Question

Riemann-Liouville fractional derivative is a nonlocal fractional derivative that doesn't vanish in general on differentiable functions. Kolwankar-Gangal fractional derivative is local but vanishes on any differentiable function. Is there some local fractional derivative that doesn't vanish on differentiable functions in general and for which $$ D^{\alpha} x^{n \alpha} = \frac{\Gamma(n\alpha+1)}{\Gamma((n-1)\alpha+1)} x^{(n-1)\alpha} $$ holds for any $x > 0$?

Answer 1

By the theorem of Peetre, a linear operator $C^\infty(\mathbb R)\to C^\infty(\mathbb R)$ which is local (=support non-increasing) is a differential operator (so involves only integer derivatives).

Answer 2

I'm not sure, but maybe you could investigate the Yang local fractional derivative?

Answer 3

Not a translation-invariant one!

Indeed, we would have:

\[ \frac{ \Gamma( 2) } {\Gamma(\frac{3}{2}) } \sqrt{x+1} = D^{1/2} (x+1)= D^{1/2}x + D^{1/2} 1 = \frac{ \Gamma( 2) } {\Gamma(\frac{3}{2}) } \sqrt{x} +\frac{\Gamma(1)} {\Gamma(\frac{1}{2} ) } \frac{1}{\sqrt{x}} \]

which is obviously false.