Diffusion on middle-$\xi$ Cantor sets

TL;DR

This paper develops a calculus framework for functions supported on middle-$\xi$ Cantor sets, enabling the solution of differential equations and analysis of diffusion processes on these fractal structures.

Contribution

It generalizes $C^{\zeta}$-calculus to middle-$\xi$ Cantor sets, allowing differential equations and diffusion behaviors to be studied on fractals with fractional dimensions.

Findings

Differential equations on middle-$\xi$ Cantor sets are solved.

Conditions for super-, normal, and sub-diffusion are established.

Illustrative examples demonstrate the calculus and diffusion analysis.

Abstract

In this paper, we study $C^{\zeta}$-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the $C^{\zeta}$-calculus on the generalized Cantor sets known as middle-$\xi$ Cantor sets. We have suggested a calculus on the middle-$\xi$ Cantor sets for different values of $\xi$ with $0<\xi<1$. Differential equations on the middle-$\xi$ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.