A fast time integral finite difference method for a space-time fractional FitzHugh-Nagumo monodomain model in irregular domains

This work aims at proposing a fast time integral (FTI) method for the space-time fractional Fitzhugh-Nagumo (FHN) monodomain model in irregular domains, which is commonly used to characterize the transmembrane potential of the heart. In order to reduce the storage and the algorithm complexity due to the geometric configuration of the heart, the derivation of the sum of the exponentials (cf. [1]) in FTI method is improved by the integral transformation, integral truncation and Gauss-Legendre quadrature. Such strategy is applied to approximate the Caputo fractional derivative. The number of the exponentials, reduced in the FTI method, is related to the calculation efficiency. The CPU time of the FHN monodomain model using FTI method is reduced by an order of magnitude. Moreover, a second-order discrete method to the Riesz space fractional derivative adopts for the spatial discretization, then an implicit-explicit scheme is derived for the nonlinear FHN model under the finite difference method. Numerical results are reported to demonstrate the convergence behaviors, robustness and high efficiency of the proposed method.