Random Boundary Value Research Articles

Linear random boundary value problems containing weakly correlated forcing functions

This paper concerns linear random boundary value problems that contain random variables In the boundary conditions and weakly correlated processes in the differential equations. When the correlation length ϵ is small the structure of the solution is pointed out, and the formulas for the density function of the solutions are derived. The discussion is given in terms of second order equations, but extensions to higher order problems are readily apparent.

Upper bounds for the means of eigenvalues of random boundary value problems with weakly correlated coefficients

This paper concerns eigenvalue problems for second-order random differential equations with weakly correlated coefficients. The random problem and the mean (deterministic) problem are embedded in a parametrized problem whose eigenvalues are expanded in a power series in the parameter. This expansion leads, via the variational characterization of the eigenvalues, to computationally accessible upper bounds for the mean values of the eigenvalues of the original problem.

Two-point boundary value problems containing a finite number of random variables

This paper concerns linear and nonlinear nth order boundary value problems that contain a finite number of random variables in the boundary conditions or in the differential equation. The results extend methods previously known for corresponding initial value problems. Numerically implementable procedures are given for the determination of the joint density of the solution at an arbitrary point. The possible use of Liouvilleapos;s equation to reduce a random boundary value problem to a random initial value problem is also indicated.

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

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 ϵ \epsilon tends to zero is generalized somewhat. Correction terms are derived that yield estimates for the distribution function when ϵ \epsilon 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.

On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems

It has been known for several years that the expected value ⟨ λ 1 ⟩ \left \langle {{\lambda _1}} \right \rangle of the smallest eigenvalue of a self-adjoint positive definite random Sturm-Liouville boundary value problem satisfies the relation ⟨ λ 1 ⟩ ≤ μ 1 \left \langle {{\lambda _1}} \right \rangle \le {\mu _1} where μ 1 {\mu _1} is the smallest eigenvalue of the corresponding deterministic problem obtained by replacing each random coefficient by its mean. It has been an open question whether similar inequalities are valid for the higher eigenvalues. The answer is negative, as shown by the counterexample given in this note.

Numerical solution of a class of random boundary value problems

This paper deals with the nonlinear two point boundary value problem y″ = f( x, y, y′, R 1,…, R n ), x 0 < x < x f S 1 y( x 0) + S 2 y′( x 0) = S 3, S 4 y( x f ) + S 5 y′( x f ) = S 6 where R 1,…, R n , S 1,…, S 6 are bounded continuous random variables. An approximate probability distribution function for y( x) is constructed by numerical integration of a set of related deterministic problems. Two distinct methods are described, and in each case convergence of the approximate distribution function to the actual distribution function is established. Primary attention is placed on problems with two random variables, but various generalizations are noted. As an example, a nonlinear one-dimensional heat conduction problem containing one or two random variables is studied in some detail.

The Method of Moments for Linear Random Boundary Value Problems

The method of moments iterative approach is applied to find the mean and autocorrelation of the solution to a random boundary value problem of the form \[ Y^{( r )} + Q_1 ( t )Y^{( {r / 2} )} + Q_2 ( t )Y^{( {r / 2 - 1} )} + \cdots + Q_{r / 2 + 1} ( t )Y = F( t ) \]\[ Y( 0 ) = Y'( 0 ) = \cdots = Y^{( {r / 2 - 1} )} ( 0 ) = 0 \]\[ Y( 1 ) = Y'( 1 ) = \cdots = Y^{( {r / 2 - 1} )} ( 1 ) = 0 \] where r is even and $Q_1 , \cdots ,Q_{r / 2 + 1} $, F belong to a class of stochastic processes defined herein as “discretizable processes”. Examples are given to illustrate the practical application of the theory for a second order random differential equation.

On a class of biharmonic random boundary value problems

On a class of biharmonic random boundary value problems

The vibrations of a random elastic string: the method of integral equations

The theory of Fredholm integral equations is applied to the problem of determining the natural frequencies of transverse vibrations of a tightly stretched elastic string whose mass per unit length varies with position in a stationary random manner. Upper and lower bounds for the statistical moments of the frequencies are given in terms of corresponding moments and appropriate correlation functions for the random linear density. The adequacy of the bounds decreases for the higher frequencies. Extensions to more general random boundary value problems are also indicated.