Research On Fast Numerical Algorithms For Some Types Of Fractional Differential Equations

Fractional differential equations,as a generalization of integer-order differential equations,have been widely used in modeling various physical and scientific phenomena in recent years.Due to the non-local characteristics of fractional differential operators,fractional differential models can describe materials and processes with hereditary and memory properties more accurately.In most cases,it is not easy to obtain the analytical solutions of fractional differential equations.Therefore,numerical methods have become common research methods.In addition,the discretizations of fractional operators usually generate large dense matrices,which lead to significant computational difficulties.Therefore,the development of corresponding high-performance algorithms is also very urgent.The main work and contributions are as follows:Firstly,we propose a fast implicit difference scheme to solve the time distributedorder and space fractional diffusion equations with variable coefficients.By performing numerical integration,the time distributed-order fractional diffusion equations are transformed into multi-term time-space fractional diffusion equations.Then,an implicit difference scheme is proposed to solve the multi-term time-space equations,and its convergence and stability are discussed.In addition,a preconditioned Krylov subspace method is constructed to solve the derived Toeplitz-like linear system.Numerical experiments are presented to confirm the theoretical findings and show that the proposed fast algorithm is more attractive than other testing methods in terms of computational cost.Secondly,we establish a fast second-order implicit difference scheme to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations.We derive the difference scheme by the fractional centered difference formula in space and the weighted and shifted Gr ¨unwald formula in time.The unconditional stability and second-order convergence of this scheme in time,space and distributed-order are proved.In the one-dimensional case,the preconditioned Krylov subspace method based on the Gohberg-Semencul formula is developed to solve the resulting Toeplitz linear system.In the two-dimensional case,a global preconditioned conjugate gradient method with a truncated preconditioner is constructed to solve the corresponding Sylvester matrix system.Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference scheme and fast solution algorithms.Thirdly,we develop a fast implicit integration factor method to solve a class of nonlinear Riesz space fractional reaction-diffusion equations.After spatially discretizing the fractional reaction-diffusion equations by the fractional centered difference formula,a nonlinear ordinary differential equation system is obtained.In order to obtain good stability and robustness,we employ the implicit integration factor scheme to solve the nonlinear system.To deal with the extremely expensive cost in the implementation of this scheme,we propose the shift-invert Lanczos method based on the Gohberg-Semencul formula to compute the exponential matrix-vector multiplication,by using the particilar property that the coefficient matrix is symmetric positive definite Toeplitz.Some numerical examples are performed to verify the correctness of the theoretical results and demonstrate the efficiency of our fast solution algorithm.Fourthly,we propose a fast compact implicit integration factor method with nonuniform time meshes to solve two-dimensional nonlinear Riesz space fractional reactiondiffusion equations.The weighted and shifted Gr ¨uwald-Letnikov approximation is employed to the spatial discretization of the equations,and a system of nonlinear ordinary differential equations in matrix form is obtained.Based on the compact implicit integration factor method,which can provide excellent stability properties,we combine it with non-uniform time meshes and diagonalization technology to construct a fast compact implicit integration factor method to solve the nonlinear system.Compared with the existing methods,the proposed method avoids the direct calculation of dense exponential matrices and requires lower computational costs.Numerical experimental results also demonstrate the effectiveness of the proposed method.