The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics

An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

“…Authors of [95] presented an implicit meshless approach based on the moving least squares (MLS) approximation for the numerical simulation of fractional advectiondiffusion equation. Authors of [60] proposed a numerical method for the solution of the time-fractional nonlinear Schrödinger equation in one and two dimensions which appears in quantum mechanics. Liu et al [48] for solving the time fractional mobile/immobile transport model proposed a finite difference method to discretize the time variable and obtained a semi-discrete scheme.…”

Galerkin finite element method for nonlinear fractional Schrödinger equations

“…In recent years, as the generalization of the standard nonlinear Schrödinger equation, there have been growing interests in the analysis and computing for the numerical solutions to nonlinear fractional Schrödinger equations (FSEs). 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].…”

Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

“…The NLSE is one of the most important equations in various areas of mathematics and physics, such as nonlinear optics, optical pulses, plasma physics and water waves [1] . Numerical methods for the NLSE have been investigated extensively, see [2][3][4][5][6] for finite difference method, [7][8][9][10][11][12] for finite element method (FEM), [13,14] for discontinuous Galerkin method, and [15][16][17][18][19] for meshless method. However, all of these studies only concentrated on the convergence analysis.…”