Semi-implicit Hermite–Galerkin Spectral Method for Distributed-Order Fractional-in-Space Nonlinear Reaction–Diffusion Equations in Multidimensional Unbounded Domains

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.