Numerical Simulation of the Time-Fractional Nonlinear Schrödinger Equation and the Time-Fractional Schrödinger–Poisson System

This paper presents an efficient localized meshless method for solving the nonlinear time-fractional Schrödinger equation and the time-fractional Schrödinger–Poisson system. The proposed approach consists of two main steps. First, the Caputo time-fractional derivative is approximated using the L1 formula, which has an accuracy of order O(δt2−β) for 0<β<1. The spatial derivatives in the coupled equations are then discretized using a localized radial basis function Hermite finite-difference (LRBF-HFD) method. In this approach, the Laplacian operator is approximated on small overlapping neighborhoods, thereby avoiding the large and ill-conditioned systems arising from global RBF methods. Numerical experiments are presented to confirm the accuracy and reliability of the proposed method.