The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients

Abstract

Abstract. A general class of linear self-adjoint random boundary value problems with weakly correlated coefficients is considered. The earlier result that the distribution function of the solution approaches the normal as the correlation length e tends to zero is generalized somewhat. Correction terms are derived that yield estimates for the distribution function when e is small but nonzero. The results are also applied to the eigenvalues and eigenfunctions of a corresponding class of random eigenvalue problems. The discussion is given in terms of second-order equations, but extensions to higher-order problems are readily apparent.

Linear random boundary value problems containing weakly correlated forcing functions

“…In 1966 Boyce [2] Purkert and vom Scheidt [3,4]; Boyce and Xia [5] found a similar and better results by combining the methods of [2] with perturbation and Chebyshev-Hermite polynomial expansion.…”

“…We introduce a new function pl (vl,v2) In order to obtain the asptotic approximation for Px(X), we need the following facts [see [5] we can use the similar way to obtain the conclusion.…”

The density function of the solution of a two-point boundary value problem containing small stochastic processes

“…For random initial value problems many important and interesting facts have been discovered; see [1,3,4,5,8], for example. However, for random boundary value problems only a relatively few results are known [2,6,9], In this paper we will consider the problem l=t(t,x(tU(t,co)), (1-la) Ax(0) + Bx(l) = <*(&>), (1-lb) where a(co) and £(t, co) are m x 1 random vectors defined on an underlying probability space (£2, 7, P), and \(t) is an unknown m x 1 random vector. We will assume that £(t,co) -£o(0 + e£i ('.…”

“…In the case that only the density function of x(?0) is concerned, then z(0 = Ax(r) + Bx(f), (1)(2)(3) where T = (1 -t)to/{\ -to) and 0 < ?0 < 1. It is easy to see that z(t) has the properties…”

“…If the density function of z(t) exists and is denoted by p(z, t), then p(z, 1) = pa{z), (1)(2)(3)(4)(5)(6) and…”