Mathematical Physics Vol 1

Chapter 8. Introduction to the Fractional Calculus

442

The fractional Riemann-Liouville integral of the order α for the function f ( t ) for α , a ∈ R can be expressed as follows

t Z a

1 Γ ( α )

α a f ( t ) ≡ RL D − α

α − 1 f

( t − τ )

( τ ) d τ .

RL I

a , t f ( t )=

(8.30)

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

t Z a

f ( τ ) ( t − τ ) 1 − α

1 Γ ( α )

α a , t f ( t )=

d τ , 0 < α < 1 , t > 0 .

RL D

(8.31)

Moreover, the left Riemann-Liouville fractional integral and the right Riemann-Liouville fractional integral are defined [5],[39],[44]respectively as

t Z a b Z a

1 Γ ( α )

α a f ( t ) ≡ RL D − α

α − 1 f

( t − τ )

( τ ) d τ .

RL I

a , t f ( t )=

(8.32)

1 Γ ( α )

α b f ( t ) ≡ RL D α

α − 1 f

( τ − t )

( τ ) d τ .

RL I

a , b f ( t )=

(8.33)

where α > 0, n − 1 < α < n . Both Gamma function and Riemann-Liouville fractional integral can be defined for an arbitrary complex order α with positive real order, as well as for purely imaginary order α . Here, the operations of only real order are considered. Furthermore, the left Riemann-Liouville fractional derivative is defined as

t Z a

d n d t n

1 Γ ( n − α )

α a , t f ( t )=

α − 1 f

( t − τ ) n −

( τ ) d τ

RL D

(8.34)

and the right Riemann-Liouville fractional derivative is defined as

b Z t

d n d t n

( − 1 ) n Γ ( n − α )

α t , b f ( t )=

α − 1 f

( τ − t )

( τ ) d τ

RL D

(8.35)

where n − 1 ≤ α < n , a , b are the terminal points of the interval [ a , b ] , which can also be ( − ∞ , ∞ ) . In the case of the α ∈ ( 0 , 1 ) the left Riemann-Liouville fractional derivative is reduced to

t Z a

1 Γ ( 1 − α )

d d t

α a , t f ( t )=

α d τ

f ( τ )( t − τ ) −

RL D

(8.36)

.

For integer values of order α the Riemann-Liouville derivative coincides with the classical, integer order one. In particular [57]

d n − 1 f ( t ) d t n − 1

α a , t f ( t )=

lim α → ( n − 1 ) +

RL D

(8.37)

and

d n f ( t ) d t n

α a , t f ( t )=

lim α → n −

RL D

(8.38)

.