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

Qinxu Ding; Patricia J Y Wong

doi:10.1186/s13662-020-03021-0

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

Adv Differ Equ. 2020;2020(1):577. doi: 10.1186/s13662-020-03021-0. Epub 2020 Oct 16.

Authors

Qinxu Ding¹, Patricia J Y Wong¹

Affiliation

¹ School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798 Singapore.

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 $(2 - γ)$ 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.

Keywords: Nonlinear Schrödinger equation; Quintic non-polynomial spline; Time-fractional derivative.