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

Mingzhu Li; Xiaohua Ding; Qiang Xu

doi:10.1186/s13662-018-1743-3

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

Mingzhu Li^*
, Xiaohua Ding
, Qiang Xu

^*Corresponding author for this work

School of Science

Harbin Institute of Technology
Qingdao University of Technology
Shandong Normal University

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

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 L₁ 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(τ² ⁻ ^α+ h⁴) 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.

Original language	English
Article number	318
Journal	Advances in Difference Equations
Volume	2018
Issue number	1
DOIs	https://doi.org/10.1186/s13662-018-1743-3
State	Published - 1 Dec 2018

Keywords

Fourier analysis
Fractional Schrödinger equation
Non-polynomial spline
Stability

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10.1186/s13662-018-1743-3

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@article{84e54c9f1d044c1d8a49be59818bd0e1,

title = "Non-polynomial spline method for the time-fractional nonlinear Schr{\"o}dinger equation",

abstract = "In this paper, we propose a cubic non-polynomial spline method to solve the time-fractional nonlinear Schr{\"o}dinger equation. The method is based on applying the L1 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(τ2 − α+ h4) 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.",

keywords = "Fourier analysis, Fractional Schr{\"o}dinger equation, Non-polynomial spline, Stability",

author = "Mingzhu Li and Xiaohua Ding and Qiang Xu",

note = "Publisher Copyright: {\textcopyright} 2018, The Author(s).",

year = "2018",

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day = "1",

doi = "10.1186/s13662-018-1743-3",

language = "英语",

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TY - JOUR

T1 - Non-polynomial spline method for the time-fractional nonlinear Schrödinger equation

AU - Li, Mingzhu

AU - Ding, Xiaohua

AU - Xu, Qiang

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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 L1 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(τ2 − α+ h4) 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.

AB - 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 L1 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(τ2 − α+ h4) 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.

KW - Fourier analysis

KW - Fractional Schrödinger equation

KW - Non-polynomial spline

KW - Stability

UR - https://www.scopus.com/pages/publications/85053179839

U2 - 10.1186/s13662-018-1743-3

DO - 10.1186/s13662-018-1743-3

M3 - 文章

AN - SCOPUS:85053179839

SN - 1687-1839

VL - 2018

JO - Advances in Difference Equations

JF - Advances in Difference Equations

IS - 1

M1 - 318

ER -

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

Abstract

Keywords

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