GL) and the Riemann-Liouville (RL) definitions,[3-5],[43]. Also, the Caputo derivative,
as a variation of the Riemann-Liouville differential operator, is used frequently. A short account of these most
frequently used operators is given next. Grunwald and Letnikov defined fractional derivative in the following
way
 'Dh f  x 
D
GL Dx f ( x) lim ,
h o0 hD
(28)
§D ·
D
'h f  x  ¦  1 j
¨ j ¸ f  x  jh  , h ! 0,
© ¹
0d j f
known as the left Grunwald-Letnikov (GL) derivative. This derivative can be seen as a limit of the fractional
order backward difference. The right sided derivative is defined accordingly

 'D h f  x 
D
GL Dx f ( x) lim ,
h o0 hD
(29)
§D ·
D
' h f  x  ¦  1 j
¨ j ¸ f  x  jh  , h  0,
© ¹
0d j f

Definitions (28) and (29) are valid for both Į > 0 (fractional derivative) and for Į < 0 (fractional integral) and,
commonly, these two notions are grouped into one single operator called GL differintegral. The GL derivative
and RL derivative are equivalent if the functions they act upon are sufficiently smooth. The generalized
binomial coefficients, calculation for D  R and k  ` 0 , is the following

§D · D D  1 ... D  j  1 * D  1
¨ j¸
© ¹
D!
j ! D  j  ! j! *  j  1 * D  j  1
,  D0  1 (30)

Let us consider n t  a / h, where a is a real constant. This constant can be interpreted as the lower terminal
(an analogue of the lower integration limit, necessary even for the derivative operator due to its non-local
properties). The GL differentigral can be expressed as a limit
ª t a º
«¬ h »¼
§D ·
D
¦
1
GL Da ,t f (t ) lim D  1 j ¨ ¸ f  t  jh  , (31)
h o0 h © j¹
j 0

where [x] means the integer part of x, a and t are the bounds of the operation for GL DaD,t f (t ) . For the numerical
calculation of fractional-order derivatives we can use the following relation (32) derived from the GL definition
(31). The relation to the explicit numerical approximation of the D -th derivative at the points kh, (k=1,2,...) has
the following form, [43]
N ( x)
¦ bj
rD BD rD 
 ( x  L  Dx f ( x ) | h f ( x  jh ) (32)
j 0

where L is the “memory length”, h is the step size of the calculation,

­ª x º ª L º ½
N (t ) min ® « » , « » ¾ , (33)
¯¬ h ¼ ¬ h ¼ ¿


&KDSWHU

 rD  is the binomial coefficient given by

> x@ is the integer part of x and b j

 rD  § 1 r D · ( rD )
b0( rD ) 1, b j ¨ 1  j ¸ b j 1 (34)
© ¹
This approach is based on the fact that (for a wide class of functions and assuming all initial conditions are
zero) the three most commonly used definitions - GL, RL, and Caputo’s - are equivalent,[48].
For expression of the Riemann-Liouville definition, we will consider the Riemann-Liouville n-fold integral
for n  N , n ! 0 defined as (this expression is usually referred to as the Cauchy repeated integration formula)
t tn tn 1 t3 t2 t
1 n 1
³³ ³ ³³ ... f  t1  dt1dt2 ...dtn 1dtn ³ t  W  f W  dW , (35)
* n
 

aa a aa a
n  fold

The fractional Riemann-Liouville integral of the order D for the function f (t ) for D ,a  R can be expressed
as follows
t
D D 1 D 1
f t  { f t  t W  f W  dW ,
* D  ³
RL I a RL Da ,t (36)
a

For the case of 0  D  1, t ! 0 , and f (t ) being a causal function of t , the fractional integral is presented as

f W 
t
D 1
f t 
* D  ³  t  W 1D
RL Da ,t dW , 0  D  1, t ! 0 (37)
a

Moreover, the left Riemann-Liouville fractional integral and the right Riemann-Liouville fractional integral are
defined [5],[38],[43]respectively as
t
D D 1 D 1
RL I a f  t { RL Da ,t f  t  ³ t  W  f W  dW , (38)
* D 
a
b
D D 1 D 1
f t  { W  t  f W  dW ,
* D  ³
RL I b RL Dt ,b f (t ) (39)
t

where D ! 0, n  1  D  n . Both Gamma function and Riemann-Liouville fractional integral can be defined for
an arbitrary complex order Į with positive real order, as well as for purely imaginary order Į. However, since
the target application area of the present book are stability issues, process control and signal processing, as well
as modeling, the operations of only real order are considered. Furthermore, the left Riemann-Liouville
fractional derivative is defined as
t
D 1 dn n D 1
RL Da ,t f t  ³ t W  f W  dW , (40)
*  n  D  dt n
a

and the right Riemann-Liouville fractional derivative is defined as

 1
n
dn
b
D
RL Dt ,b f  t  W  t n D 1 f W  dW ,
*  n  D  dt n ³
(41)
t

where n  1 d D  n , a, b are the terminal points of the interval > a, b@ , which can also be f, f . In the very
important case of D  (0,1) where the above definition of the left Riemann-Liouville fractional derivative is
reduced to


