  To achieve efficient and accurate long-time integration, we propose a fast,
accurate, and stable high-order numerical method for solving
fractional-in-space reaction-diffusion equations. The proposed method is
explicit in nature and utilizes the fourth-order compact finite difference
scheme and matrix transfer technique (MTT) in space with FFT-based
implementation. Time integration is done through the accurate fourth-order
modified exponential time differencing Runge-Kutta scheme. The linear stability
analysis and various numerical experiments including two-dimensional (2D)
Fitzhugh-Nagumo, Gierer-Meinhardt, Gray-Scott and three-dimensional (3D)
Schnakenberg models are presented to demonstrate the accuracy, efficiency, and
stability of the proposed method.
