  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.
