Study Of Anomalous Diffusion In Comb Model And Heat Conduction Based On Fractional Constitutive Model

Anomalous diffusion and heat conduction are important research fields and have wide applications in science and engineering.The study of anomalous diffusion in this paper is based on the comb model.It is a special case of random walk and its special behavior is that the particle transport along the x direction is possible only along the backbone.The transport along the y direction is perpendicular to the backbone.It has broad application prospects which are related to simulate the transport of cancer cells,the diffusion in percolation clusters,the transport along spiny dendrites,the research of quantum mechanics,and so on.It attracts many scholars to study.The Fick's law and Fourier's law are the the basic laws to study heat and mass transfer problem.Both the constitutive models satisfy linear relation.However,they correspond to the infinite propagation velocity,which contradicts with the principle of causality.In this paper,we modify the classical constitutive models from two aspects,one is the introduction of relaxation parameter,which overcomes the disadvantages of classical constitutive models and makes the governing equation have both the characteristics of parabolic and hyperbolic.The other is the introduction of fractional order operator,that the governing equation changes from local differential form to nonlocal integral one and makes transport process have a memory characteristic and a nonlocal characteristic.This paper applies the modified constitutive models to the study of the anomalous diffusion in comb model and heat conduction.It mainly contains two parts,one is to apply the time and space fractional Fick's model,the time fractional Cattaneo model,the one-dimensionl and two-dimensional fractional Cattaneo-Christov model to the anomalous diffusion in comb model.The other is to apply the one-dimensionl fractional Cattaneo-Christov model to the study of heat conduction model.The solution of governing equation is obtained by the analytical method and numerical method.Analytic method is using the integral transform method,including Laplace transform method and Fourier transform method.Numerical method is using the numerical difference method,the time fractional order is discreted by L1-approximation and L2-approximation while the space fractional derivative is discretized by shifted Grunwald definition.The paper is mainly using the figures which is drawing from the solutions to analyze the influences of different parameters on the particle or temperature distribution,the particles numbers and mean square displacement on the x axis.The detailed analysis and discussion on physical characteristics are given.