Mathematical Physics Vol 1

Chapter 8. Introduction to the Fractional Calculus

448

8.4.2 Hilfer fractional derivative Natural question is if there exists an operator with an additional parameter that has two derivatives RL and Caputo as particular cases. The probably simplest construction of such an operator was suggested by Hilfer in [53]. The so-called generalized Riemann–Liouville fractional derivative (nowadays referred to as the Hilfer fractional derivative ) or order α , n − 1 < α ≤ n ∈ N , and type β , 0 ≤ β ≤ 1, is defined by the following composition of three operators: ( D α , β a f )( t )= RL I β ( n − α ) a d n d t n h RL I ( 1 − β )( n − α ) a f i ( t ) (8.79) In particular, if n = 1, the previous definition is equivalent with The fractional operator ( D α , β a f ) given by (8.80) was firstly introduced by Hilfer [53]. For β = 0, this operator is reduced to the Riemann–Liouville fractional derivative D α , 0 a ≡ RL D α a and the case β = 1 corresponds to the Caputo fractional derivative D α , 1 a ≡ C D α a . Also, one can observe the following properties of Hilfer fractional derivative: a) The Hilfer derivative ( D α , β a f ) , can be written as: where γ = α + β − αβ . The parameter γ satisfies 0 < γ ≤ 1, γ ≥ α , γ > β , 1 − γ < 1 − β ( 1 − α ) . The ( D α , b a f ) derivative is considered as an interpolator between the Riemann–Liouville and Caputo derivative since ( D α , b a f )= ( RL D α a f , β = 0 , C D α a f , β = 1 . (8.82) 8.4.3 Marchaud fractional derivative For a function f ∈ C 1 [ a , b ] , − ∞ < a < b < ∞ , the left Marchaud fractional derivative M D α a , t , 0 < α < 1, is defined and given as an equivalent form of the Riemann–Liouville derivative: ( D α , β a f )( t )= RL I β ( 1 − α ) a d d t RL I 1 − γ a f ( t )= RL I γ − α a D γ a f , ( t ) (8.81) ( D α , β a f )( t )= RL I β ( 1 − α ) a d d t h RL I ( 1 − β )( 1 − α ) a f i ( t ) . (8.80)

t Z a

α Γ ( 1 − α )

f ( t ) − f ( τ ) ( t − τ ) 1 + α

f ( t ) Γ ( 1 − α )( t − a ) α

α a , t f ( t )=

d τ , 0 < α < 1 .

M D

(8.83)

+

Also, the right Marchaud fractional derivative is defined as

b Z t

α Γ ( 1 − α )

f ( t ) − f ( τ ) ( τ − t ) 1 + α

f ( t ) Γ ( 1 − α )( b − t ) α

α b , t f ( t )=

d τ , 0 < α < 1 .

M D

(8.84)

+

For a more general class of functions f , the left Marchaud derivative is defined as a limit of restricted Marchaud derivatives, i.e the integrals in previous definitions are assumed to be convergent. Namely, M D α a , t f ( t )= f ( t ) Γ ( 1 − α )( t − a ) α + lim ε → 0 ψ ε ( t ) (8.85) where the function ψ ε ( t ) has to be defined separately for a < t < a + ε

t − ε Z a

f ( t ) − f ( τ ) ( t − τ ) 1 + α

ψ ε ( t )=

d τ , ε > 0 .

(8.86)