Fractional Calculus - Variable order derivatives and integrals

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SUMMARY

The discussion focuses on fractional calculus, specifically variable order derivatives and integrals, highlighting the equation F'1/2(u) + F'x(u) = F'1/3(u). Participants seek research on solving integral and differential equations where the unknown is the order of the derivative or integral. Key resources mentioned include a Wikipedia article on fractional calculus and a paper from Reed College's Physics Department, which provide foundational knowledge and further references in this expansive field.

PREREQUISITES
  • Understanding of fractional calculus concepts
  • Familiarity with differential and integral equations
  • Knowledge of variable order derivatives
  • Basic mathematical analysis skills
NEXT STEPS
  • Research the various named theorems in fractional calculus
  • Study the implications of initial conditions in fractional differential equations
  • Explore advanced applications of fractional calculus in physics
  • Examine numerical methods for solving fractional differential equations
USEFUL FOR

Mathematicians, physicists, and engineers interested in advanced calculus, particularly those working with fractional derivatives and integrals in theoretical and applied contexts.

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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?
 
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