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

Nurlybek Zhumatayev (Silkway International University, Shymkent, Kazakhstan), Zhanat Umarova (South Kazakhstan State University, Shymkent, Kazakhstan), Gani Besbayev (South Kazakhstan State University, Shymkent, Kazakhstan) and Almira Zholshiyeva (Silkway International University, Shymkent, Kazakhstan)
DOI: 10.4018/IJDST.2020040105
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## Abstract

In this work, an attempt has been made to eliminate the contradiction of the Smoluchowski equation, using modern distributed algorithms for creating calculation algorithm and implementation a program for building a more perfect model by changing the type of the kinetic equation of aggregation taking into account the relaxation times. On the basis of the applied Mathcad package, there is a developed computer model for calculating the aggregation of dispersed systems. The obtained system of differential equations of the second order is solved by the Runge-Kutt method. The authors are presetting the initial conditions of the calculation. A subsequent analysis was made of the obtained non-local equations and the study of the behavior of solutions of different orders. Also, this research can be aimed at the generalization of the proposed approach for the analysis of aggregation processes in heterogeneous dispersed systems, involving the creation of aggregation models, taking into account both time and space non-locality.
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## 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:

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

• Modeling of systems with uncertain structure;

• 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.

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## The Main Part

The analysis of related works convincingly shows that on elementary examples it is possible to identify certain general qualitative features of equations with aftereffect. Also, to realize the specifics of their use in the construction of mathematical models and the subsequent analysis of specific systems in the situation of incompleteness of the initial information about the system. (Clive, 2014)

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