the global analysis on the spectral collocation method for time fractional schrodinger equation abstract in this paper a spectral collocation method is proposed and analyzed for solving the time fractional schrodinger equation the space derivative is discretized using the collocation method and the time fractional derivative using grunwald letnikov formulation the stability and convergence of the full discretization scheme are analyzed based on the z transform the global behavior of the finite difference spectral collocation method is derived numerical examples show a good agreement with the theoretical analysis

collocation method for fractional quantum mechanics we show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional laplacian using a collocation approach based on little sinc functions which discretizes the schrodinger equation on a uniform grid the different boundary conditions are naturally implemented using sets of functions with the appropriate behavior good convergence properties are observed a comparison with results based on a wentzel kramers brillouin analysis is performed

on the l convergence of a difference scheme for coupled nonlinear schrodinger equations in this article a finite difference scheme for coupled nonlinear schrodinger equations is studied the existence of the difference solution is proved by brouwer fixed point theorem with the aid of the fact that the difference solution satisfies two conservation laws the finite difference solution is proved to be bounded in the discrete l norm then the difference solution is shown to be unique and second order convergent in the discrete l norm finally a convergent iterative algorithm is presented

a fourth order explicit schemes for the coupled nonlinear schrodinger equation abstract in this paper we derive numerical methods for solving the coupled nonlinear schrodinger equation we discretize the space derivative by central difference formulas of fourth order the resulting ordinary differential system is solved by the fourth order explicit runge kutta method neumann and periodic boundary conditions are used the method is tested for accuracy and the conserved quantities these methods conserve the three conserved quantities conserve exactly for at least five decimal places a comparison has been made with some existing methods

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

on convergence and stability of the explicit difference method for solution of nonlinear schrodinger equations the first boundary value problem for a nonlinear schrodinger equation is investigated the conditional convergence and stability on the initial data of the explicit three level difference scheme of dufort frankel type in c and w 2 1 norms are proved grid analogues of energy conservation laws and grid multiplicative inequalities are used

analysis of a symplectic difference scheme for a coupled nonlinear schrodinger system in general proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations in this paper a symplectic difference scheme is proposed for an initial boundary value problem of a coupled nonlinear schrodinger system an important lemma and an induction argument are used to prove the unique solvability convergence and stability of numerical solutions an iterative algorithm is also proposed for the symplectic scheme and its convergence is proved numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

two novel linear implicit momentum conserving schemes for the fractional korteweg de vries equation abstract we propose two conservative linear implicit schemes for the space fractional korteweg de vries fkdv equation one is the linear implicit crank nicolson scheme and the other is the linear implicit leap frog scheme in order to obtain a high order discretization in the space direction we adopt the fourier pseudospectral method the crank nicolson scheme and leap frog scheme are used for temporal discretization and those two schemes are efficient in practical computations because of their linear property furthermore we analyse the uniqueness boundness convergence of the two schemes numerical experiments are presented to validate the theoretical analysis

a compact difference scheme for a two dimensional fractional klein gordon equation with neumann boundary conditions in this paper a high order finite difference scheme for a two dimensional fractional klein gordon equation subject to neumann boundary conditions is proposed the difficulty induced by the nonlinear term and the neumann conditions is carefully handled in the proposed scheme the stability and convergence of the finite difference scheme are analyzed using the matrix form of the scheme numerical examples are provided to demonstrate the theoretical results

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

on partially inexact hss iteration methods for the complex symmetric linear systems in space fractional cnls equations the space fractional coupled nonlinear schrdinger cnls equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal plus toeplitz matrix the hermitian and skew hermitian splitting hss method and the partially inexact hss pihss method are employed to solve the discretized linear system in the inner iteration processes of the hss method we only need to solve the linear sub systems associated with the hermitian part inexactly by the conjugate gradient cg method resulting in pihss iteration method theoretical analyses show that both hss and pihss methods are unconditionally convergent numerical examples are given to demonstrate the effectiveness of the hss iteration and the pihss iteration

