The Method of Moments for Linear Random Boundary Value Problems
Melvin D. Lax · SIAM Journal on Applied Mathematics · 1976
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.