The Equations for Large Vibrations of Strings

Antman, Stuart S.

doi:10.2307/2321203

Cited by 60 publications

(21 citation statements)

References 9 publications

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“…We did not study this problem. Detailed computations related to it and concerning the nonlinear beam model can be found in [6]. In solving eq.…”

Section: Resultsmentioning

confidence: 99%

Asymptotic models for a long, elastic cylinder

Karwowski

1990

J Elasticity

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We study asymptotic expansions for the displacement field of a long elastic cyfinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.

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“…We did not study this problem. Detailed computations related to it and concerning the nonlinear beam model can be found in [6]. In solving eq.…”

Section: Resultsmentioning

confidence: 99%

Asymptotic models for a long, elastic cylinder

Karwowski

1990

J Elasticity

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“…The problem was posed explicitly for the first time by James Bernoulli in 1690, when he asked to find the curve assumed by a flexible chain which hangs freely, fixed between two points [8][9][10][11].…”

Section: Solution Based On Newton's Lawsmentioning

confidence: 99%

Interweaving the Principle of Least Potential Energy in School and Introductory University Physics Courses

2017

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Abstract:Understanding advanced physical phenomena such as vertically hanging elastic column, soap bubbles, crystals and cracks demands expressing and manipulating a system's potential energy under equilibrium conditions. However, students at schools and universities are usually required to consider the forces acting on a system under equilibrium conditions, rather than taking into account its potential energy. As a result, they find it difficult to express the system's potential energy and use it for calculations when they do need to do so. The principle of least potential energy is a powerful idea for solving static equilibrium physics problems in various fields such as hydrostatics, mechanics, and electrostatics. In the current essay, the authors describe this principle and provide examples where students can apply it. For each problem, the authors provide both the force consideration solution approach and the energy consideration solution approach.

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“…is a weak solution to the Cauchy problem for the parabolic equation (2) with initial data u(s, x) = ϕ(x). In fact, G is nothing but the transition density of the symmetric (strong) Feller process associated with the Dirichlet form E (cf., e.g., [18,27,28]) or equivalently generated by A via martingale problem (cf., e.g., [31]).…”

Section: Symmetric Integro-differential Operatorsmentioning

confidence: 99%

“…An introductory account of integro-differential scalar conservation laws can be found in [2,30]. Recently, there has been increasing interest in studying integro-differential conservation laws of nonlocal type involving generators of Lévy type (see, e.g., [6][7][8][9][10][11][12] and references therein) as well as in studying white noise perturbations of Burgers type nonlinear PDEs with random initial data, see, e.g., [4,5,19,33,37,38] and references therein, or white noise driven stochastic Burgers and fractal Burgers equations, respectively, in [34,35], where the initial problem for both the stochastic Burgers equation and stochastic fractal Burgers equation with Lévy space-time white noise is examined in the mild formulation.…”

Section: Introductionmentioning

confidence: 99%

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

Truman

2006

Journal of Functional Analysis

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In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integrodifferential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:where A is an integro-differential operator associated with a symmetric, nonlocal, regular Dirichlet form, and F t,x stands for a Lévy space-time white noise. The problem is interpreted as a stochastic integral equation of jump type involving certain convolution kernels. Existence of a unique local (in time) L 2 (R)-valued solution is obtained.

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The Equations for Large Vibrations of Strings

Cited by 60 publications

References 9 publications

Asymptotic models for a long, elastic cylinder

Asymptotic models for a long, elastic cylinder

Interweaving the Principle of Least Potential Energy in School and Introductory University Physics Courses

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

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