Mathematical Physics Vol 1

A.3 Fractional Derivatives and Integrals

461

Only one of the two operators I and D needs to be used, since it is all a matter of changing the signof α . In practice D is the one more often used.

A.3.2 Properties

Semigroup properties For ℜ ( α ) ∈ ( n − 1 , n ] , ℜ ( β ) ∈ ( m − 1 , m ] , m , n ∈ N , and for suitable functions, we have: RL I α a + RL I β a + f ( x )= RL I α + β a + f ( x ) and RL I α b − RL I β b − RL f ( x )= RL I α + β b − f ( x ) . (A.45)

( x − a ) − j − α Γ ( 1 − j − α )

m ∑ j = 1

RL D β

RL D α + β

RL D β − j

RL D α

RL f

( x )=

a + f ( x ) −

a + f ( a )

(A.46)

a +

a +

RL D β

RL I α − β

RL D β

RL I α − β b −

RL I α

RL I α

a + f ( x )=

a + f ( x ) and

f ( x )=

f ( x ) .

(A.47)

a +

b −

b −

RL D α

RL I α

RL D α

RL I α

a + f ( x )=

f ( x )= f ( x . )

(A.48)

a +

b −

b −

RL I n − α

a + f

( n − j )

( a )

n ∑ j = 1

RL I α

RL D α

α − j

a + f ( x )= f ( x ) −

( x − a )

(A.49)

a +

Γ ( α − j + 1 )

( − 1 ) n

f

( n − j )

RL I n − α b −

( a )

n ∑ j = 1

RL I α

RL D α

α − j

f ( x )= f ( x ) −

( b − x )

(A.50)

.

b −

b −

Γ ( α − j + 1 )

RL D m RL D α

RL D α + m

a + f ( x )=

a + f ( x ) , m ∈ N .

(A.51)

RL D m RL D α

RL D α + m b −

f ( x )=( − 1 ) m

f ( x ) , m ∈ N .

(A.52)

b −

C D α

RL I α

C D α

RL I α

b − f ( x )= f ( x ) , ℜ ( α ) / ∈ N ∨ α ∈ N .

a + f ( x )=

(A.53)

a +

b −

1 Γ ( n − α ) 1 Γ ( n − α )

C D α

RL I α

RL I α + 1 − n a +

n − α

, ℜ ( α ) ∈ N ∧ I ( α )̸= 0 . (A.54)

a + f ( x )= f ( x ) −

f ( a )( x − a )

a +

C D α

RL I α

RL I α + 1 − n a +

n − α

, ℜ ( α ) ∈ N ∧ I ( α )̸= 0 . (A.55)

a + f ( x )= f ( x ) −

f ( b )( b − x )

b −

n − 1 ∑ k = 0

f ( k ) ( a ) k !

RL I α

C D α

( x − a ) k .

a + f ( x )= f ( x ) −

(A.56)

a +

n − 1 ∑ k = 0

( − 1 ) k f ( k ) ( b ) k !

RL I α

C D α

( b − x ) k .

f ( x )= f ( x ) −

(A.57)

b −

b −

L I β

L I α + β +

L I β −

α + β −

L I α +

L I α −

+ f ( x )=

f ( x ) and

f ( x )= L I

f ( x )

(A.58)

L I β −

L D α +

L I α + f ( x )= f ( x ) and

L D α −

f ( x )= f ( x ) .

(A.59)