Fractional derivatives (a derivative with order $\alpha$ such as $d^{\alpha}/dt^{\alpha}$ where $\alpha$ is a real or complex value) are used in many fields of science and in particular in modeling the behavior of visco-elastic materials i.e defining stress ($\sigma$) - strain ($\varepsilon$) relations.
Indeed, a fractional derivative provides an additional degree of freedom without adding a new element to simple conventional visco-elastic models like e.g. the ones by Maxwell, Kelvin & Voigt or Zener.

$\sigma = E \varepsilon$

$\sigma = \eta d\varepsilon/dt$

$1/E \times d\sigma/dt + \sigma/\eta = d\varepsilon/dt$

$\sigma = E\varepsilon + \eta d\varepsilon/dt$

$\sigma + \eta/E_2 \times d\sigma/dt$

$= E_1\varepsilon + \eta(E_1+E_2)/E_2 \times d\varepsilon/dt$

In a thread following a tweet by Gabriel Peyré, I discovered **an original way of using** (and further familiarize oneself with) **fractional derivatives**, illlustrated by Dillon Berger (and further explained by Gen Kuroki - in Japanese) **: to smooth** the **animation** between powers of the Taylor series of a function.

Compare the two graphs on this page to see the effect.

As an example, the Taylor series expansion of $\sin(x)$, around $x=0$ (a.k.a. the MacLaurin series), is :

Below, use the + and - buttons to increase or decrease the power expansion for $\sin(x)$ Taylor series. Note that the terms with even numbers for the powers of $x$ are not used in the equation above and thus they change nothing to the approximation of $\sin(x)$ :

$\sin(x)$

$\simeq$

$x$

Below is the animation for $\sin(x)$ with a step of 0.2 for $\nu$ between two plots. The value of the maximum power is

The Gamma function implemented in this page is an approximation proposed by Gergo Nemes (see the wikipedia page on Stirling's approximation). It was reported as a simpler and faster but not as much accurate as Lanczos approximation by Peter Olson in his answer to a stackoverflow question on the Gamma function implementation in Javascript.

The content of this page is copyleft under :

the creative commons license Attribution 3.0 Unported (CC BY 3.0).

Luc Jaouen (@ljaouen), ISSN 2606-4138.

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