An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function

Abstract

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear Riesz-space and Caputo-time fractional reaction–diffusion equation with prehistory. The problem is first approximated by the L1 difference method in the temporal direction, and then the Galerkin–Legendre spectral method is applied for the spatial discretization. The key advantage of the proposed method is that the implementation of the iterative approach is linear. The stability and the convergence of the semi-discrete approximation are proved by invoking the discrete fractional Halanay inequality. The stability and convergence of the fully discrete scheme are also investigated utilizing discrete fractional Grönwall inequalities, which show that the proposed method is stable and convergent. Furthermore, to verify the efficiency of our method, we provide numerical results that show a satisfactory agreement with the theoretical analysis.