High-Accuracy Algorithms For Fractional Nonlinear Reaction-Diffusion Equations

This article is mainly divided into three parts.In the first part,we aim at the initialboundary value problem of one-dimensional two-side space fractional nonlinear reactiondiffusion equation.The temporal partial derivative is discretized by implicit midpoint formula,the Riemann-Liouville spatial fractional partial derivatives are approximated by quasi-compact difference operators,thus,a class of new high-accuracy difference scheme is constructed.The stability and convergence of the proposed scheme are proved by energy method.Numerical experiments demonstrate that the proposed numerical method is effective.In the second part,we devote into the initial-boundary value problem of twodimensional Riesz space fractional nonlinear reaction-diffusion equation.The temporal partial derivative is discretized by the second-order difference formula(abbr.BDF2),the Riesz spatial fractional partial derivatives are approximated by fourth-order compact difference operators,thereby,a class of new high-accuracy alternating-direction implicit(abbr.ADI)scheme is constructed.The stability and convergence analysis results of numerical method are obtained.Numerical results verify the effectiveness of the proposed numerical scheme.In the third part,we aim at the initial-boundary value problem of Riesz tempered fractional nonlinear reaction-diffusion equation.The temporal partial derivative is discretized by implicit midpoint formula,the Riesz tempered fractional derivative is evaluated by the modified second-order Lubich tempered difference operator,thus,a class of new finite difference method is constructed.The analysis of stability and convergence of the proposed numerical scheme are discussed by energy method.Numerical results show that the proposed method is effective.