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

Abstract

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and $(2-\gamma )$ ( 2 − γ ) th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

Numerical Solution of Time-Fractional Schrödinger Equation by Using FDM

“…Cen et al also adopted the L2-1 σ formula on graded meshes for solving the TFKBE with an initial singularity [18]. Ding et al solved a nonlinear TFSE by using the quintic non-polynomial spline in the spatial term and the L1 formula in the time term [19]. Mokhtari, R. et al constructed three finite difference schemes by adopting different L-type formulas to approximate the Caputo derivatives in the time direction and the central difference format in the space direction, respectively.…”

Parametric quintic spline for time fractional Burger’s and coupled Burgers’ equations

“…Secondly, the nonpolynomial splines [26][27][28][29][30][39][40][41][42][43][44] are used to solve many fractional order partial differential equations such as fractional subdiffusion problems [26,27,39,40], fractional diffusion-wave problems [41,42], fractional Schrödinger equation [28,29],…”

Numerical treatments of the nonlinear coupled time‐fractional Schrödinger equations

“…Recently, several numerical and analytical methods for solving time and time‐space fractional Schrödinger equations have been developed, such as the hyperbolic function method, 13 q‐homotopy analysis transform method, 14 quintic non‐polynomial spline, 15 homotopy perturbation Sumudu transform method, 16 homotopy analysis method, 17 the Laplace Adomian decomposition method, 18 and the Crank‐Nicolson scheme 19 . Bhrawy et al 20 developed the Jacobi spectral collocation method to solve fractional Schrödinger equations with spatial and time‐fractional derivatives.…”