A generalized non-local Fick’s law on fractal-dimension is derived. Using modified Fick’s law a time-space fractional diffusion model with a fractional oscillator term is built. The solution is obtained in terms of a Mittag-Leffler function using a finite Hankel integral transformation and Laplace transformation. In addition, numerical simulations are discussed. The results show that the effect range of the time-fractional derivative
on the probability density is greater than that of the fractional oscillator parameter
. The effect range of
on a probability density is opposite to that of
. This paper provides a new analytical tool to develop fluid mechanics, heat conduction and other engineering sciences.