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A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology

Published online by Cambridge University Press:  14 November 2011

A. S. Lodge ,
J. B. McLeod  and
J. A. Nohel
A. S. Lodge
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
J. B. McLeod
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
J. A. Nohel
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, U.S.A.
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Synopsis

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 99 - 137
Copyright
Copyright © Royal Society of Edinburgh 1978

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