Numerical solutions of two-dimensional fractional Schrodinger equation

In this article, the authors proposed Chebyshev pseudospectral method for numerical solutions of two-dimensional nonlinear Schrodinger equation with fractional-order derivative in both time and space. Fractional-order partial differential equations are considered as generalizations of classical integer-order partial differential equations. The proposed method is established in both time and space to approximate the solutions. The Caputo fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to diagonally block system of nonlinear algebraic equations, which will be solved using Newton–Raphson iterative method. Some model examples of the equations, defined on a rectangular domain, have tested with various values of fractional order α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}. Moreover, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. For the proposed method, highly accurate numerical results are obtained which are compared with the analytical solution to confirm the accuracy and efficiency of the proposed method.