A second‐order, chaos‐free, explicit method for the numerical solution of a cubic reaction problem in neurophysiology

Abstract

AbstractA second order explicit method is developed for the numerical solution of the initialvalue problem w′(t) ≡ dw(t)/dt = ϕ(w), t > 0, w(0) = W0, in which the function ϕ(w) = αw(1 − w) (w − a), with α and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first‐order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. Soc. (London) A 430, 541–576 (1990)]. In addition to being chaos free and of higher order, the method is seen to converge to one of the correct steady‐state solutions at w = 0 or w = 1 for any positive value of α. Convergence is monotonic or oscillatory depending on W0, α, a, and l, the parameter in the discretization of the independent variable t. The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial‐value boundary‐value problem ∂u/∂t = κ∂2u/∂x2 + αu(1 − u)(u − a) in which u = u(x,t). The numerical method so developed obtained the solution by solving a single linear algebraic system at each time step. © 1993 John Wiley & Sons, Inc.