Diffusion on middle-$\xi$ Cantor sets

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.