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Existence of traveling wavefront solutions for the discrete Nagumo equation. (English) Zbl 0752.34007
It is known that the Nagumo equation u/t+D 2 u/x 2 +f(u)=0 has a so-called travelling wave front. This means that there exists a function U such that U(-)=0, U()=1 and that u(x,t)=U(x/D+ct), c>0 is a solution. In this paper so-called “discrete Nagumo equation” is considered. In fact it is an infinite system of ODE’s of the form u ˙ n =d(u n-1 -2u n +u n+1 )+f(u n ), n, d>0. The author proves (in a rigorous way) that under certain conditions on f a similar travelling wave front exists also for the discrete case. Namely it is proved that there exists a function U, satisfying the conditions U(-)=0, U()=1, U(x)>0, x, and such that u n (t)=U(n+ct), c>0, is a solution of the discrete Nagumo equation, provided that d is large enough. It is to be stressed that the proof given here has a clear approximational aspect.
MSC:
34A35ODE of infinite order
34A45Theoretical approximation of solutions of ODE
35K57Reaction-diffusion equations
65M06Finite difference methods (IVP of PDE)