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D d t
f (W )(t  W ) D dW .
1
RL Da ,t f (t )
*(1  D ) dt a ³ (42)

A very important fact is that for integer values of order Į the Riemann-Liouville derivative coincides with the
classical, integer order one. In particular [48]

D d n 1 f (t )
lim RL Da ,t f (t ) (43)
D o( n 1)  dt n 1
and
d n f (t )
lim  RL DaD,t f (t ) (44)
D on dt n
A very interesting property of the fractional derivative is that the fractional derivative of a constant is not equal
to zero. The RL fractional derivative of a constant C takes the form

D  t  a D z0
RL Da ,t C C (45)
* 1  D 

However, definitions of the fractional differentiation of Riemann-Liouville type create 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. A solution to this conflict was proposed by Caputo, see [33,34]. The left Caputo fractional derivative
is
t
f   W  dW ,
D 1 n D 1
C Da ,t f  t  ³ t  W 
n
(46)
* n  D 
a

and the right Caputo fractional derivative is

 1
n b
D
C Dt ,b f  t  ³ W  t n D 1 f ( n ) W  dW , (47)
* n  D 
t

where f ( n ) (W ) d n f (W ) / dW n and n  1 d D  n  ]  . It is obvious from the definition (47) that the Caputo
fractional derivative of a constant is zero. Regarding continuity with respect to the differentiation order,
Caputo derivative satisfies the following limits

D d n 1x (t )
lim C Da ,t x (t ) n 1
 D ( n 1) x ( a ) (48)
D o( n 1) 
dt
and

d n x (t )
lim  C DaD,t x (t ) . (49)
D on dt n
Obviously, Riemann-Liouville operator RL Dan , n   f, f  , varies continuously with n . This is not the
case with the Caputo derivative. Obviously, Caputo derivative is stricter than Riemann-Liouville derivative;
one reason is that the n-th order derivative is required to exist. On the other hand, the initial conditions of
fractional differential equations with Caputo derivative have a clear physical meaning and Caputo derivative is
extensively used in engineering applications. The left and right Riemann-Liouville and Caputo fractional
derivatives are interrelated by the following expressions
n 1
 1k f ( k ) (a ) t  a k D
D
RL Da ,t f  t 
D
C Da ,t f  t   ¦ *  k  D  1   , (50)
k 0


&KDSWHU

n 1
 1 f ( k ) (b)
k
RL Dt ,b f  t  C Dt ,b f  t   ¦
D D
 b  t k D . (51)
k 0
*  k  D  1

1. 3 Basic Properties of Fractional Order Differintegrals

As stated previously, for a wide class of functions, Grunwald-Letnikov definition of the fractional derivative
operator coincides with the Riemann-Liouville definition. Thus, in the present section only Riemann-Liouville
and Caputo derivatives will be considered. Also, left-side operators are used primarily in the following
chapters. Thus, all of the properties presented next will be accounted for this kind of fractional operators only.
Similar properties can be formulated and proven for the right-sided operators accordingly. The reader is
referred to the available literature [3-5],[43].
Similar to the classical, integer-order integral, the Riemann-Liouville fractional integral satisfies the semi-
group property,[38] i.e. for any positive orders D and E
E
D
RL I t ,a RL I t ,a f (t ) RL I tE,a RL I tD,a f (t ) RL I tD,a E f (t ) . (52)

Interestingly, the same does also hold for integer order derivatives, but not for fractional order ones. Let us
introduce the following notation
n j
§d· n D
f n(nD j ) (t ) ¨ ¸ RL I a ,t f (t ) . (53)
© dt ¹
A combination of Riemann-Liouville derivatives, for example, results in the following expression
n f n(n E j ) ( a )
D
RL Da ,t RL Da ,t
E
f (t ) RL Da ,t
D E
f (t )  ¦ *(1  j  D ) (t  a ) j D , (54)
j 1

with n being the smallest integer bigger then E . Thus, in general,

E
D
RL Da ,t RL Da ,t f (t ) z RL DaE,t RL DaD,t f (t ) z RL DaD,t E f (t ) . (55)

A similar result can be obtained for the Caputo derivative. Fractional derivatives do not commute!
It is a well-known fact that the classical derivative is the left inverse of the classical integral. The similar
relation holds for the Riemann-Liouville derivative and integral
D D
RL Da ,t RL I a ,t f (t ) f (t ) . (56)

The opposite, however, is not true (in both the fractional and integer order case)

f n(nD j ) ( a )
n
D D
RL I a ,t RL Da ,t f (t )
D E
RL Da ,t f ( t )  ¦ * ( D  j  1)
(t  a )D  j . (57)
j 1

