What's your intuitive understanding of fractional derivatives?

One way to think of it is analytic continuation.

Consider squares of natural numbers, you have

  1-> 1^2 = 1
  2-> 2^2 = 4
  3-> 3^2 = 9

This is some relation from all natural numbers to some (infinite) subset of N which are called perfect squares, where the relationship is specified by y = x^2 where x is the input and y is the perfect square. You can easily generalize the relationship (function) here by instead noting that another relation, y = x^2, but where x from the set of real numbers and the relation maps all real numbers to another subset of the real numbers (it maps all to all positive real numbers actually). This new relationship is a "more general" relationship because the original relationship from integers to perfect squares is a subset of this new map, or rather, a restriction of the domain to integers gives the old relationship.

You can always make generalizations of a given concept in mathematics, and fractional derivatives are an example. Here, the order of differentiation a is identified in standard calculus by an integer, but you can invent a new definition "generalizes" the idea of a derivative by defining new operations that when a is restricted to an integer, yield the original operation. There are a handful of methods people utilize in such situations, for an example, converting a factorial to a gamma function is a common tactic used in generalizing certain types of functions (in fact, the gamma function itself is one of the best examples of analytic continuation).

The fact that the wiki article later points out many definitions exist is due to the fact the only real requirement is that their restriction to integer orders reduce to the original derivatives, hence the possibility of many different distinct generalizations (it's important to realize a mathematical generalization is different in nature to "generalizations" in vernacular speech, usually generalizations are more abstract, and thus are a larger encompassing group but are fewer in number, but it's not like inheritance in Java or something). Anyway, this explains what they are and they do seem rather abstract without a direct physical intuition about them, but they like many things that seem like just play time for mathematicians end up having real applications in actual PDEs in physics and elsewhere.

I like the treatment in this book

https://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp...

Definitely getting Frequency Illusion here [1]. This video [2] came up in my recommendations a few days ago and now it's here. The video does have some good visualizations though.

First time I had ever considered fractional derivatives.

[1] https://en.m.wikipedia.org/wiki/Frequency_illusion

[2] https://youtu.be/2dwQUUDt5Is