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A kind of anomalous diffusion and heat transfer on a comb structure with anisotropic relaxation are studied, which can be used to model many problems in Biologic and Nature in fractal porous media. The Hausdorff derivative is introduced and new governing equations is formulated in view of fractal dimension. Numerical solutions are obtained and the Fox H-function analytical solutions is given for special cases. The particles spatial-temporal evolution (STE) and the mean square displacement (MSD) versus time are presented. The effects of backbone and finger relaxation parameters, and the time fractal parameter are discussed. Results show that the MSD decreases with the increase of backbone parameter or the decrease of finger relaxation parameter in a short of time, but they have little effect on MSD in a long period. Particularly, the MSD has time dependence in the form of /2 t ( 0 1    ) when t >τ, which indicates that the diffusion is an anomalous sub-diffusion and heat transfer.

Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media