Mathematical Physics Vol 1

8.4 Some other types of fractional derivatives

447

and

d n d t n  

( τ ) d τ 

∞ Z t

d n d t n t

f ( t ) =

( − 1 ) n Γ ( n − α )

D − ( n − α ) + ∞

α + ∞ f ( t )=

LW D α −

α − 1 f

( τ − t ) n −

f ( t )=( − 1 ) n

 ,

t D

(8.69) respectively, where t ∈ R and n =[ α ]+ 1, α ≥ 0. In particular, when α = n ∈ N 0 , then − ∞ D 0 t f ( t )= t D 0 + ∞ f ( t )= f ( t ) , (8.70) − ∞ D n t f ( t )= f ( n ) ( t ) and t D n + ∞ f ( t )=( − 1 ) n f ( n ) ( t ) (8.71) where f ( n ) ( t ) denotes the classical (integer) derivative of f ( t ) of order n . If 0 < α < 1 and t ∈ R , then

d d t  

( τ ) d τ 

∞ Z 0

t Z − ∞

α Γ ( 1 − α )

f ( t ) − f ( t − τ ) τ α + 1

1 Γ ( n − α )

D α

α − 1 f

( t − τ ) n −

d τ ,

 =

t f ( t )=

− ∞

(8.72)

and

d d t  

( τ ) d τ 

∞ Z t

∞ Z 0

α Γ ( 1 − α )

f ( t ) − f ( t + τ ) τ α + 1

1 Γ ( n − α )

α − 1 f

n + ∞ f ( t )= −

( τ − t ) n −

d τ .

 =

t D

(8.73) For the Liouville-Weyl fractional integrals we can also state the corresponding semigroup property for α ≥ 0, β ≥ 0 LW I α + f ( t ) · LW I β + f ( t )= LW I α + β + f ( t ) (8.74) and LW I α − f ( t ) · LW I β − f ( t )= LW I α + β − f ( t ) , (8.75) where, for complementation, we have defined LW I 0 + = LW I 0 − = I ( identity operator ) and LW D 0 + = LW D 0 − = I . In fact, we easily recognize the fundamental property LW D α + · LW I α + =( − 1 ) nLW D α − · LW I α − = I . (8.76) Because of the unbounded intervals of integration, fractional integrals and derivatives of Liouville Weyl type can be successfully handled via the Fourier transform and the related theory of pseudo-differential operators, that simplifies their treatment. We assume that the integrals in their definitions are in a proper sense, in order to ensure that the resulting functions can be Fourier transformable in the ordinary or generalized sense. Let

+ ∞ Z

e − i ω t f

ˆ F ( ω )=

( t ) d t , ω ∈ R

(8.77)

− ∞

be the Fourier transform of a function f ( t ) of real variable t ∈ R . Let f ( t ) be defined on ( − ∞ , + ∞ ) and 0 < α < 1. Then the Fourier transform of the Liouville-Weyl integral and differential operator satisfies ˆ F − ∞ D − α t f ( t ) =( i ω ) − α ˆ F ( ω ) , ˆ F − ∞ D + α t f ( t ) =( i ω ) + α ˆ F ( ω ) , ˆ F tD − α ∞ f ( t ) =( − i ω ) − α ˆ F ( ω ) , ˆ F t D + α ∞ f ( t ) =( i ω ) + α ˆ F ( ω ) (8.78)