A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.
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 . 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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