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

• We propose L 2 − 1 σ /Hermite–Galerkin spectral method to solve nonlinear fractional differential equation in R d. • Arbitrary time-step size of our scheme can be used without resulting in the nonsense numerical results. • Our method can arrive at the spectral (resp. second-order) accuracy for spatial (resp. temporal) discretization. • Effects of the fractional orders on the pattern formations of fractional Allen–Cahn and Gray–Scott models are studied. • Comparisons between the fractional models and the corresponding integer-order ones are investigated. 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 L 2 − 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. [ABSTRACT FROM AUTHOR]