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 rate r>0 and ρ∈(0,1) that governs the overall dynamics of the problems, this study investigates the nonnegativity and boundedness of the exact and numerical solutions. It also generalizes the results from the classical FHN equation to the time fractional FHN equation in addition to indicating when ρ limits the solutions from above. Further, we present the theoretical and numerical asymptotic stability of the zero solutions with respect to r-values, in the context of the L2-norm. Additionally, we study the unconditional long time behavior regardless of r-values if ρ bounds the initial solutions from above. In our numerical investigation, we implement the Grünwald-Letnikov scheme to approximate the Caputo fractional derivative of order α∈(0,1) and the backward difference scheme for the first-order partial derivative operator with respect to t, together with the central finite difference method for spatial discretization. Moreover, we explore the nonlinear function from a linearly implicit scheme. We also examine the numerical scheme’s solvability. Finally, numerical applications are performed to validate theoretical results.