Development and Calculation of a Computer Model and Modern Distributed Algorithms for Dispersed Systems Aggregation: Modern Distributed Algorithms

Introduction

Under the conditions of incomplete definiteness, more attention is paid to modeling and researching systems. The reason for the incomplete certainty is the lack of reliable information about the value of the object of modeling. This information can be predetermined by many factors. The degree of incomplete certainty depends on the methods of obtaining information and its reliability. It is customary to call the incompletely determined data, regardless of degree, simply uncertain. The functioning of a variety of systems with incompletely defined parameters, structure, algorithm, or influential quantities is called functioning under conditions of uncertainty.

Mathematical modeling and computer research of various technogenic and natural processes is associated with ambiguities of a twofold kind. This is, firstly, the incompleteness of ideas about the essence of the process and its mechanisms. Secondly, the incompleteness of the initial information to calculate the dynamics of the process and the evolution of the system. Both factors influence the structure of the model and the reliability of the analysis.

The main directions of modeling systems in conditions of uncertainty:

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  Modeling of systems with uncertain parameters (operations on undefined operands);

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  Modeling of systems with uncertain structure;

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  Modeling of decision making under uncertainty conditions (undefined algorithms).

The necessity of taking into account the relaxation phenomena in deriving the kinetic equations of aggregation has been observed by many researchers (Arnold, 1992; Arnold, 1981; Bastin & Dochain, 1990; Cluade, 2009). Indeed, a certain internal contradiction, inherent in the known kinetic equations of aggregation, is that the rate of evolution of the concentration of clusters of a certain order is assumed to be dependent on the concentrations of clusters of lower orders at the same time moment.

Here and further could be talking about the time non-locality, that is, the delay of the process, with taking into account the hierarchy of relaxation times. The choice of the model equation in this case is a technical question. Here can use other models. More principal is that the application of the methodology of relaxation transfer nuclei (Christof, Harald, & Peter, 2017) for modification of kinetic equations is more formal than for transfer equations. This approach is, of course, controversial.

Another aspect of the problem is related to the fact that the Smoluchowski kinetic equation is written for the medium, which is assumed to be absolutely homogeneous with respect to the volume concentration of clusters of different orders. The same assumption is accepted by authors in the new model.

The Main Part