DISCRETE AND CONTINUOUS MODELS AND APPLIED COMPUTATIONAL SCIENCE

The problem of finding equilibrium configurations of one-component charged particles, induced by external electrostatic fields in planar systems, is a subject of active studies in fundamental as well in experimental investigations. In this paper the results of numerical analysis of the equilibrium configurations of charged particles (electrons), confined in a circular region by an infinite external potential at its boundary are presented. Equilibrium configurations with minimal energy are searched by means of special calculation scheme. This computational scheme consists of the following steps. First, the configuration of the system with the energy as close as possible to the expected energy value in the ground equilibrium state is found using a model of stable configurations. Next, classical Newtonian molecular dynamics is used using viscous friction to bring the system into equilibrium with a minimum energy. With a sufficient number of runs, we obtain a stable configuration with an energy value as close as possible to the global minimum energy value for the ground stable state for a given number of particles. Our results demonstrate a significant efficiency of using the method of classical molecular dynamics (MD) when using the interpolation formulas in comparison with algorithms based on Monte Carlo methods and global optimization. This approach makes it possible to significantly increase the speed at which an equilibrium configuration is reached for an arbitrarily chosen number of particles compared to the Metropolis annealing simulation algorithm and other algorithms based on global optimization methods.

For one-dimensional inhomogeneous (with respect to the spatial variable) linear parabolic equations, a combined approach is used, dividing the original problem into two subproblems. The first of them is an inhomogeneous one-dimensional Poisson problem with Dirichlet–Robin boundary conditions, the search for a solution of which is based on the Chebyshev collocation method. The method was developed based on previously published algorithms for solving ordinary differential equations, in which the solution is sought in the form of an expansion in Chebyshev polynomials of the 1st kind on Gauss–Lobatto grids, which allows the use of discrete orthogonality of polynomials. This approach turns out to be very economical and stable compared to traditional methods, which often lead to the solution of poorly defined systems of linear algebraic equations. In the described approach, the successful use of integration matrices allows complete elimination of the need to deal with ill-conditioned matrices.

The second, homogeneous problem of thermal conductivity is solved by the method of separation of variables. In this case, finding the expansion coefficients of the desired solution in the complete set of solutions to the corresponding Sturm–Liouville problem is reduced to calculating integrals of known functions. A simple technique for constructing Chebyshev interpolants of integrands allows to calculate the integrals by summing interpolation coefficients.

The analysis of trajectory dynamics and the solution of optimization problems using computer methods are relevant areas of research in dynamic population-migration models. In this paper, four-dimensional dynamic models describing the processes of competition and migration in ecosystems are studied. Firstly, we consider a modification of the “two competitors—two migration areas” model, which takes into account uniform intraspecific and interspecific competition in two populations as well as non-uniform bidirectional migration in both populations. Secondly, we consider a modification of the “two competitors—two migration areas” model, in which intraspecific competition is uniform and interspecific competition and bidirectional migration are non-uniform. For these two types of models, the study is carried out taking into account the variability of parameters. The problems of searching for model parameters based on the implementation of two optimality criteria are solved. The first criterion of optimality is associated with the fulfillment of such a condition for the coexistence of populations, which in mathematical form is the integral maximization of the functions product characterizing the populations densities. The second criterion of optimality involves checking the assumption of the such a four-dimensional positive vector existence, which will be a state of equilibrium. The algorithms developed on the basis of the first and second optimality criteria using the differential evolution method result in optimal sets of parameters for the studied population-migration models. The obtained sets of parameters are used to find positive equilibrium states and analyze trajectory dynamics. Using the method of constructing self-consistent one-step models and an automated stochastization procedure, the transition to the stochastic case is performed. The structural description and the possibility of analyzing two types of population-migration stochastic models are provided by obtaining Fokker–Planck equations and Langevin equations with corresponding coefficients. Algorithms for generating trajectories of the Wiener process, multipoint distributions and modifications of the Runge–Kutta method are used. A series of computational experiments is carried out using a specialized software package whose capabilities allow for the construction and analysis of dynamic models of high dimension, taking into account the evaluation of the stochastics influence. The trajectory dynamics of two types of population-migration models are investigated, and a comparative analysis of the results is carried out both in the deterministic and stochastic cases. The results can be used in the modeling and optimization of dynamic models in natural science.

In recent years, artificial intelligence methods have been developed in various fields, particularly in education. The development of computer systems for student learning is an important task and can significantly improve student learning. The development and implementation of deep learning methods in the educational process has gained immense popularity. The most successful among them are models that consider the multimodal nature of information, in particular the combination of text, sound, images, and video. The difficulty in processing such data is that combining multimodal input data by different channel concatenation methods that ignore the heterogeneity of different modalities is an inefficient approach. To solve this problem, an inter-channel attention module is proposed in this paper. The paper presents a computer vision-linguistic system of student learning process based on the concatenation of multimodal input data using the inter-channel attention module. It is shown that the creation of effective and flexible learning systems and technologies based on such models allows to adapt the educational process to the individual needs of students and increase its efficiency.

Our group has been investigating kinetic models for quite a long time. The structure of classical kinetic models is described by rather simple assumptions about the interaction of the entities under study. Also, the construction of kinetic equations (both stochastic and deterministic) is based on simple sequential steps. However, in each step, the researcher must manipulate a large number of elements. And once the differential equations are obtained, the problem of solving or investigating them arises. The use of symbolic-numeric approach methodology is naturally directed. When the input is an information model of the system under study, represented in some diagrammatic form. And as a result, we obtain systems of differential equations (preferably, in all possible variants). Then, as part of this process, we can investigate the resulting equations (by a variety of methods). We have previously taken several steps in this direction, but we found the results somewhat unsatisfactory. At the moment we have settled on the package Catalyst. jl, which belongs to the Julia language ecosystem. The authors of the package declare its relevance to the field of chemical kinetics. Whether it is possible to study more complex systems with this package, we cannot say. Therefore, we decided to investigate the possibility of using this package for our models to begin with standard problems of chemical kinetics. As a result, we can summarize that this package seems to us to be the best solution for the symbolic-numerical study of chemical kinetics problems.

Earlier we developed a stable fast numerical algorithm for solving ordinary differential equations of the first order. The method based on the Chebyshev collocation allows solving both initial value problems and problems with a fixed condition at an arbitrary point of the interval with equal success. The algorithm for solving the boundary value problem practically implements a single-pass analogue of the shooting method traditionally used in such cases. In this paper, we extend the developed algorithm to the class of linear ODEs of the second order. Active use of the method of integrating factors and the d’Alembert method allows us to reduce the method for solving second-order equations to a sequence of solutions of a pair of first-order equations. The general solution of the initial or boundary value problem for an inhomogeneous equation of the second order is represented as a sum of basic solutions with unknown constant coefficients. This approach ensures numerical stability, clarity, and simplicity of the algorithm.