An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

Abstract

In this work, we consider numerical solutions of the FitzHugh-Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The sub equation approach is applied to derive dynamically consistent schemes for the sub models. This is followed by a consistent and systematic merging of the sub schemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

“…The first step in this analysis of (1) involved discretizing the system. Since the available research includes some work on discretization, such as in [6], an innovative technique has been attempted. Pott [7] identifies the subtleties of the numerator and the denominator in the standard definition of a derivative and generalizes (2) to find that: (i) the rate at which the numerator and the denominator terms reach the base point need not be uniform and (ii) any increment or decrement therein need not be strictly linear initially.…”

A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

“…κ is a physical constant which represents diffusion coefficient , Ψ accounts for all local reactions and γ is the fractional order and 0 < γ < 1.In order to calculate wave solutions, the function Ψ (u (x, t)) has to be specified; if κ = 1 and Ψ (u (x, t)) = u (x, t) (1 − u (x, t)) (u (x, t) − r) , the equation (2.1) is called as time fractional Fitzhugh-Nagumo model. Thus, time fractional initial-boundary Fitzhugh-Nagumo model [24,26,27] will be considered and it is formulated as…”

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

“…Nowadays, NSFD schemes have been widely used as a powerful and efficient class of numerical methods for solving differential equations arising in real-world situations. We refer the readers to [32][33][34][35][36][37][38][39] and [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] for good reviews and some recent notable works related to NSFD schemes, respectively. Recently, we have successfully developed the Mickens' methodology to construct NSFD schemes for differential equation models arising in real-world applications [55][56][57][58][59][60].…”