Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions. (English) Zbl 1482.65153
Summary: We investigate the impact of spatial-temporal discretisation schemes on the dynamics of a class of reaction-diffusion equations that includes the FitzHugh-Nagumo system. For the temporal discretisation we consider the family of six backward differential formula (BDF) methods, which includes the well-known backward-Euler scheme. The spatial discretisations can feature infinite-range interactions, allowing us to consider neural field models. We construct travelling wave solutions to these fully discrete systems in the small time-step limit by viewing them as singular perturbations of the corresponding spatially discrete system. In particular, we refine the previous approach by Hupkes and Van Vleck for scalar fully discretised systems, which is based on a spectral convergence technique that was developed by Bates, Chen and Chmaj.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35C07 | Traveling wave solutions |
| 92C20 | Neural biology |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Keywords:
travelling waves; FitzHugh-Nagumo system; singular perturbation; spatial-temporal discretisationReferences:
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