Mathematical Physics Vol 1

8.2 Basic Definitions of Fractional Order Differintegrals

443

Moreover, very interesting property of the RL fractional derivative is that the fractional derivative of a constant C is not equal to zero. The RL fractional derivative of a constant C takes the form

( t − a ) − α Γ ( 1 − α )̸

α a , t C = C

RL D

= 0 .

(8.39)

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 [41,42]. The left Caputo fractional derivative is

t Z a

d n f ( τ ) d τ n

1 Γ ( n − α )

α a , t f ( t )=

α − 1

( t − τ ) n −

d τ , n − 1 ≤ α < n ∈ Z +

C D

(8.40)

and the right Caputo fractional derivative is

b Z t

d n f ( τ ) d τ n

1 Γ ( n − α )

α t , b f ( t )=

α − 1

( τ − t ) n −

d τ , n − 1 ≤ α < n ∈ Z + .

C D

(8.41)

It is obvious from the definition (8.40) that the Caputo fractional derivative of a constant is zero. Regarding continuity with respect to the differentiation order, Caputo derivative satisfies the following limits lim α → ( n − 1 ) + C D α a , t x ( t )= d n − 1 x ( t ) d t n − 1 − D ( n − 1 ) x ( a ) (8.42) and lim α → n − C D α a , t x ( t )= d n x ( t ) d t n (8.43) Obviously, Riemann-Liouville operator RL D n a , n ∈ ( − ∞ , ∞ ) , 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 ∑ k = 0 n − 1 ∑ k = 0

f ( k ) ( a ) Γ ( k − α + 1 ) f ( k ) ( b ) Γ ( k − α + 1 )

α a , t f ( t )= C D α

α

( t − a ) k −

RL D

a , t f ( t )+

(8.44)

α t , b f ( t )= C D α

α

( b − t ) k −

RL D

t , b f ( t )+

(8.45)

.

Grunwald and Letnikov defined fractional derivative in the following way

( ∆ α

h f ( x )) h α ( − 1 ) | j |

α x f ( x )= lim h → 0 h f ( x )= ∑

GL D

j

(8.46)

α

∆ α

f ( x + jh ) , h > 0 ,

0 ≤| j | < ∞