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 $x > 0$?

  • $\begingroup$ I'm not sure, but maybe you could investigate the Yang local fractional derivative? $\endgroup$ – Petern Jan 25 '13 at 17:04

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).


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.

  • $\begingroup$ Does not exist. $\endgroup$ – Dimiter P May 24 '16 at 20:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.