A Fast Implicit Difference Scheme For Sloving A Class Of Variable-Order Time-Space Fractional Diffusion Equations

In recent years,variable-order fractional partial differential equations have been used to model dynamic physical systems,such as the geometry of media.Considering that the variable-order time-space fractional order diffusion equation is of great significance for solving anomalous diffusion problems,a fast implicit difference method for solving a class of variable-order time-space fractional diffusion equations is proposed.L1 formula and shifted Grünwald formula are employed to discretize the temporal and spatial derivatives respectively.The scheme achieves first-order convergence in both time and space.The stability and convergence of the implicit difference schemes proposed in this paper are rigorously proved by matrix analysis.The preconditioned conjugate gradient normal residual method is prosed to solve the linear system.The coefficient matrix of this system of linear equations is similar to the Toeplitz matrix and the fast Fourier transform can be used to reduce the computational cost of the matrix–vector multiplication.The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners.Finally,numerical results are presented to show the utility of the proposed methods.