The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes

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 Mittag–Leffler function using finite Hankel integral transformation and Laplace transformation. In addition, numerical simulations are discussed. The results show that the effect range of time-fractional derivative νν on probability density is greater than that of fractional oscillator parameter ββ. The effect range of νν on probability density is opposite to that of ββ. This paper provides a new analytical tool to develop fluid mechanics, heat conduction and other engineering science.