Legendre spectral methods based on two families of novel second-order numerical formulas for the fractional activator-inhibitor system

Abstract

In this paper, a novel numerical scheme is proposed to numerically solve the fractional activator-inhibitor system, which is a coupled nonlinear model. In the temporal direction, we employ two families of novel fractional θ-methods, the FBT-θ and FBN-θ methods, in spatial direction, the Legendre spectral method is used. Based on some positivity properties of the coefficients of both methods, the stability and convergence of the numerical scheme are proved. We derive an optimal convergence rate in space and second order accuracy in time. Finally, some numerical results are given to confirm the theoretical analysis.