Circulant And Skew-circulant Splitting Iteration For The Tempered Fractional Diffusion Equation

The tempered fractional diffusion equation is used to amend the fractional diffusion equation by multiplying the exponential tempered factor to better describe the anomalous diffusion in the bounds of space or limited vitality in nature.The exponential tempered power-law jump distribution leads to the tempered space fractional derivatives.The exponential tempered power-law wait time leads to the tempered time fractional derivatives,improving the defect of the fractional diffusion equations.In recent years,the numerical algorithms of the fractional diffusion equation have attracted people’s research interest.In this paper,we mainly study the two kinds of different parameters circulant and skew-circulant splitting iteration method to solve the numerical solutions of the tempered fractional diffusion equation.The main work of this paper is as follows:(1)For the tempered fractional two-point boundary value problem,the tempered and weighted and shifted Grünwald difference operator(tempered-WSGD)is used to approximate the tempered Riemann-Liouville fractional derivative,and the coefficient matrix of linear systems is a dense,asymmetric and with Toeplitz structure.Solving the Toeplitz system by circulant and skew-circulant splitting(CSCS)iteration method and it is proved that the linear system is unconditionally convergent by CSCS iteration method.At each iteration,the linear system is solved by using the fast Fourier transform,which required only computational cost O(Nlog N),where N represent the number of mesh nodes in spatial.The numerical example shows that the fast algorithm is feasible and effective.(2)For the tempered fractional two-point boundary value problem with the diffusion coefficients equivalent,the tempered-WSGD is employed to discretize the tempered fractional derivative.Obtain symmetric,positive definite and linear systems with Toeplitz structure.The two-parameter circulant and skew-circulant splitting iteration method are used to solve the linear system.The convergence of two-parameter circulant and skew-circulant splitting iteration method is analyzed.The selection of two parameters is analyzed.Numerical examples show that the convergence rate of fast algorithm is fast.(3)The implicit second-order finite difference scheme,which is used to discretize the tempered fractional diffusion equation.Symmetric and positive definite linear system with Toeplitz structure is obtained.The two-parameter circulant and skew-circulant splitting iteration method is employed to solve the Toeplitz system.The method is proved to be unconditionally convergent the unique solution of the linear system.And also achieve fast and efficient numerical results.