Crank–Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation

ABSTRACT

In this paper, the Crank–Nicolson Fourier spectral approximations for solving the space fractional nonlinear Schrödinger equation are proposed. Firstly, the numerical formats of the Crank–Nicolson Fourier Galerkin and Fourier collocation methods are established. The fast Fourier transform technique is applied to practical computation. Secondly, Convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Moreover, a rigorous analysis of the conservation for the Crank–Nicolson Fourier Galerkin fully discrete system is derived. Thirdly, the Bayesian method is presented to estimate the fractional derivative order and the coefficient of nonlinear term based on the spectral format of the direct problem. Finally, some numerical examples are given to confirm the theoretical analysis.

KEYWORDS:

• Space fractional nonlinear Schrödinger equation
• Crank–Nicolson Fourier spectral methods
• convergence
• mass and energy conservation
• parameter estimation
• Bayesian method

2010 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 65M06
• 65M12
• 65M15
• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been supported by the National Natural Science Foundation of China [Grants Nos. 11472161, 11102102], and the Natural Science Foundation of Shandong Province [Grant ZR2014AQ015].