In the study, I attempt to set intuition and describe mathematical foundations for this intriguing paradigm of calculus, called fractional calculus, and apply it to various common functions.
Transition across rational derivative orders for \( x^3 \)
1
Phase space evolution plotting \((f(x), D^\pi f(x))\) for \(f(x) = \sin(x)\)
1
Evolution of \(D^\alpha x^3\) over range of orders \(\alpha \in [0.5, \pi]\).
Memory contributions with weighting kernel \(1 / (x - t)^{1-\alpha}\) for \(\alpha = \pi\)