Non-polynomial spline method for the time-fractional nonlinear Schr\xf6dinger equation

Mingzhu Li,Qiang Xu,Xiaohua Ding

doi:10.1186/s13662-018-1743-3

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(tau^{2-alpha }+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.

Highlights

In this paper, we consider the following time-fractional nonlinear Schrödinger equation: ∂ α u(x, i ∂tα t) + ∂ 2 u(x, ∂ x2 t) λ u(x, t) 2u(x, t) = f (x, t),(x, t) ∈ [a, b] × [0, T], (1)
In [14], the stability analysis was presented for a first order difference scheme applied to a nonhomogeneous time-fractional Schrödinger equation
In [29], Ding et al proposed two classes of difference schemes for solving the fractional reaction-subdiffusion equations based on a mixed spline function

Summary

Introduction

We consider the following time-fractional nonlinear Schrödinger equation:. Numerical methods have become important for the approximate solutions of time-fractional Schrödinger equations. In [14], the stability analysis was presented for a first order difference scheme applied to a nonhomogeneous time-fractional Schrödinger equation. In [27], Talaat et al presented a general framework of the cubic parametric spline functions to develop a numerical method for the time-fractional Burgers’ equation. In [29], Ding et al proposed two classes of difference schemes for solving the fractional reaction-subdiffusion equations based on a mixed spline function. In [32,33,34,35], the spline method was employed for the numerical solution of time-fractional fourth order partial differential equation. We apply the spline method based on a cubic non-polynomial spline function to the time-fractional nonlinear Schrödinger equation.

Cubic non-polynomial spline function

Conclusion

Methods