What are some genuine ways to define the derivative of a fractal?

Question

Seeing the success of applying measure theory to generalize integration to fractals, I wonder whether or not there is a method to generalize the derivative to a fractal. Most courses start off fractal courses mentioning that derivatives aren't defined on fractals, but that seems odd considering that the next thing they do is define objects with fractional dimensionality.

I'd like references to derivatives that generalize to fractals. Referencing nebulous relations to "fractional" calculus and papers that don't give numerical or non-theoretical methods to determine values for derivatives on actual fractals won't suffice.

I'll also accept papers that just generalize the derivative to generally considered "now-where" differentials functions.

For reference: I know of the weak derivative, it really doesn't apply to fractals. Fractional calculus is non-local so that doesn't work either. I've tried defining concepts of average derivative, but that just leads back to the integral (still working though). "Fractal" derivative on Wikipedia is just a change of variables with no actual interpretation so definitely a no on that one. Perhaps one can skip a derivative and just define the second derivative on fractals?

Or do something similar to the arithmetic derivative and define it as ${\text df(x)\over \text dx}=f(x)g(x)$ for some pre-defined function $g(x)$... — abiessu, Apr 25 '15 at 00:50
@abiessu of course then you'd have to define g(x) so it satisfies properties at least reminiscent of a typical derivative. Perhaps g(x) could be some kind of analog to measure. Although this is very dissimilar to the arithmetic derivative sense it needs to work over the reals. — Zach466920, Apr 25 '15 at 00:59
Maybe something listed at Is there a garden of derivatives? is what you want? See also my answer to Most general $A \subseteq \mathbb R$ to define derivative of $f: A \to \mathbb R$?. — Dave L. Renfro, Apr 27 '15 at 20:04
@DaveL.Renfro Thanks for the references (I have a lot of reading to do!). If this derivative exists, it'll definitely be among at least one of those references. — Zach466920, Apr 27 '15 at 20:19

user48672 · Accepted Answer · 2015-08-05T15:07:29.470

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Derivatives on fractals sets was considered by several authors:

Jiang, H. & Su, W. Some fundamental results of calculus on fractal sets Communications in Nonlinear Science and Numerical Simulation , 1998, 3, 22 - 26;
Parvate, A. & Gangal, A. Fractal differential equations and fractal-time dynamical systems Pramana J Phys, Springer India, 2005, 64, 389 - 409;
Parvate, A. & Gangal, A. D. Calculus on fractal subsets of real line - I: formulation, Arxiv math-ph/0310047;
Bongiorno, D & Corrao G, On the fundamental theorem of calculus for fractal sets, Fractals, 2015
Escribano C, Reyes M, Analysis of Functions Defined on Fractal Sets, - Fractals, 1999

edited Aug 05 '15 at 15:07

answered Aug 02 '15 at 19:33

user48672

1,132

I'm not on a PC, are these public access? – Zach466920 Aug 02 '15 at 19:41
The Arxiv reference is. For the others I am not sure. – user48672 Aug 02 '15 at 20:55
well, something's better than nothing :) – Zach466920 Aug 02 '15 at 21:16
The reference of Bongiorno can be accessed from Google Scholar. – user48672 Aug 05 '15 at 15:11
That last one is very good :) – Zach466920 Aug 05 '15 at 15:30
Do you have original research in that direction? – user48672 Aug 05 '15 at 16:05
Nope, just really curious. I've been trying to look deeper into the subject, to see how to get an intuitive notion of the derivative. – Zach466920 Aug 05 '15 at 17:03
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I am studying the so-called alpha derivative or local fractipnal derivative. It seems to be able to handle fractqls. – user48672 Aug 05 '15 at 19:32

What are some genuine ways to define the derivative of a fractal?

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