Numerical simulation for a time-fractional coupled nonlinear Schrödinger equations

Abstract

In this paper, we attempt to find an approximate solution of time-fractional coupled nonlinear Schrödinger equations (TFCNLS) through one of the meshless approach based on radial basis functions (RBFs) collocation. The time-fractional derivative is described in terms of the Caputo derivative. Discretizing the time-fractional derivative of the mentioned equation, we first use a scheme of order $O\left(\mathrm{\Delta }{t}^{2-\alpha }\right)$, $0<\alpha \le 1,$ and then the average value of the function in a consecutive time step is used for other terms. Also, we use the RBFs collocation method to approximate the spatial derivative. On the other hand, the stability analysis of the suggested scheme is investigated in a similar way to the classic von Neumann technique for TFCNLS equations. This present paper is to indicate that the meshfree methods are appropriate and reliable to obtain a numerical solution of fractional partial differential equations. This efficiency and accuracy of the present method are verified by solving two examples. We obtain the numerical results from solving this problem on the rectangular domain. All obtained numerical experiments are presented in tables and figures. Finally, it can be said that the main advantage of the mentioned scheme is that the algorithm is very simple and easy to apply.

Keywords:

• Time fractional coupled nonlinear Schrödinger equations (TFCNLS)
• Radial basis functions (RBFs) approximations
• Caputo derivative
• stability

2010 Mathematics Subject Classifications:

• 65M12
• 65M70

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which helped improve our manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).