Macroscopic and microscopic anomalous diffusion in comb model with fractional dual-phase-lag model

A novel constitutive equation which considers the macroscopic and microscopic relaxation characteristics and the memory and nonlocal characteristics is proposed to describe the anomalous diffusion in comb model. Formulated governing equation with the fractional derivative of order 1 + α corresponds to a diffusion-wave one and solutions are obtained analytically with the Laplace and Fourier transforms. As the solutions show, the existence of macroscopic relaxation parameter makes the expression of mean square displacement contain an integral form and the specific value for the microscopic relaxation parameter and macroscopic one changes the coefficient of fractional integral. The particle distribution and mean square displacement of Fick's model and the dual-phase-lag model are same at the short and long time behaviors and the special case of equal macroscopic and microscopic relaxation parameters. The particle distributions and mean square displacement with the effects of different parameters are presented graphically. Results show that the wave characteristic becomes stronger for a larger α, a larger τq or a smaller τP. For mean square displacement, the magnitude is larger at the short time behavior and smaller at the long time behavior for a smaller α. Besides, for a smaller τq or a larger τP, the magnitude is larger.