A nonlinear singularly perturbed volterra functional differential equation

Abstract

We study the Cauchy problem for the nonlinear Volterra f.d.e. with infinite "memory" $(V)\left\{ {\begin{array}{*{20}c}{ - \mu y'(t) = \int_{ - \infty }^t {a(t - s){\mathbf{ }}F{\mathbf{ }}(y(t),{\mathbf{ }}y(s)){\mathbf{ }}ds{\mathbf{ }}(t > 0)} } \\{y(t) = g(t){\mathbf{ }}( - \infty < t \leqslant 0),} \\\end{array} } \right.$ where μ>0 is a small parameter, ′=d/dt, a is a given real kernel, and F, g are given real functions; (V) models the elongation ratio of a homogeneous filament of a molten polyethelene which is stretched on the time interval (−∞, 0], then released to undergo elastic recovery for t>0. Under physically reasonable assumptions concerning a, F, g we present results on qualitative behaviour of solutions of (V) and of the corresponding reduced equation when μ=0, as well as the relation between them as μ → 0⁺, both for t near zero and for large t. These results, obtained jointly with A.S. Lodge and J.B. Mc Leod, show that in general the filament of polyethelene never recovers its original length, and that the affect of the Newtonian term −μy'(t) in (V) is highly significant during the early part of the recovery, but not in the ultimate recovery. We also present an implicit finite difference scheme, recently developed by O. Nevanlinna, to obtain a discretization of (V) which exhibits most of the same qualitative properties as the continuous model.

Sponsored by the United States Army under Grant Number DAAG 29-776-0004 and Contract Number DAAG 29-75-C-0024.