On Time-Splitting Spectral Method For The Space Fractional Schrodinger Equations In The Semiclassical Regime

At present,fractional differential equations have been widely used in the fields of physic-s,engineering,control robotics,molecular dynamics and image processing.Many physical phenomena in reality can only be obtained by combining fractional calculus with quantum mechanics.Accurate representation.The fractional schrodinger equation is established un-der the concept of introducing fractional calculus.the fractional schrodinger equations in the semiclassical regime is used to describe a specific quantum phenomenon in quantum physics.The standard schrodinger equation is a special case of the equation.Due to the existence of the singular properties of the fractional derivative and the very small Planckian constant,It is found that the fractional order s of he space fractional schrodinger equations in the semiclassical regime has a significant effect on the numerical solution and error of the equation.This paper mainly considers the space fractional schrodinger equations in the semi-classical regime with very small Planckian constants,including the Riesz space fractional linear schrodinger equation in the semiclassical regime and the Riesz space fractional non-linear schrodinger equation in the semiclassical regime.Firstly,the semiclassical limit of the space fractional schrodinger equations in the semiclassical regime is given,and the first-order time splitting spectral method and the second-order Strang splitting spectral method are given.Then the numerical discrete process is given for the properties of the equation.And proves the numerical stability and convergence of the time-split fourier spectral method.The processing of the fractional derivative term is the difficulty of this paper.In this paper,a lot of numerical experiments are carried out in combination with the splitting technique.The nu-merical simulations of the space fractional schrodinger equations in the semiclassical regime meaning with different fractional orders are carried out.The experimental results show the high-precision and unconditional stability of the time-split spectral method.It is found that the fractional order of the space fractional schrodinger equation has a significant effect on the stability and error of the time-split spectral method.This article is divided into five chapters,the framework of which is as follows.The first chapter is the introduction.It summarizes the theoretical and practical sig-nificance of the fractional order schrodinger equation,the research status and development trend,the research content and innovation of this thesis.The second chapter is the preliminary knowledge of the fractional derivative and the space fractional schrodinger equations in the semiclassical regime.In the third chapter,the space fractional linear schrodinger equations in the semiclassi-cal regime is proposed.The first-order time splitting spectral method and the second-order Strang splitting method are given,and the first-order numerical method stability and con-vergence is proved.In the fourth chapter,the semi-classical limit of the space fractional nonlinearschrodinger equations in the semiclassical regime is proposed,and the high-precision time splitting spec-tral method is used to solve the equation,and the stability of the numerical method is proved.The fifth chapter is a numerical experiment.The numerical experiments of the space fractional linear schrodinger equations and the space fractional nonlinear schrodinger equa-tion in the semiclassical regime are given.The numerical results under different fractional orders are listed and analyzed.