Block-centered Difference Schemes For Two Kinds Of Fractional Differential Equations

Fractional partial differential equations are used to accurately describe anomalous diffusion phenomena.But unlike integral partial differential equations,most fractional partial differential equations need to obtain numerical solutions by constructing difference schemes due to the nonlocality of fractional operators.As an extension of classical fractional partial differential equations,solving tempered fractional partial differential equations will be more complex and challenging in form.Up to now,there are still many deficiencies and areas to be improved in the study of tempered fractional partial differential equations.Based on existing work,this thesis established differential schemes for two types of models using two types of tempered L1 formulas and fast L1 formulas,and completed theoretical analysis and numerical experiments.The first kind of model was tempered subdiffusion model with variable coefficients,and the tempered L1 block-centered difference schemes of the one-dimensional and two-dimensional tempered subdiffusion models were established.The second kind of model was Webster-Lokshin model.Block-centered difference scheme and fast difference scheme were established respectively.The main research work are as follows:(1)Block-centered difference for two-dimensional diffusion models with variable coefficient.On a non-uniform grid in the spatiotemporal direction,a second-order backward difference formula is used to discretize the time derivative and establish a block-centered difference scheme for diffusion model with variable coefficients.The stability and error estimation of the difference scheme were derived.Theoretical analysis is consistent with numerical experimental results,verifying the effectiveness of the difference scheme.(2)Block-centered difference for tempered subdiffusion models with variable coefficients.On a non-uniform grid in the spatiotemporal direction,the tempered implicit L1 formula and the tempered explicit-implicit L1 formula are used to discretize the tempered Caputo fractional derivative.The tempered implicit L1 block-centered difference schemes and tempered explicit-implicit L1 block-centered difference schemes for one and two dimensional cases are established respectively.Theoretical analysis of two types of tempered L1 block-centered difference schemes is provided.The consistency between numerical experiments and theoretical analysis demonstrates the rationality of the two types of difference schemes.(3)Fast difference of Webster-Lokshin model.In time direction,L1 formula and fast L1 formula are used to discretize Caputo fractional derivative,and in space direction,blockcentered difference is used to discrete integer derivative to establish block-centered difference scheme and fast difference scheme.The stability and error estimation of the two types of difference schemes are consistent with numerical experiments,indicating the feasibility of the difference schemes.