I was looking for a geometrical interpretations of fractional derivatives and fractional integrals.
I would be glad to see any kind of intuitive and preferably visual interpretation of the objects of fractional calculus.
Can anyone recommend the source or share personal opinion on the topic?
On
$$\left(\frac{d}{dx}\right)^{1/2} x^a = \frac{\Gamma(a+1)}{\Gamma(a+1/2)} x^{a-1/2}$$ and $$\left(\frac{d}{dx}\right)^{1/2} e^{ax} = \sqrt{a} e^{ax}$$ so it's look cool, but unfortunately it's not local : the distribution filter to get $\left(\frac{d}{dx}\right)^{1/2} f$ from $f$ is not point-wise as $\delta'(x)$ :
$$f'(x) = [f \ast \delta'](x)$$
$$\left(\frac{d}{dx}\right)^{1/2}f(x) = [f \ast d_{1/2}](x)$$ (where $\ast$ is the convolution) but $d_{1/2}(x)$ is not even supported on a finite interval. it means that from the Taylor series or the Fourier series of a function it's easy to calculate the $1/2$th derivatives, but from a simple function such as $1/\sin(x)$ it becomes much more difficult, and even impracticable.
Please see the following link
www.scielo.org.mx/pdf/rmf/v60n1/v60n1a6.pdf
about "A physical interpretation of fractional calculus" and,
vectron.mathem.pub.ro/dgds/v15/D15-ta.pdf
entitled "The geometric and physical interpretation of fractional order."