CRANK-NICOLSON DIFFERENCE SCHEME FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH THE RIESZ SPACE FRACTIONAL DERIVATIVE

Changhong Guo; Shaomei Fang

doi:10.11948/20180178

2021 Volume 11 Issue 3

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Changhong Guo, Shaomei Fang. CRANK-NICOLSON DIFFERENCE SCHEME FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH THE RIESZ SPACE FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1074-1094. doi: 10.11948/20180178

Citation:

Changhong Guo, Shaomei Fang. CRANK-NICOLSON DIFFERENCE SCHEME FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH THE RIESZ SPACE FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1074-1094. doi: 10.11948/20180178

CRANK-NICOLSON DIFFERENCE SCHEME FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION WITH THE RIESZ SPACE FRACTIONAL DERIVATIVE

Changhong Guo¹,
Shaomei Fang^2, ,

1.
School of Management, Guangdong University of Technology, Guangzhou 510520, China
2.
Department of Mathematics, South China Agricultural University, Guangzhou 510640, China

Corresponding author: Email address: fangsm90@163.com(S. Fang)
Fund Project: This research was supported by the National Natural Science Foundation of China(No. 71974038), China Scholarship Council(No. 201708440509), and Natural Science Foundation of Guangdong Province, China(No. 2017A030310564)

Abstract

Abstract

This paper studied the Crank-Nicolson(CN) difference scheme for the derivative nonlinear Schrödinger equation with the Riesz space fractional derivative, which generalized the classical Schrödinger equation that was used as a model in quantum mechanics. The existence of this difference solution is proved by the Brouwer fixed point theorem. Since the difference solution of the equation satisfies the mass conservation law, the corresponding convergence is also investigated in the $ L_2 $ norm, which turns out to be the second order accuracy in both temporal and space directions. Especially when the fractional order equals to two, all those results are in accordance with the conclusions for the difference solution developed for the non-fractional derivative Schrödinger equation. Finally, some numerical examples are carried out and further verified the theoretical results.
- Derivative Schrödinger equation /
- Riesz space fractional derivative /
- Crank-Nicolson scheme /
- convergence
MSC: 35Q55, 65R10, 47B06

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References

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