Analysis of approximate solutions for neural signal propagation via the fractional Fitzhugh-Nagumo model

The Fitzhugh- Nagumo model (FNM) is essential for explaining how electrical signals travel through excitable material, such as nerve fibers, and how impulses are transmitted from the nerves. So, the key purpose of this article is to investigate a nonlinear transmission of the fractional order FNM using a computationally efficient analytical approach named the modified generalized Mittag-Leffler function method (MGMLFM). A more realistic formulation of the FNM with memory effects and non-local behavior has been obtained by generalizing it using the Caputo fractional operator. Furthermore, we study special cases derived from this fractional FNM that lead to other famous equations such as the fractional Newell-Whitehead model (NWM) and the fractional Zeldovich model (ZM). The MGMLFM approach to solving general fractional partial differential equations (FPDEs) is described. Additionally, the convergence and error analysis for this method are demonstrated. We illustrate the behavior of the approximate solution using graphical representations for varying values of the fractional operator $\alpha$ which converges to the exact solution when $\alpha=1$. Additionally, the MGMLFM approximate values are contrasted with the known precise values in some tables which are exactly consistent with it when $\alpha=1$, as well as, we compare the estimated absolute error from MGMLFM with other published methods under the same circumstances which is found to be much lower than comparable methods. The findings show the MGMLFM's effectiveness and advantages, which include its easily calculable components, direct implementation of the problems, satisfactory approximate solutions, small absolute error, and no need for linearization, perturbation, or transformations. So, the MGMLFM is a useful instrument for determining the outcomes of any more nonlinear problems that may arise in science and engineering.