Gaussian waves in the Fitzhugh–Nagumo equation demonstrate one role of the auxiliary function H(x, t) in the homotopy analysis method

We apply the method of homotopy analysis to study the Fitzhugh–Nagumo equation. u t = u xx + u ( u - α ) ( 1 - u ) , which has various applications in the fields of logistic population growth, flame propagation, neurophysiology, autocatalytic chemical reaction, branching Brownian motion process and nuclear reactor theory. In particular, we focus on the case of Gaussian wave forms, which has not previously been discussed in the literature. Through an application of homotopy analysis, we are able to demonstrate an interesting role of the auxiliary function H(x, t). In particular, we show that while we are free to pick specific forms of H(x, t), there are certain forms which permit both (i) regularity of solutions by cancelling possible singularities and (ii) ease the process of integrating when computing the higher order terms in the homotopy expansion. In addition to the discussion on the choice of H(x, t), we are able to deduce the behavior of Gaussian waveform solutions in the Fitzhugh–Nagumo model for various values of both the wave speed and the ambient concentration parameter (α).