Non-polynomial spline method for the time-fractional nonlinear Schrödinger equation

Abstract

In this paper, we propose a cubic non-polynomial spline method to solve the time-fractional nonlinear Schrödinger equation. The method is based on applying the L 1 formula to approximate the Caputo fractional derivative and employing the cubic non-polynomial spline functions to approximate the spatial derivative. By considering suitable relevant parameters, the scheme of order O(τ 2-α + h 4) has been obtained. The unconditional stability of the method is analyzed by the Fourier analysis. Numerical experiments are given to illustrate the effectiveness and accuracy of the proposed method.

Quintic non-polynomial spline for time-fractional nonlinear Schrödinger equation

“…As shown in Theorem 5.1 , the numerical scheme ( 2.25 )–( 2.27 ) achieves sixth order convergence in the spatial dimension. This improves the work of [ 36 ] where the spatial convergence order is observed to be four through numerical experiment. Furthermore, in [ 36 ] the authors have only proved the stability of their scheme, while we have proven the stability, unique solvability and convergence of our method by the Fourier method.…”

“…For the time-fractional Schrödinger equation: Khan et al [ 32 ] have applied the homotopy analysis method; Mohebbi et al [ 47 ] have used the Kansa approach to approximate the spatial derivative and L 1 discretization to approximate the Caputo time-fractional derivative; a Krylov projection method has been developed in [ 22 ]; a Jacobi spectral collocation method has been applied to a multi-dimensional time-fractional Schrödinger equation in [ 4 ]; a quadratic B-spline Galerkin method combined with L 1 discretization scheme has been proposed in [ 16 ]; a linearized L 1-Galerkin finite element method has been used in [ 35 ] for a multi-dimensional nonlinear time-fractional Schrödinger equation; a cubic non-polynomial spline method combined with L 1 discretization has been proposed in [ 36 ] and the stability has been shown by the Fourier method, the convergence order is not proved but is observed from numerical experiments to be .…”

On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

“…which show that (P 1 ) is satisfied. Hence, by Corollary 5, we deduce that problems (15) and (16) admits no global weak solution.…”

“…For further interesting contributions to the analysis of the blow-up behavior of solutions to fractional nonlinear Schrödinger problems, we mention the papers of Fino-Dannawi-Kirane [12] (semilinear equation with fractional Laplacian), Ionescu-Pusateri [13] (equation in dimension one with cubic nonlinearities) and Kirane-Nabti [14] (nonlocal in time equation). Finally, we recall the paper of Li-Ding-Xu [15] where a cubic non-polynomial spline method is implemented to solve the time-fractional nonlinear Schrödinger equation. Furthermore, the stability of the method is analyzed by Fourier analysis.…”

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

“…Mohyu-Din et al [15] investigated the extended Bspline solution of time fractional advection diffusion equation by means of a fully implicit finite difference scheme. Li et al [16] developed a non-polynomial spline scheme to solve time fractional nonlinear Schrodinger equation. In [17], Pezza and Pitolli used a fractional spline collocation Galerkin scheme to develop series solution for time fractional diffusion equation.…”