Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

Wang, Dongling; Xiao, Aiguo; Yang, Wei

doi:10.1016/j.jcp.2013.02.037

Cited by 176 publications

(75 citation statements)

References 27 publications

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“…When α becomes smaller, the shape of the soliton will change more quickly. As shown in [30], this property of the fractional Schrödinger equation can be used in physics to modify the shape of wave without the change of the nonlinearity and dispersion effects. From Figs.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…When α = 2, the problem becomes the classic integer-order Schrödinger equation and by [30,32], we know the exact solution is given by…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For the time-fractional Schrödinger equations, Mohebbi et al [23] employed a meshless technique based on collocation methods and radial basis functions, Khan et al [16] derived approximating solutions by homotopy analysis methods, and Wei et al [35] gave discrete solution via a rigorous analysis of implicit fully discrete local discontinuous Galerkin method. For the space-fractional Schrödinger equations, some fully or linearly implicit difference methods were introduced and discrete conservation properties were analyzed in [30,31,33]. Two-dimensional problems were considered and a fourth-order ADI scheme was presented in [40].…”

Section: Introductionmentioning

confidence: 99%

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Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…When α = 2, the problem becomes the classic integer-order Schrödinger equation and by [30,32], we know the exact solution is given by…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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“…The Crank-Nicolson difference scheme for the coupled nonlinear fractional Schrödinger equations was proposed by Wang et al [20] in 2013. Subsequently, Atangana and Cloot [21] studied its stability and convergence for the space fractional variable-order Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

Wen¹,

Zhao²

2018

Bound Value Probl

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The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix D and a complex symmetric Toeplitz matrix T which is just the symmetric real Toeplitz T plus an imaginary identity matrix iI. In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix A. Finally, compared with the other algorithms by numerical experiments, the new method is more effective.

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“…Nowadays, higher order approximations [13], [14], ADI methods [15] are available for the corresponding numerical simulations. Moreover, recently, several kinds of linear [16], [17] and non-linear problems are studied containing fractional order Laplacian operators [18], [19].…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of fractional order elliptic boundary value problems

Szekeres

Izsák

2016

Journal of Computational and Applied Mathematics

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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.

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Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

Cited by 176 publications

References 27 publications

Galerkin finite element method for nonlinear fractional Schrödinger equations

Galerkin finite element method for nonlinear fractional Schrödinger equations

A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

Finite element approximation of fractional order elliptic boundary value problems

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