Anomalous diffusion in comb model with fractional dual-phase-lag constitutive relation

A fractional dual-phase-lag constitutive relation is proposed to describe the anomalous diffusion in comb model. A novel governing equation with the Dirac delta function is formulated and the highest order is 1 + alpha which corresponds to a diffusion-wave equation. Solutions are obtained analytically with Laplace and Fourier transforms. Dynamic characteristics for the spatial and temporal evolution of particle distribution and the mean square displacement versus time with the effects of different parameters such as the fractional parameters and the relaxation parameters are analyzed and discussed in detail. Results show that the wave characteristic becomes stronger for a larger fractional parameter, a smaller microscopic relaxation parameter or a larger macroscopic one. For a larger alpha, a smaller beta, a larger macroscopic relaxation parameter or a smaller microscopic one, a novel oscillating distribution versus time is presented, and at this condition, the magnitude of mean square displacement is larger at the smaller time while larger at the larger time. Most important of all, the anomalous diffusion in comb model with a diffusion-wave equation corresponds to a subdiffusion behavior because of its special structure. (C) 2018 Elsevier Ltd. All rights reserved.