An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

Abstract

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as $\left(0<{\beta }_0<{\beta }_1<{\beta }_2<\cdots <{\beta }_m<1\right)$ . The problem is first approximated by the $L1$ difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of $2-{\beta }_m$ in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.