Abstract
Comparisons are made between three different methods for computing the stable solitary wave solution for the FitzHugh-Nagumo equations which consist of a nonlinear diffusion equation coupled to an ordinary differential equation in time. They model the Hodgkin-Huxley equations which describe the propagation of the nerve impulse down the axon. Two of the methods involve the travelling wave equations. Previous accurate numerical computations of these equations as an initial-value problem using a shooting method lead to inaccurate values for the wave speed; however, nonlinear corrections to the initial values are shown to yield accurate values. A boundary-value method applies “asymptotic boundary conditions” and uses a spline-collocation code called COLSYS for numerical solution of boundary-value problems which leads to accurate wave profiles and speeds. The third method is to solve an initial-boundary-value problem with an adaptive outgoing wave condition for the partial differential equations where the solitary wave emerges as the stable long time solution. The concept of a “wave integral” is introduced and they are derived to determine the wave speed used in the adaptive boundary condition and to measure the closeness of the computed solutions to the exact solitary wave solution.
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This work was supported in part by the Natural Sciences and Engineering Research Council Canada under Grant A4559 and by the John Simon Guggenheim Memorial Foundation
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Miura, R.M. Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations. J. Math. Biology 13, 247–269 (1982). https://doi.org/10.1007/BF00276063
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DOI: https://doi.org/10.1007/BF00276063