Asymptotic analysis for time fractional FitzHugh-Nagumo equations

In the present study, our focus is on the classical and time fractional FitzHugh-Nagumo(FHN) equations. Regarding the growth rater>0 and rho is an element of(0,1)that governs theoverall dynamics of the problems, this study investigates the nonnegativity and bound-edness of the exact and numerical solutions. It also generalizes the results from theclassical FHN equation to the time fractional FHN equation in addition to indicatingwhen rho limits the solutions from above. Further, we present the theoretical and numer-ical asymptotic stability of the zero solutions with respect tor-values, in the context oftheL2-norm. Additionally, we study the unconditional long time behavior regardlessofr-values if rho bounds the initial solutions from above. In our numerical investigation,we implement the Gr & uuml;nwald-Letnikov scheme to approximate the Caputo fractionalderivative of order alpha is an element of(0,1)and the backward difference scheme for the first-orderpartial derivative operator with respect tot, together with the central finite differencemethod for spatial discretization. Moreover, we explore the nonlinear function from alinearly implicit scheme. We also examine the numerical scheme's solvability. Finally,numerical applications are performed to validate theoretical results.