Fractional Calculus - Variable order derivatives and integrals

[s00mb]

TL;DR

Looking for papers and research

Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the first term is F to the 1/2 derivative in fractional calculus and the second term is to an unknown power of derivative and the last term is a 1/3 fractional derivative. I am curious if anyone knows of any work on this kind of topic? What would be the initial conditions needed to solve such an equation?

 

[jedishrfu]

First there’s Wikipedia:

https://en.wikipedia.org/wiki/Fractional_calculus

from that article you can find many additional named theorems to investigate as its a much bigger field than you may think.

and this paper from Reed College Physics Dept:

https://www.reed.edu/physics/facult...ractional Calculus/A. Fractional Calculus.pdf

Heres a brief introduction to it: