Computational aspects of some autonomous differential equations

Abstract

This paper deals principally with the differential equation ϵ d 4 y / d x 4 + d 2 y / d x 2 = y - y 2 (0 ≤ x < ∞) Subject to the conditions d y (0) / d x = 0, d y / d x < 0 (0 < x < ∞), lim x→∞ y ( x ) = 0. where ϵ > 0 is a prescribed constant. The equation has served as a model for water waves with surface tension. Interest centres on the behaviour of d 3 y / d x 3 at the point x = 0; and we prove that this quantity is strictly positive and investigate its numerical behaviour as a function of ϵ . A summary of principal formulae appears in §4 at the end of the paper, and table 1 gives numerical results. The introductory section of the paper speculates on whether our method might have wider application to similar autonomous differential equations, which involve a small positive multiple of the highest derivative.