Utilizing expression (50), similar expressions can be obtained relating the Riemann-Liouville integral and
derivative of Caputo type. In particular, assuming that the integrand is continuous or, at least, essentially
bounded function, Caputo derivative is also the left inverse of the fractional integral.
It is rather important to notice that the Caputo and the Riemann-Liouville formulations coincide when the
initial conditions are zero [43]. Besides, the RL derivative is meaningful under weaker smoothness
requirements. In fact, assuming that all initial conditions are zero, a number of relations between the fractional
order operators is greatly simplified. In such a case, both fractional integral and fractional derivatives possess
the semi-group property; the fractional derivative is both left and right inverse to the fractional integral of the
same order; and the operations of fractional integration and differentiation can exchange places freely. In the
symbolic notation, for any 0  D  E
D E E D D E
RL Da ,t RL Da ,t f RL Da ,t RL Da ,t f RL Da ,t f, (58)

D D D D
RL I a ,t RL Da ,t f RL Da ,t RL I a ,t f f (t ) , (59)


 ,QWURGXFWLRQWR)UDFWLRQDO&DOFXOXVZLWK%ULHI+LVWRULFDO%DFNJURXQG

D E E D D E
C Da ,t C Da ,t f C Da ,t C Da ,t f C Da ,t f , (60)

D D D D
RL I a ,t C Da ,t f C Da ,t RL I a ,t f f (t ) , (61)

Laplace transform is one of the major formal tools of science and engineering, especially when modeling
dynamical systems. Also, Laplace transform is also usually used for solving fractional integro-differential
equations involved in various engineering problems. The Laplace transform L {.} of the RL fractional
derivative is
f n 1
D
L { RL D0,t f (t )} ³e
 st
RL D0,t
D
f (t )dt D
s F ( s)  ¦ sk RL D0,D tk 1 f (t ) t 0 (62)
0 k 0

Laplace transform of the Riemann-Liouville fractional integral (38) of f  t  is

L { RL I 0D f  t }
1
F ( s) , (63)
sD
The terms appearing in the sum on the right hand side of the expression (62) involve the initial conditions and
these conditions must be specified when solving fractional differential equations. Laplace transform of Caputo
fractional derivative is
f n 1
³e
 st
C D0,t
D D
f (t )dt s F ( s )  ¦ sD k 1 f (k ) (0), n 1  D  n (64)
0 k 0

which implies that all the initial conditions required by a fractional differential equation are presented by a set
of only classical integer-order derivatives. Note also that the assumption of zero initial conditions is perfectly
sensible when implementing fractional order controllers and filters. However, when attempting to simulate a
fractional order system, the effect of initial conditions must be taken into consideration. In such a case, also, the
difference between various definitions of fractional operators cannot be neglected. Besides that, the geometric
and physical interpretations of fractional integration and fractional differentiation can be found in Podlubny’s
work,[43].Assuming that all initial conditions are equal to zero the fractional differintegral can be exactly
represented by its transfer function
1
G( s) (65)
sD
which corresponds to the fractional derivative for negative values of the exponent D and to the fractional
integral for the positive ones. By substituting s jZ into (65) one obtains the frequency characteristic of
fractional operators. Thus, the important difference between integer-order and fractional-order systems is
revealed,[49,50]. A well-known fact is that the slope of the amplitude characteristics of the integer order
systems is always an integer multiple of 20 dB/decade. This is not true for fractional order systems which can,
in general, have amplitude characteristics of arbitrary slope. Similarly, an integer order system can have
constant phase only if it is a multiple of pi/2, while the fractional order systems can have arbitrary constant
phase. Thus, sometimes,fractional systems are referred to as ideal Bode systems. Amplitude and phase
characteristics of fractional differintegrals of different order are shown in the following Figures 1,2.


&KDSWHU

80

60 D = -2

D = -1.5
40

20 log|G(j Ȧ)|
D = -1

20 D = -0.5

0 D =0

-20 D = 0.5

D =1
-40
D = 1.5

-60 D =2

-80 -2 -1 0 1 2
10 10 10 10 10
Ȧ
Fig. 1: Logarithmic amplitude characteristics of few fractional differintegrator (65)

1 D = -2

0.8
D = -1.5

0.6
D = -1
0.4

D = -0.5
0.2
arg(G(j Ȧ)|

0 D =0

-0.2
D = 0.5

-0.4
D =1
-0.6

D = 1.5
-0.8

-1 D =2

-2 -1 0 1 2
10 10 10 10 10
Ȧ
Fig. 2: Phase characteristics of few fractional differintegrators (65)

In the field of control theory, many aspects of linear systems have been investigated, in particular, different
forms of stability and robustness criteria are developed. An in-depth generalization of various aspects of control
theory to fractional order systems has been presented in [49-51].


 ,QWURGXFWLRQWR)UDFWLRQDO&DOFXOXVZLWK%ULHI+LVWRULFDO%DFNJURXQG

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