Nonlinear Volterra integrodifferential systems with L1-kernels

Grossman, Stanley I.; Miller, Richard K.

doi:10.1016/0022-0396(73)90011-9

Cited by 91 publications

(34 citation statements)

References 5 publications

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“…The behaviour of the perturbed solution for t > to determines the stability of the unperturbed solution: If the perturbed solution remains bounded (or tends to zero) then the unperturbed solution is (asymptotically) stable. The definitions of stability of solutions of integrodifferential equations used here are similar to the usual definitions for ordinary differential equations (see [5], [6] and [7]). …”

Section: T Ih() -H(w(t -R))le-(t-t)dr~it Li(r T R) -W(r)(t -R)lementioning

confidence: 99%

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

Aarts

Ven

1995

J Eng Math

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The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe ('spurt') and hysteresis. Our theoretical results are confirmed by numerical calculations of the velocity gradient.

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Section: T Ih() -H(w(t -R))le-(t-t)dr~it Li(r T R) -W(r)(t -R)lementioning

confidence: 99%

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

Aarts

Ven

1995

J Eng Math

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“…Results of S. I. Grossman and R. K. Miller [2] (for a E Lx(0, oo)) and of D. F. Shea and S. Wainger [7] show that u, u' E Lx(0, oo) if a has no representation (1.3) satisfying (1.4) and (1.5). Here we establish that Jo M0|+|«i(0|<ft<°o (1.7) in the asymptotically periodic case if I a(t) dt <oo, (1.8)…”

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confidence: 99%

An 𝐿¹ remainder theorem for an integrodifferential equation with asymptotically periodic solution

Hannsgen¹

1979

Proc. Amer. Math. Soc.

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“…Among previous studies of stability theory for integrodifferential equations, involving the resolvent formula, we mention those of S. I. Grossman and R. K. Miller [1], [2] and of…”

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confidence: 99%