Numerical treatment of the nerve conduction equation

Abstract

This paper is devoted to study the numerical solution of a pair of differential equations which are of the FitzHugh–Nagumo (F–N) type. Existence and uniqueness for such system has been given in 1978 by Rauch and Smoller. Our main concern in this article is to use high ordered splines as basis for the collocation method to solve numerically such differential equations. Finite difference methods were used in 1979 by Khalifa as well as collocation with cubic and quadratic splines but higher order ones were not used because of the discontinuity in the second derivative of the solution. In this paper, we considered the problem under the same basis in three separate regions and end up with a forced parameter in the quintic splines to satisfy the solution in the first and third subregions after getting the solution in the middle region, in which we overcome the difficulty of the discontinuity in the second derivative. Numerical results obtained are good and more accurate than the previous ones.