Mathematical Physics Vol 1

Chapter 8. Introduction to the Fractional Calculus

446

Laplace transform is one of the major formal tools of science and engineering, especially when modeling dynamical systems. Also, Laplace transform is also usually used for solving fractional integro-differential equations involved in various engineering problems. The Laplace transform L {·} of the RL fractional integral (8.32) of f ( t ) is L { RL I α 0 , t f ( t ) } = 1 s α F ( s ) . (8.63) Laplace transform of the RL fractional derivative is The terms appearing in the sum on the right hand side of the expression (8.64) involve the initial conditions and these conditions must be specified when solving fractional differential equations. Laplace transform of Caputo fractional derivative is ∞ Z 0 e − st C D α 0 , t f ( t ) d t = s α F ( s ) − n − 1 ∑ k = 0 s α − k − 1 f ( k ) ( 0 ) , n − 1 < α < n (8.65) which implies that all the initial conditions required by a fractional differential equation are presented by a set of only classical integer-order derivatives. Note also that the assumption of zero initial conditions is perfectly sensible when implementing fractional order controllers and filters. However, when attempting to simulate a fractional order system, the effect of initial conditions must be taken into consideration. In such a case, also, the difference between various definitions of fractional operators cannot be neglected. Besides that, the geometric and physical interpretations of fractional integration and fractional differentiation can be found in Podlubny’s work,[52]. The fractional integrals and derivatives, defined on a finite interval [ a , b ] of R , are naturally extended to whole axis R . Namely, we can also define the fractional integrals over unbounded intervals and, as left inverses, the corresponding fractional derivatives. L { RL D α 0 , t f ( t ) } = ∞ Z 0 e − st RL D α 0 , t f ( t ) d t = s α F ( s ) − n − 1 ∑ k = 0 RL D α − k − 1 0 , t f ( t ) | t = 0 . (8.64)

8.4 Some other types of fractional derivatives The left and right Liouville-Weyl fractional integrals − ∞ D − α t on the whole axis R are defined by

f ( t ) and t D − α

+ ∞ f ( t ) of order α > 0

t Z − ∞

1 Γ ( α )

D − α t

LW I α

α − 1 f

( t − τ )

( τ ) d τ ,

f ( t )=

+ f ( t )=

(8.66)

− ∞

and

∞ Z t

1 Γ ( α )

α + ∞ f ( t )=

LW I α −

α − 1 f

( τ − t )

( τ ) d τ ,

t D −

f ( t )=

(8.67)

respectively, where t ∈ R and α > 0. 8.4.1 Left and right Liouville-Weyl fractional derivatives on the real axis The left and right Liouville-Weyl fractional derivatives − ∞ D α t f ( t ) and t D α

+ ∞ f ( t ) of order α

on the whole axis R are defined by

d n d t n  

( τ ) d τ 

t Z − ∞

d n d t n − ∞

f ( t ) =

1 Γ ( n − α )

D − ( n − α ) t

D α

LW D α

α − 1 f

( t − τ ) n −

 , (8.68)

t f ( t )=

+ f ( t )=

− ∞