Mathematical Physics Vol 1

A.3 Fractional Derivatives and Integrals

459

A.3 Fractional Derivatives and Integrals

Let α ∈ C : ℜ ( α ) ∈ ( n − 1 , n ] , n ∈ N , and let [ a , b ] be a finite interval in R .

A.3.1 Definitions of some unidimensional fractional operators Riemann-Liouville Left-sided Integral RL I α a + f ( x )= 1 Γ ( α ) Z x a f ( u ) ( x − u ) 1 − α d u , x ≥ a .

(A.28)

Riemann-Liouville Right-sided Integral RL I α b − f ( x )= 1 Γ ( α ) Z b x Riemann-Liouville Left-sided Derivative

f ( u ) ( x − u ) 1 − α

d u , x ≤ b .

(A.29)

x Z a

d n d x n

1 Γ ( n − α )

f ( u ) ( x − u ) 1 − n + α

RL D α

nRL I n − α

a + f ( x )= D

a + f ( x )=

d u , x ≥ a .

(A.30)

Riemann-Liouville Right-sided Derivative

b Z x

d n d x n

1 Γ ( n − α )

f ( u ) ( u − x ) 1 − n + α

RL D α

f ( x )= D nRL I n − α b −

f ( x )=

d u , x ≤ b .

(A.31)

b −

Caputo Left-sided Derivative

x Z a

d n f ( u ) d u n

1 Γ ( n − α )

1 ( x − u ) 1 − n + α

c D α

RL I n − α

n f ( x )=

a + f ( x )=

a + D

d u , x ≥ a .

(A.32)

Caputo Right-sided Derivative

∞ Z x

d n d x n

( − 1 ) n Γ ( n − α )

f ( u ) d u ( u − x ) 1 − n + α

c D α −

f ( x )=( − D ) nL I n − α −

x < ∞ .

f ( x )=

(A.33)

,

Left-sided Finite-Difference

( − 1 ) k

α k

⌊ x − a

h ⌋

∑ k = 0

∆ α

h , a + f ( x )=

f ( x − kh ) , x ≥ a .

(A.34)

Right-sided Finite-Difference

( − 1 ) k

α k

⌊ b − x

h ⌋

∑ k = 0

∆ α

f ( x + kh ) , x ≤ b .

(A.35)

f ( x )=

h , b −

Grünwald-Letnikoff Left-sided Derivative GL D α a + f ( x )= lim h → 0 + ∆ α Grünwald-Letnikoff Right-sided Derivative

h , a + f ( x ) h α

x ≥ a .

(A.36)

,

∆ α

f ( x )

h , b −

GL D α

f ( x )= lim h → 0 +

x ≤ b .

,

b −

h α