Fox H functions in fractional diffusion - FRActional CALculus ...

326 F. Mainardi et al. / Journal of Computational and Applied Mathematics 178 (2005) 321–331
For = 0 we have a symmetric operator with respect to x, that can be interpreted as
xD 0 =−
− d2
dx2 /2
. (3.6)
This can be formally deduced by writing −|| =−( 2 ) /2 . For 0 < < 2 and || min {, 2 − }, the
Riesz–Feller derivative can be shown to admit the integral representation in the x domain,
xD
(1 + )
∞ f(x + ) − f(x)
f(x) = sin [( + )/2]
0 1+
d
∞
+ sin[( − )/2]
d . (3.7)
0
f(x − ) − f(x)
1+
By denoting the Laplace transform of a sufficiently well-behaved (generalized) function f(t), f(s) =
L {f(t); s}= ∞
0 e −st f(t) dt,R(s)>af , the Caputo time-fractional derivative of order (m−1 < m,
m ∈ N) turns out to be defined through
L{ tD ∗ f(t); s}=s m−1
f(s) −
k=0
s −1−k f (k) (0 + ), m − 1 < m. (3.8)
This leads to define, see e.g. [7,31],
tD ⎧
⎪⎨
1 t f
∗ f(t) := (m − ) 0
⎪⎩
(m) () d
,
+1−m (t − )
d
m−1 <