Fractal derivatives, fractional derivatives and $q$-deformed calculus

TL;DR

This paper analyzes fractional and fractal derivatives, highlighting their differences, similarities, and continuous approximations like $q$-calculus and Caputo derivatives, with implications for fractal systems and differential equations.

Contribution

It clarifies the relationship between fractal derivatives and fractional derivatives, introducing continuous approximations and their relevance to fractal geometry and differential equations.

Findings

$q$-calculus derivative approximates fractal derivatives

Caputo's derivative is proportional to a fractal derivative

Implications for fractional differential equations and fractal systems

Abstract

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the $q$-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo's derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.