Numerical Methods Of Several Kinds Of Time Fractional Differential Equations

In recent years,fractional calculus has attracted the attention of researchers because of its wide application in simulating the phenomena with memory and heredity.The numerical solution of fractional differential equations has also become one of the hot topics.Fractional differential operators usually contain kernel functions with weak singularity.The influence of singularity should be considered in the construction and analysis of numerical methods.On the other hand,because the fractional differential equation is globally related,its computational complexity is high.Fast and stable calculation method is also an important research content.Based on the above two aspects,this thesis studies the numerical methods of four kinds of fractional differential equations.The full text is divided into five chapters.The first chapter introduces the historical background of fractional calculus,the research status of numerical solution of fractional differential equations at home and abroad,and the four models studied.In Chapter 2,the fourth-order multi-term fractional equation is studied.Based on the order reduction method,a fast compact difference scheme is proposed.The unique existence,stability and convergence of the numerical scheme are proved by energy method.Numerical experiments verify the theoretical estimation results and the effectiveness of numerical meth-ods.In Chapter 3,the time fractional Kd V-Burgers’equation is studied.Use2-1_σformula ap-proximates the Caputo fractional derivative to obtain a stable second-order temporal accuracy difference scheme.In order to improve the spatial accuracy,a second-order spatial accuracy difference scheme is established based on the weighted operator.The theoretical estimation accuracy is consistent with the numerical experimental results.In Chapter 4,the two-dimensional semilinear fractional sub-diffusion equation is studied.Firstly,a nonlinear difference scheme with second-order accuracy in time and space is estab-lished.In order to improve the efficiency of computation,the coarse grid nonlinear system and fine grid linear system are established by using the two grid technology.Based on the energy method,the stability and convergence results of the numerical method are obtained.The numerical results show the effectiveness of the two grid difference scheme.In Chapter 5,the fractional predator-prey model is studied.A couple of coupled integro-differential equations are obtained by integral transformation technique.Based on the vari-ational principle,the variational iterative scheme is established,and the convergence of the scheme is proved.Finally,the numerical results verify the effectiveness of the scheme.