In this paper, a stochastic method of characteristics (MOC) solver is developed based on polynomial chaos expansion (PCE) to quantify the uncertainty in water-hammer equations describing transient flow in a simple reservoir-pipeline-valve system. The randomness is considered due to boundary and initial conditions. The Galerkin scheme is used for the projection of equations onto the stochastic dimension, and the governing equations are solved for the expansion coefficients. These coefficients are then used to reconstruct the mean solution of pressure wave perturbation as a result of valve closure, in addition to the calculation of other higher-order statistical moments. The computed results are in excellent agreement with those calculated by using the traditional MOC over a wide range of system parameters including steady and unsteady friction. The stochastic solution has the advantage of being robust and more efficient than other nonintrusive methods, such as Monte Carlo simulations (MCS).

On the basis of Awad sup-entropy, the efficiency function for type-I censored sample from the Weibull distribution is numerically introduced. The properties of the derived efficiency are discussed. Furthermore, for a given efficiency, the termination time of the experiment, and the maximum likelihood estimates for the Weibull parameters, are proposed. Simulation results are tabulated and discussed. Censored and complete samples are compared for a wide range of the efficiency. The comparisons show the quality of the developed algorithms and the effectiveness of using censoring in estimating with the Weibull distribution.

The population models allow for a better understanding of the dynamical interactions with the environment and hence can provide a way for understanding the population changes. They are helpful in studying the biological invasions, environmental conservation and many other applications. These models become more complicated when accounting for the stochastic and/or random variations due to different sources. In the current work, a spectral technique is suggested to analyze the stochastic population model with random parameters. The model contains mixed sources of uncertainties, noise and uncertain parameters. The suggested algorithm uses the spectral decompositions for both types of randomness. The spectral techniques have the advantages of high rates of convergence. A deterministic system is derived using the statistical properties of the random bases. The classical analytical and/or numerical techniques can be used to analyze the deterministic system and obtain the solution statistics. The technique presented in the current work is applicable to many complex systems with both stochastic and random parameters. It has the advantage of separating the contributions due to different sources of uncertainty. Hence, the sensitivity index of any uncertain parameter can be evaluated. This is a clear advantage compared with other techniques used in the literature.