Mohamed El-Beltagy

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.

In the current work, the stochastic point reactor model is analyzed using the Wiener-Hermite expansion (WHE). The simplified stochastic point reactor model (Ayyoubzadeh and Vosoughi, 2014), at which no matrix square root, is considered. The stochastic system is reduced to a set of deterministic equations that are solved to get the mean and variance of the neutron and precursor populations. The well-known numerical deterministic techniques are used to get the solution without the need for the time-consuming sampling techniques. Estimations of the neutron and precursor groups fluctuations at the startup are quantified. Many cases are tested and compared with the results in the literature. The current technique is shown to be efficient, accurate and simple compared with the available techniques.