The application of truncated hierarchy techniques in the solution of a stochastic linear differential equation

Richardson, Jonathan M.

doi:10.1090/psapm/016/0193684

Cited by 24 publications

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“…It is known [1,2,3,6,7], that for the moments of the solution to (1) we have an infinite chain of equations. To close this chain, we consider the so called closure problem.…”

Section: Consider the Following Linear Ordinary Equation In R D Dx (1mentioning

confidence: 99%

“…To close this chain, we consider the so called closure problem. Several methods of closure were proposed (see, for example, [6,8]) but only few of them are mathematically justified.…”

Section: Consider the Following Linear Ordinary Equation In R D Dx (1mentioning

confidence: 99%

“…where the operator F and g are determined by the right hand side of (6). Let c := sup |C(i)| and A > -a.…”

Section: T E [0t\ K Tmentioning

confidence: 99%

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A Closure Method for Randomly Perturbed Linear Systems

Bobryk¹,

Stettener²

2001

Demonstratio Mathematica

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“…It is known [1,2,3,6,7], that for the moments of the solution to (1) we have an infinite chain of equations. To close this chain, we consider the so called closure problem.…”

Section: Consider the Following Linear Ordinary Equation In R D Dx (1mentioning

confidence: 99%

“…To close this chain, we consider the so called closure problem. Several methods of closure were proposed (see, for example, [6,8]) but only few of them are mathematically justified.…”

Section: Consider the Following Linear Ordinary Equation In R D Dx (1mentioning

confidence: 99%

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A Closure Method for Randomly Perturbed Linear Systems

Bobryk¹,

Stettener²

2001

Demonstratio Mathematica

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“…The differential equation which it must satisfy is EI<^ + P^ + KX)U = ° (2"1} where x is the axial coordinate, E is Young's modulus, and I is beam moment of inertia. Let the mean stiffness of the elastic foundation be K, that is </(*)> = K (2.2) where the angular bracket is the average over the length of the beam, namely [3], [4], in random initial value problems [5],2 and more recently in an eigenvalue problem [6]. The method consists, in general, of multiplying the stochastic differential equation an arbitrary number of times by the stochastic function or by the dependent variable evaluated at different values of its argument.…”

mentioning

confidence: 99%

“…sRichardson [5] discusses various related truncated hierarchy methods and cites references to other applications of these techniques. [Vol.…”

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confidence: 99%

Buckling under axial compression of long cylindrical shells with random axisymmetric imperfections

Amazigo¹

1969

Quart. Appl. Math.

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Abstract.The buckling of long cylinders with homogeneous random axisymmetric geometric imperfections under uniform axial compression is studied by means of a modified truncated hierarchy technique. It is found that the buckling load of the cylinder depends only on the spectral density of the random imperfections. In particular, for small values of the standard deviation of the axisymmetric imperfection the buckling load depends only on the value of the spectral density at a specific wave number.1. Introduction. It is generally known that the buckling strengths of some structures are greatly reduced by the presence of small imperfections. However, the number of quantitative studies is limited. Furthermore, although it is recognized that these imperfections-geometric or otherwise-are generally random, very few investigators have used a statistical approach to the problem. Then it becomes routine to calculate the probability density of X* if a joint probability density is assigned to . In general, the analytic determination of the relation (1.1) is difficult if not impossible.Fraser calculates the buckling load of infinitely long imperfect beams on nonlinear foundations by assuming that the imperfections are homogeneous random functions and hence are characterized by their mean and autocorrelation. By means of the method of equivalent linearization he obtains the dependence of the buckling load on the spectral density of the imperfection.Like Fraser we assume that the cylinder imperfections are homogeneous random functions of the axial coordinate and hence are characterized by their mean and autocorrelation. The buckling of infinitely long imperfect cylindrical shells under axial compression is analyzed on the basis of a modified truncated hierarchy method.A model problem (the buckling of axially compressed infinite beams on linear stochastic foundations) is used to assess the validity of this technique. This problem is governed by a differential equation which is similar to the Karman-Donnell equations for cylinders when the latter equations have been linearized for investigating buckling phenomenon.

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Response of a randomly inhomogeneous layer overlying a homogeneous half‐space to surface harmonic excitation

Hryniewicz

Hermans

1988

Earthq Engng Struct Dyn

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SUMMARYThe paper is concerned with the study of the effect of the randomness of material parameters on mean wave propagation in a semi-infinite viscoelastic medium. The medium considered in this paper is a configuration of a randomly nonhomogeneous layer overlying a homogeneous half-space and is loaded harmonically on the top surface. The method used is that of Karal and Keller and is based on the idea of fundamental matrix and Bourret approximation. The integrodifferential equation obtained in this paper is solved by the Laplace transform method. By using boundary and continuity conditions the mean wave solution is obtained. Numerical results show that the correlation functions, which introduce long-range interactions, can be the source of the wave amplification.

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The application of truncated hierarchy techniques in the solution of a stochastic linear differential equation

Cited by 24 publications

References 0 publications

A Closure Method for Randomly Perturbed Linear Systems

A Closure Method for Randomly Perturbed Linear Systems

Buckling under axial compression of long cylindrical shells with random axisymmetric imperfections

Response of a randomly inhomogeneous layer overlying a homogeneous half‐space to surface harmonic excitation

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