In fractional calculus, the Caputo derivative of a monomial has the following form:
$$\operatorname{\mathit D}_t^\alpha\,t^\beta = \frac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}t^{\beta-\alpha}$$
I wish to compute the Caputo derivative of $x(1+t^2)$ with respect to $t$.
I tried the following code:
β = 2;
u[x_, t_] = x*(t^0 +t^β);
u[x, t] /. {x -> x, t^0 -> t^α/Gamma[1 - α],
t^β ->
Gamma[β + 1]/
Gamma[β - α + 1] t^(β - α)}
and obtain the following output:
But I think this code is not correct. Any suggestion?