Artificial Boundary Method And Numerical Solution Of Time Fractional Schr(?)dinger Equation

As a basic equation in quantum mechanics,the Schr(?)dinger equation mainly reveals the basic law of the movement of matter in the microscopic physical world.The time fractional Schr(?)dinger equation,as its generalized equation,is an equation that describes the evolution of Femarkov in quantum physics by considering the non-Gaussian evolution of quantum mechanical particles and is widely used.This article is divided into two parts.The first part is devoted to the artificial boundary method and its numerical solution of the time fractional linear Schr(?)dinger equation on the unbounded region.Firstly,the Laplace transform is applied to the equation to obtain its precise artificial boundary conditions,and the problem on the unbounded region is transformed into an initial boundary value problem with precise artificial boundary conditions.It should be noted here that the precise artificial boundary we obtain is the generalized Caputo-type fractional derivative of the generalized Mittag-Leffler function.In theoretical analysis,the difficulties caused by the generalized Mittag-Leffler function and fractional derivatives and the complexity caused by the potential function V(x)? 0.Secondly,the unconditional stability of the solution of the time fractional-order linear Schr(?)dinger equation with precise artificial boundary conditions is strictly proved.In order to solve the numerical solution of the equation,the L1 difference scheme of the precise artificial boundary conditions is first constructed and its truncation error is analyzed.The difference scheme of the time fractional linear Schr(?)dinger equation is established and the stability and error estimates are proved.Finally,numerical examples are used to verify the feasibility of theoretical analysis.The second part is devoted to the artificial boundary method and its numerical solution of the time fractional nonlinear Schr(?)dinger equation on the unbounded region.First we use the unified method and linearization process to derive the absorbing artificial boundary of the equation.Second,the well-posedness of the solution of the time fractional nonlinear Schr(?)dinger equation with absorbing artificial boundary conditions is proved.Finally,the finite difference scheme of the time fractional nonlinear Schr(?)dinger equation is constructed,and the linearization method and iterative method are used to solve the difference scheme,and the error analysis is carried out.At the same time,the feasibility of the theoretical analysis is verified by a numerical example.It should be emphasized that in the time fractional linear and nonlinear Schr(?)dinger equations on the two types of unbounded regions studied in this paper,the potential function V(x)can be taken to be non-zero outside the artificial boundary,which is more suitable for the actual situation.