Crank-Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh-Nagumo monodomain model

In this paper, a two-dimensional fractional FitzHugh-Nagumo monodomain model (2D-FFHNMM) with zero Dirichlet boundary condition is considered. The model consists of a coupled two-dimensional space fractional nonlinear reaction-diffusion model (2D-SFNRDM) and an ordinary differential equation. The 2D-SFNRDM and ordinary differential equation are decoupled at each time step. A new Crank-Nicolson alternating direction implicit (ADI) Galerkin finite element method for the 2D-SFNRDM is developed. The stability and convergence of the numerical method are discussed. Finally, some numerical examples on 2D-SFNRDM and 2D-FFHNMM are given for verification of our theoretical analysis.

[1]  David A. Benson,et al.  Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data , 2007 .

[2]  Fawang Liu,et al.  Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation , 2010, Numerical Algorithms.

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  Fawang Liu,et al.  Galerkin finite element approximation of symmetric space-fractional partial differential equations , 2010, Appl. Math. Comput..

[5]  Fawang Liu,et al.  A numerical method for the fractional Fitzhugh–Nagumo monodomain model , 2013 .

[6]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[7]  Zhengang Zhao,et al.  A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation , 2011 .

[8]  D. Benson,et al.  Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications , 2009 .

[9]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[10]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[11]  Graeme Fairweather,et al.  An alternating direction Galerkin method for a class of second-order hyperbolic equations in two space variables , 1991 .

[12]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[13]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[14]  Jianfei Huang,et al.  Two finite difference schemes for time fractional diffusion-wave equation , 2013, Numerical Algorithms.

[15]  Fawang Liu,et al.  A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation , 2014, SIAM J. Numer. Anal..

[16]  Ningming Nie,et al.  Solving spatial-fractional partial differential diffusion equations by spectral method , 2014 .

[17]  S. K. Chung,et al.  Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term , 2012 .

[18]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[19]  Yangquan Chen,et al.  Computers and Mathematics with Applications Numerical Approximation of Nonlinear Fractional Differential Equations with Subdiffusion and Superdiffusion , 2022 .

[20]  Weihua Deng,et al.  Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation , 2013, BIT Numerical Mathematics.

[21]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[22]  Hong Wang,et al.  A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations , 2012, SIAM J. Sci. Comput..

[23]  Graeme Fairweather,et al.  Alternating-Direction Galerkin Methods for Parabolic and Hyperbolic Problems on Rectangular Polygons , 1975 .

[24]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[25]  K. Burrage,et al.  Fourier spectral methods for fractional-in-space reaction-diffusion equations , 2014 .