Mathematical Physics Vol 1

8.3 Basic Properties of Fractional Order Differintegrals

445

fractional operators only. Similar properties can be formulated and proven for the right-sided operators accordingly [3-5],[52]. Similar to the classical, integer-order integral, the Riemann-Liouville fractional integral satisfies the semi-group property ,[5] i.e. for any positive orders α and β

β t , a f ( t )= RL I β

α + β t , a RL

α t , aRL I

α t , a f ( t )= RL I

RL I

t , aRL I

(8.53)

Interestingly, the same does also hold for integer order derivatives, but not for fractional order ones. Fractional derivatives do not commute! Let us introduce the following notation

( t )=

d d t

n − j

f ( n − j ) n − α

n − α a , t

RL I

f ( t ) .

(8.54)

A combination of Riemann-Liouville derivatives, for example, results in the following expression

f ( n − j ) n − β ( a ) Γ ( 1 − j − α )

n ∑ j = 1

β a , t f ( t )= RL D

α + β a , t f ( t ) −

α a , t RL D

α

( t − a ) − j −

RL D

(8.55)

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

β a , t f ( t )̸= RL D β

α + β a , t f ( t ) .

α a , t RL D

α a , t f ( t )̸= RL D

RL D

a , t RL D

(8.56)

A similar result can be obtained for the Caputo derivative. 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

α a , t RL I

α a , t f ( t )= f ( t ) .

RL D

(8.57)

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

f ( n − j ) n − α ( a ) Γ ( α − j + 1 )

n ∑ j = 1

α + β a , t f ( t ) −

α a , t RL D

α a , t f ( t )= RL D

α − j

RL I

( t − a )

(8.58)

.

Utilizing expression (8.43), 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 [52]. 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 < α < β

β a , t f ( t )= RL D β α a , t f ( t )= RL D α β a , t f ( t )= C D β α a , t f ( t )= C D α

α + β a , t f ( t ) ,

α a , t RL D α a , t RL D α a , t C D α a , t C D

α a , t f ( t )= RL D α a , t f ( t )= f ( t ) , α a , t f ( t )= C D α a , t f ( t )= f ( t ) .

RL D

a , t RL D a , t RL I

(8.59) (8.60) (8.61) (8.62)

RL I

α + β a , t f ( t ) ,

C D RL I

a , t C D a , t RL I