### A numerical method for the fractional Fitzhugh–Nagumo monodomain model

Fawang Liu, Ian Turner, Vo Anh, Qianqian Yang, Kevin Burrage

#### Abstract

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.

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#### Keywords

fractional FitzHugh--Nagumo Monodomain Model;fractional Riesz space nonlinear reaction-diffusion model; stability and convergence

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DOI: http://dx.doi.org/10.21914/anziamj.v54i0.6372

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