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

A 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) = W 0 , in which the function ϕ( w ) = α w (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 W 0 , 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∂ 2 u /∂ x 2 + α u (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.