CaputoD¶

CaputoD[f, {x, α}] gives the Caputo fractional differintegral of f. For a power it is a single Gamma ratio, Gamma[p + 1]/Gamma[p - α + 1] x^(p - α), and the operator is linear, so a polynomial goes term by term.

$ wo 'CaputoD[t^2, {t, 1/2}]'
(8*t^(3/2))/(3*Sqrt[Pi])

What sets Caputo apart from Riemann–Liouville is the constant: the function is differentiated ⌈α⌉ times before the rest of the order is integrated away, so a constant vanishes under any positive order.

$ wo 'CaputoD[1, {t, 1/2}]'
0

A whole order is the ordinary derivative, and a negative one integrates:

$ wo 'CaputoD[t^2, {t, 2}]'
2

$ wo 'CaputoD[t^2, {t, -1}]'
t^3/3

The order may be symbolic:

$ wo 'CaputoD[t^2, {t, alpha}]'
(2*t^(2 - alpha))/Gamma[3 - alpha]