Modelling And Computational Methods For Anomalous And Tempered Anomalous Dynamics

Anomalous diffusion widely appears in nature;anomalous diffusion has been widely used in physics,chemistry,biology,engineering and so on.Fractional calculus and fractional differential equations are popular in describing anomalous diffusion.The establishment of the model of anomalous and tempered anomalous dynamics are helpful for us to understand the phenomenon of anomalous diffusion.The main research contents of the study are as follows.Firstly,the diffusion process is studied by the microscopic and macroscopic models of the anomalous and tempered anomalous dynamics,and the microscopic aspects of the research object are the random walk model,specifically including,continuous time random walk(CTRW),Lévy walk,and Lévy flight.Based on these models,we discuss the statistical properties of the random walk trajectory of the particles in detail,and the numerical simulation is carried out,and an algorithm for generating efficient random variables is proposed.The macroscopic aspect of the study is that the particle satisfies the deterministic equation of power law distribution,namely,the time-dependent fractional partial differential equation(PDE).Secondly,the numerical calculation method of the time dynamic evolution equation is discussed.We dig out the potential of the short memory principle of fractional operators and apply it felicitously to reduce the computation cost by using the idea of equidistributing meshes for numerically solving the initial value problem.The equidistributing predictor-corrector approach is proposed.The algorithms are described in detail by pseudo codes.Numerical experiment results confirm that the designed numerical schemes have linearly increasing computation cost with time but not losing the accuracy at the same time.The algorithm is able to efficiently solve the time dependent fractional PDEs.Then,we introduce the techniques of equidistributing the meshes and present the detailed numerical schemes.Error estimates for the proposed schemes are performed;and the numerical examples demonstrate the efficiency of our algorithms.Thirdly,based on the research of the second part,we introduce the tempered fractional order operator and dig out the potential of the short memory principle of tempered fractional operators and apply it felicitously to reduce the computation cost by using the idea of equidistributing meshes for numerically solving the initial value problem with the predictor-corrector approach.This chapter provides the predictor-corrector approach with theoretically proved convergence order for the tempered fractional differential equation.The numerical results show that the computation precision is not lost,and the computation cost is reduced.The larger the time is,the more benefits the equidistributing methods obtain.Fourthly,the comparison principle of the tempered fractional order derivative is derived.Using Lyapunov direct method to study the tempered fractional order system and extending the Lyapunov direct method,the stability of the Mittag-Leffler is derived,the stability of the actual system is more convenient to verify by the method proposed in this chapter.