The control of an elastic fluid

Bateman, H.

doi:10.1090/s0002-9904-1945-08413-4

Cited by 23 publications

(12 citation statements)

References 9 publications

(10 reference statements)

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“…Bateman reviews the history of the subject in sec. 3.1 of [35], though he is unaware of the early work of Willis and Airy. Recently, Atiyah and Moore have speculated on the possible use of time-shifted equations of motion in relativistic field theories [36].…”

Section: A Forced Resonancementioning

confidence: 99%

Self-oscillation

2013

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Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain dynamical systems that gives rise to a great variety of vibrations, both useful and destructive. In a self-oscillator, the driving force is controlled by the oscillation itself so that it acts in phase with the velocity, causing a negative damping that feeds energy into the vibration: no external rate needs to be adjusted to the resonant frequency. The famous collapse of the Tacoma Narrows bridge in 1940, often attributed by introductory physics texts to forced resonance, was actually a self-oscillation, as was the swaying of the London Millennium Footbridge in 2000. Clocks are self-oscillators, as are bowed and wind musical instruments. The heart is a "relaxation oscillator," i.e., a non-sinusoidal self-oscillator whose period is determined by sudden, nonlinear switching at thresholds. We review the general criterion that determines whether a linear system can self-oscillate. We then describe the limiting cycles of the simplest nonlinear self-oscillators, as well as the ability of two or more coupled self-oscillators to become spontaneously synchronized ("entrained"). We characterize the operation of motors as self-oscillation and prove a theorem about their limit efficiency, of which Carnot's theorem for heat engines appears as a special case. We briefly discuss how self-oscillation applies to servomechanisms, Cepheid variable stars, lasers, and the macroeconomic business cycle, among other applications. Our emphasis throughout is on the energetics of self-oscillation, often neglected by the literature on nonlinear dynamical systems.Comment: 68 pages, 33 figures. v4: Typos fixed and other minor adjustments. To appear in Physics Report

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Section: A Forced Resonancementioning

confidence: 99%

Self-oscillation

2013

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“…Assuming

\sup_{s \in false[0, t false]} false| f^{'} false(s false) false| < \infty

for all t >0 and zero initial conditions, we obtain the following explicit representation of the solution

u (n, t) = - \int_{0}^{t} (g_{β - 1} * G_{1, β}) (n, t - s) f^{'} (s) d s,

where 1< β ≤ 2. In the bordeline case β =2, we get after integration by parts

u (n, t) = \int_{0}^{t} J_{2 n} (2 (t - s)) f (s) d s,

so that we recover the explicit solution of proposed by Bateman, p.644 and give new insight on the representation in the super diffusive case.…”

Section: Special Casesmentioning

confidence: 53%

“…Example The partial differential/difference equation

u_{t t} false(n, t false) = u false(n + 1, t false) - 2 u false(n, t false) + u false(n - 1, t false) + g false(n, t false), n \in double-struckZ, t > 0

was studied by Bateman, Section 4.9 with forcing term g ( n , t )=− δ 0 ( n ) f ′ ( t ), where

f \in W^{1, q} false(R_{+} false), 1 < q < \infty

, and

δ_{0} false(n false) = ({, \begin{matrix} left left \\ array 1, & array n = 0, \\ array 0 & array otherwise. \end{matrix})

Physically, this means that the motion of the particle labeled 0 is forced while the motion of the other particles is free. We consider their super diffusive version, ie, Equation with g as just described.…”

Section: Special Casesmentioning

confidence: 99%

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

González-Camus

Keyantuo

Lizama

et al. 2019

Math Methods in App Sciences

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We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem double-struckDtβufalse(n,tfalse)=−false(−Δdfalse)αufalse(n,tfalse)+gfalse(n,tfalse), where 0<β ≤ 2, 0<α ≤ 1, n∈double-struckZ, (−Δd)α is the discrete fractional Laplacian, and double-struckDtβ is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.

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“…The first term is the input augmentation, or "feedback," which is of course absent from eq. (2)(3)(4)(5). This is the reciprocal of the familiar RF/RG gain ap proximation for this circuit, with Rp = 1/G.…”

Section: Anticausal Analysis 1351mentioning

confidence: 99%