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In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis. Introduction.In the past decades, fractional calculus has been used to model particle transport in porous media. Recently, there has been increasing interest in the study of fractional calculus for its wide application in many fields of science and engineering, such as the physical and chemical processes, materials, control theory, biology, finance, and so on (see [3,9,25,23,33,35]). In physics, fractional derivatives are used to model anomalous diffusion, where particles spread differently than the classical Brownian motion model [23]. Kinetic equations of the diffusion, diffusionadvection, and Fokker-Planck equations with partial fractional derivatives were recognized as a useful approach for the description of transport dynamics in complex systems. Reaction-diffusion models have been used for numerous applications in patten formation in biology, chemistry, physics, and engineering. These systems show that diffusion can produce the spontaneous formation of spatial-temporal patterns. The idea is to use a fractional-order density gradient to recover, at least at a phenomenological level, the nonhomogeneities of the porous media. Given the structural

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In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

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