CISM LECTURE NOTES International Centre for Mechanical Sciences

For convenience, we define the operator J α −∞ by
R.Gorenflo 279
J α −∞u(t) := 1
t
(t − τ)
Γ(α) −∞
α−1 u(τ) dτ , t ∈ IR , α > 0 . (1.6)
If u(t) isacausal function, i.e. u(t) =0for−∞ −1, t>0. We have, for α ≥ 0, the relations
J α t γ =
Γ(γ +1)
Γ(γ +1+α) tγ+α , D α t γ =
Γ(γ +1)
Γ(γ +1− α) tγ−α . (1.8)
For proofs consult [3], [4] or [8].
Note the remarkable fact that the fractional derivative Dαu is not zero for the
constant function u(t) ≡ 1ifα∈ IN . In fact, (1.8) with γ = 0 teaches us that
D α 1=
1
Γ(1 − α) t−α , α ≥ 0, t > 0. (1.9)
This, of course, is ≡ 0forα ∈ IN, due to the poles of the gamma function in the
points 0, −1, −2,.... Furthermore, we observe, again by looking at (1.8), that
D α t α−1 ≡ 0 for t>0, α > 0,
which implies that D α is not right-inverse to J α .Wehave
J α D α t α−1 ≡ 0, but D α J α t α−1 = t α−1
for t>0, α > 0.
These matters cause some problems in numerical treatment of fractional
differential and integral equations and require great care in analytical investigations.
Everything would be more coherent if we would consistently work with the
, m − 1 < α ≤ m, m ∈ IN, and with
operators J α −∞ and Dα −∞ = DmJ m−α
−∞
generalized functions in the sense of Gel’fand and Shilov [9] instead of functions,
these generalized functions vanishing for t