Riemann–Liouville and Caputo local fractional operators in Yang's calculus with applications to linear fractional differential equations

This paper developed Riemann–Liouville and Caputo-type operators within Yang's local fractional calculus on the formal fractional-number set $ \mathbb{R}^{r} $, where $ 0 < r\leq1 $. Here $ r $ is the local/fractal parameter associated with the underlying fractal geometry (for example, $ r = \log 2/\log 3 $ for the ternary Cantor set), and it is kept distinct from the independent operator order $ \alpha $. Starting from Yang's local fractional integral, derivative, and Laplace transform, we introduced left- and right-sided two-parameter operators of order $ (\alpha, r) $. We established power rules, transform formulas, semigroup and composition properties, and the relation between the Riemann–Liouville and Caputo versions. A generalized local Mittag–Leffler function associated with the local gamma function was introduced, leading to distinct and mathematically consistent solution kernels for Riemann–Liouville and Caputo initial-value problems. Several linear examples were presented together with Cantor-supported graphical illustrations for different values of $ r $ and $ \alpha $, including the smooth classical limit $ r = 1 $. When $ \alpha = 1 $, the proposed construction reduces to Yang's basic local fractional operator.