Mean and covariance equations for boundary-value problems

Differential equations are developed for the mean and covariance functions of the solution of one-dimensional boundary-value problems with white noise input. The equations represent an extension of the classical mean and covariance equations used in random vibration to analyze the response of linear systems subjected to white noise input. The differences between the classical and the proposed mean and covariance equations are caused by the incomplete specification of the state vector at the left end of the domain of integration and the dependence of the state vector at boundaries on the input. A statically determined and a statically indetermined beam subjected to white noise are used to illustrate the proposed method.