Application of Improved SPH Method in Solving Time Fractional Schrödinger Equation

Abstract

In this paper, a pure meshless method for solving the time fractional Schrödinger equation (TFSE) based on KDF-SPH method is presented. The method is used for the first time to numerically solve the TFSE. The method utilizes the finite difference method (FDM) to approximate the time fractional-order derivative defined in the Caputo sense. The spatial derivatives are discretized by the KDF-SPH meshless method. Expressions for the kernel approximation and the particle approximation are provided. To ensure the validity and flexibility of the numerical calculations, we conducted numerical simulations of one- and two-dimensional linear/nonlinear time Schrödinger equations (1D/2D TFLSE/TF-NLSE) in both bounded and unbounded regions. We also examined nonlinear time fractional Schrödinger equations that lack analytical solutions and compared our method with other meshless methods. Numerical results show that the proposed method can approximate to the second-order precision in space, which verifies the effectiveness and accuracy of the proposed method.