A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

In this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in $L^{\infty }$ L ∞ -norm is obtained, and the spectral scheme is shown to be unconditionally convergent in $L^{2}$ L 2 -norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.

Numerical analysis of stochastic SIR model by Legendre spectral collocation method

This article represents Legendre spectral collocation method based on Legendre polynomials to solve a stochastic Susceptible, infected, Recovered (SIR) model. The Legendre polynomials on stochastic SIR model that convert it to a system of equations has been applied and then solved by the Legendre spectral method, which leads to excellent accuracy and convergence by implementing Legendre–Gauss–Lobatto collocation points permitting to generate coarser meshes. The numerical results for both the deterministic and stochastic models are presented. In case of probably small noise, the verge dynamics is analyzed. The large noise will show eradication of disease, which controls disease spreading. Various graphical results demonstrate the effectiveness of the proposed method to SIR model.