novel numerical methods for solving the time space fractional diffusion equation in two dimensions in this paper a time space fractional diffusion equation in two dimensions tsfde 2d with homogeneous dirichlet boundary conditions is considered the tsfde 2d is obtained from the standard diffusion equation by replacing the first order time derivative with the caputo fractional derivative t d gamma gamma in 0 1 and the second order space derivatives with the fractional laplacian delta alpha 2 alpha in 1 2 using the matrix transfer technique proposed by ilic et al m ilic f liu i turner and v anh fract calc appl anal 9 2006 pp 333 349 the tsfde 2d is transformed into a time fractional differential system as t d gamma mathbf u k alpha mathbf a alpha 2 mathbf u where mathbf a is the approximate matrix representation of delta traditional approximation of mathbf a alpha 2 requires diagonalization of mathbf a which is very time consuming for large sparse matrices the novelty of our proposed numerical schemes is that using either the finite difference method or the laplace transform to handle the caputo time fractional derivative the solution of the tsfde 2d is written in terms of a matrix function vector product f mathbf a mathbf b at each time step where mathbf b is a suitably defined vector depending on the method used to generate the matrix mathbf a the product f mathbf a mathbf b can be approximated using either the preconditioned lanczos method when mathbf a is symmetric or the mathbf m lanzcos method when mathbf a is nonsymmetric which are powerful techniques for solving large linear systems we give error bounds for the new methods and illustrate their roles in solving the tsfde 2d we also derive the analytical solution of the tsfde 2d in terms of the mittag leffler function finally numerical results are presented to verify the proposed numerical solution strategies

linearized crank nicolson scheme for the nonlinear time space fractional schrodinger equations abstract in this paper a crank nicolson difference scheme is first derived for solving the nonlinear time space fractional schrodinger equations the truncation error and stability of the scheme are discussed in detail the existence of the numerical solution is shown by the brouwer fixed point theorem for improving the calculating efficiency a three level linearized difference scheme is also proposed and analyzed both schemes are subsequently extended to the nonlinear coupled equations and some similar results are given and proved several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes and comparison with the related work is presented

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

a computationally effective alternating direction method for the space and time fractional bloch torrey equation in 3 d the space and time fractional bloch torrey equation st fbte has been used to study anomalous diffusion in the human brain numerical methods for solving st fbte in three dimensions are computationally demanding in this paper we propose a computationally effective fractional alternating direction method fadm to overcome this problem we consider st fbte on a finite domain where the time and space derivatives are replaced by the caputo djrbashian and the sequential riesz fractional derivatives respectively the stability and convergence properties of the fadm are discussed finally some numerical results for st fbte are given to confirm our theoretical findings

an efficient split step quasi compact finite difference method for the nonlinear fractional ginzburg landau equations abstract in this paper we propose a split step quasi compact finite difference method to solve the nonlinear fractional ginzburg landau equations both in one and two dimensions the original equations are split into linear and nonlinear subproblems the riesz space fractional derivative is approximated by a fourth order fractional quasi compact method furthermore an alternating direction implicit scheme is constructed for the two dimensional linear subproblem the unconditional stability and convergence of the schemes are proved rigorously in the linear case numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method

finite element approximation of fractional order elliptic boundary value problems a finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous dirichlet type boundary conditions it is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non fractional elliptic operators it is proved that this approach which is also called the matrix transformation or matrix transfer method delivers optimal rate of convergence in the l 2 norm

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

an inverse mapping theorem for h differentiable set valued maps taking advantage of recent developments in the theory of generalized differentiation we present an inverse mapping theorem for set valued maps and prove its stability under small linear perturbations using variational convergences of set valued mappings we present as well an approximate version of our inverse theorem

the global solution for a class of systems of fractional nonlinear schrodinger equations with periodic boundary condition in this paper we consider a class of systems of fractional nonlinear schrodinger equations we prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the faedo galerkin method

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

new conservative difference schemes for a coupled nonlinear schrodinger system abstract in this paper two conservative difference schemes for solving a coupled nonlinear schrodinger cnls system are numerically analyzed firstly a nonlinear implicit two level finite difference scheme for cnls system is studied then a linear three level difference scheme for cnls system is presented an induction argument and the discrete energy method are used to prove the second order convergence and unconditional stability of the linear scheme numerical examples show the efficiency of the new scheme

existence of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation in this paper we prove the existence and uniqueness of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation by using energy method

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

an efficient conservative difference scheme for fractional klein gordon schrodinger equations abstract in the paper we give an efficient conservative scheme for the fractional klein gordon schrodinger equations based on the central difference scheme the crank nicolson scheme and leap frog scheme first we use central difference scheme for discretizing the system in space direction second we use crank nicolson and leap frog scheme for discretizing the system in time direction we find that the scheme can be decoupled linearized and suitable for parallel computation to increase computing efficiency and preserve mass and energy conservation laws the convergence of the scheme is discussed and it is shown that the scheme is of the accuracy o 2 h 2 the numerical experiments are given and verify the correctness of theoretical results and the efficiency of the scheme

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders