Space-time Spectral Collocation Methods For Two Kinds Of Fractional Differential Equations

In recent years,fractional differential equations have been widely applied in the fields of mechanics,biochemistry and electrical engineering.Because the frac-tional order differential equation is global,it is more advantageous to depict some problems with memory process than integer order differential equations.At the same time,due to the existence of fractional integral operators,it becomes more difficult to obtain the analytic solution of the equation.Therefore,how to solve the fractional order differential equation efficiently is the subject of many scholars'r e-search.In this paper,two kinds of fractional order differential equations are s olved by the space-time spectral collocation method,namely the two-dimensional linear fractional diffusion equation and the two-dimension nonlinear space fractional Riesz equation.First,for the study on the two-dimensional linear fractional order diffusion equation,based on the discrete form of the spectral collocation method in space,we transform the original equation into the linear ordinary differential equations of time.Then in time using the spectral allocation method to discrete,using the matrix re-duction for the matrix expression,and combining with the method of the searching minimum of the square of the linear system of equations,we get an efficient solu-tion.The priori error estimation of the semi-discrete method norm is given,and the stability and convergence of the full-discrete method are proved.Experimental re-sults show that the method can obtain spectral precision in time and space.Second,for the study of the two-dimension nonlinear space fractional Riesz equation,we use the space-time spectral collocation method and the Newton meth-od of nonlinear equations.The optimal priori error estimation of the spatial semi-discrete scheme in L~2 norm is given,and the uniqueness of the numerical solution of the full-discrete method is proved.The error accuracy of our method under coarse meshes is higher than that in other papers.The numerical examples verify the conclusion of the theory and show that our method is effective.For the two kinds of fractional differential equations,we develop two different spectral collocation methods.The proposed method has achieved good precision in both theoretical proof and numerical experiment results,and has a small amount of calculation,which can be applied to the numerical solution of similar equations.