Anomalous Diffusion On A Grid Comb-like Structure And Modelling Of The Distribution Of Particle Trajectory And Its Functionals

Nowadays, comb models and distributed-order diffusion equations have many ap-plications, we can get many interesting results from these equations as well. This paper is composed of five chapters. In the first chapter, we mainly give a brief overview of comb models and distributed-order diffusion equations. Some fundamental results are also illustrated. The second chapter of this paper mainly discusses fractional diffusion on a grid comb-like structure. In this chapter we analyse time fractional diffusion along the teeth. And the ways of diffusion along the backbone we choose time fractional or-der, temper time fractional order, space fractional order. We also show the relationship between the comb model and continuous time random walk model (CTRW) in the final part of this chapter. By using CTRW, we can get some properties of comb-like structure. In the third chapter of this paper, we mainly study distributed-order diffusion equations. In this chapter, we use stochastic representation to derive distributed-order temper time fractional diffusion equation. Then we also study some properties of this equation. In the next chapter we also derive distributed-order time fractional Feynman-Kac equations. With the help of these equations we study some properties as well. In the last chapter we give the summary and the defect of this paper.