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A second‐order, chaos‐free, explicit method for the numerical solution of a cubic reaction problem in neurophysiology
Author(s) -
Price W. G.,
Wang Yigong,
Twizell E. H.
Publication year - 1993
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690090302
Subject(s) - mathematics , chaos (operating system) , order (exchange) , mathematical analysis , computer science , finance , economics , computer security
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.