A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation

Fei, Mingfa; Wang, Nan; Huang, Chengming; Ma, Xiaohua

doi:10.1016/j.apnum.2020.03.004

Cited by 12 publications

(8 citation statements)

References 43 publications

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“…In addition, spectral methods have also been applied in solving the nonlocal FSE, including spectral Galerkin schemes [24][25][26][27][28][29][30] and collocation schemes [31][32][33][34][35]. On the other hand, numerical studies of the FSE with Caputo fractional derivative in time were considered in [36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

et al. 2020

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In this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in $L^{\infty }$ L ∞ -norm is obtained, and the spectral scheme is shown to be unconditionally convergent in $L^{2}$ L 2 -norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

et al. 2020

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“…In literature, fractional Schrödinger equations exist with a fractional derivative appearing in time, space or both time and space variables [1][2][3][4]. In this study, we consider a nonlinear fractional Schrödinger equation which is established by adding a nonlinear term to the linear form of Schrödinger equation as…”

Section: Introductionmentioning

confidence: 99%

“…Analysis of various Schrödinger equations in an abstract form exists in literature widely [5]. Here, we will consider some applications of the theoretical results for abstract nonlinear fractional Schrödinger equation (1) in [6].…”

Section: Introductionmentioning

confidence: 99%

“…where L, M,M are positive constants. Then, a unique bounded solution u(t) for problem (1) exists in H α .…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, first and second order of accuracy difference schemes are established for abstract problem (1) in [6]. Discretization through time variable in N steps starts with denoting a step size τ = T /N where [0, T ] is the time interval for problem (1). The set of grid points is given as…”

Section: Introductionmentioning

confidence: 99%

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Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition

Hicdurmaz

2020

E-Journal of Analysis and Applied Mathematics

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In this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form of the nonlinear fractional Schrödinger equation are revisited to derive new applications of these theorems on the nonlinear fractional Schrödinger problems with Neumann boundary condition. Consequently, first and second order of accuracy difference schemes are constructed for the nonlinear fractional Schrödinger problem with Neumann boundary condition. Stability analysis show that the constructed difference schemes are stable. Stability theorems for the stability of the nonlinear fractional Schrödinger problem with Neumann boundary condition are presented. Additionally, applications of the new theoretical results are presented on a one dimensional nonlinear fractional Schrödinger problem and a multidimensional nonlinear fractional Schrödinger problem with Neumann boundary conditions. Numerical results are presented on one and multidimensional nonlinear fractional Schrödinger problems with Neumann boundary conditions and different orders of derivatives in fractional derivative term. Numerical results support the validity and applicability of the theoretical results. Numerical results present the convergence rates are appropriate with the theoretical findings and construction of the difference schemes for the nonlinear fractional Schrödinger problem with Neumann boundary condition.

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Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

Zhang

Jiang

2022

Numerical Methods Partial

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In this article, we consider the two‐dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second‐order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time‐stepping numerical method is also proved in a simple manner without imposing the Courant‐Friedrichs‐Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.

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A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation

Cited by 12 publications

References 43 publications

A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition

Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

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