Finite difference/Hermite–Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction–diffusion equation in unbounded domains

The aim of this paper is to develop an efficient finite difference/Hermite–Galerkin spectral method for the time-fractional nonlinear reaction–diffusion equation in unbounded domains with one, two, and three spatial dimensions. For this purpose, we employ the L2−1 _σ formula to discretize the temporal Caputo derivative. Additionally, we apply the Hermite–Galerkin spectral method with scaling factor for the approximation in space. The stability of the fully discrete scheme is established to show that our method is unconditionally stable. Numerical experiments including one-, two-, and three-dimensional cases of the problem are carried out to verify the accuracy of our scheme. The scheme is show-cased by solving two problems of practical interest, including the fractional Allen–Cahn and Gray–Scott models, together with an analysis of the properties of the fractional orders.