×

Existence of traveling wavefront solutions for the discrete Nagumo equation. (English) Zbl 0752.34007

It is known that the Nagumo equation \(\partial u/\partial t+D\partial^ 2 u/\partial x^ 2+f(u)=0\) has a so-called travelling wave front. This means that there exists a function \(U\) such that \(U(-\infty)=0\), \(U(\infty)=1\) and that \(u(x,t)=U(x/\sqrt D+ct)\), \(c>0\) 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 \(\dot u_ n=d(u_{n-1}- 2u_ n+u_{n+1})+f(u_ n)\), \(n\in\mathbb{Z}\), \(d>0\). 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 \(U(-\infty)=0\), \(U(\infty)=1\), \(U(x)>0,\) \(\forall x\in\mathbb{R}\), 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:

34A35 Ordinary differential equations of infinite order
34A45 Theoretical approximation of solutions to ordinary differential equations
35K57 Reaction-diffusion equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, Siam Rev., 18, No. 4 (1976) · Zbl 0345.47044
[2] Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve propagation, (Lecture Notes in Mathematics, Vol. 446 (1975), Springer-Verlag: Springer-Verlag Berlin), 5-49 · Zbl 0325.35050
[3] Bell, J., Some threshold results for models of myelinated nerves, Math. Biosc., 181-190 (1981) · Zbl 0454.92009
[4] Bell, J.; Cosner, C., Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42, 1-14 (1984) · Zbl 0536.34050
[5] Britton, N. F., Traveling wave front solutions of a differential-difference equation arising in the modelling of myelinated nerve axon, (Sleeman, B.; Jarvis, R., Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1151 (1984), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0574.92012
[6] Chi, H.; Bell, J.; Hassard, B., Numerical solution of a nonlinear advances-delay-differential equation from nerve conduction theory, J. Math. Biol., 24, 583-601 (1986) · Zbl 0597.92009
[7] Fife, P. C.; McLeod, J. B., The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65, 333-361 (1977) · Zbl 0361.35035
[8] Hugh, R. Fritz, Impulse and physiological states in models of nerve membrane, Biophys. J., 1, 445-466 (1961)
[9] McKean, H. K., Nagumo’s equation, Adv. in Math., 4, 209-223 (1970) · Zbl 0202.16203
[10] Nagumo, J.; Ariomot, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, (Proc. IRE, 50 (1962)), 2061-2070
[11] Keener, J. P., Propagation and its failure in the discrete Nagumo equation, (Proceedings Conf. Ordinary and Partial Differential Equations. Proceedings Conf. Ordinary and Partial Differential Equations, Dundee (1986)) · Zbl 0939.34049
[12] Keener, J. P., Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47, 556-572 (1987) · Zbl 0649.34019
[13] Keener, J. P., On the formation of circulation patterns of excitation in anisotropic excitable media, J. Math. Biol., 26, 41-57 (1988) · Zbl 0631.92004
[14] Scott, A. C., Active and Nonlinear Wave Propagation in electronics (1970), Wiley-Interscience: Wiley-Interscience New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.