Jalalabad State University

This is an example of a paper that will provide a numerical simulation study on analytical solutions for optimal control of solar heat supply systems (SHSS). It will develop and solve a mathematical model for the “solar collector - storage tank” subsystem using a finite element technique and standard k-epsilon turbulence closure, derive an optimal pump speed control law for maximizing heat delivery to the storage tank, through a collection of output heat functions demonstrated as quadratic with respect to flow rate G opt for each instantaneous solar insolation level J, continuously determine G opt through algorithm methodology based on microcontroller maximum heat exchange outlet temperature and non-volatile “J-G” maps in memory for rapid response to non-stationary irradiance conditions. The simulation results demonstrated that the proposed strategy can achieve a seasonal thermal efficiency increase of 11-22 percentage points compared to conventional ON/OFF differential temperature control while increasing the pressure drop only 10.3%. In addition, the thermal-enhancement of peak Nusselt number (18.9%) and Performance Evaluation Criterion (1.15) indicates that there is a net thermohydraulic benefit to implementing the proposed strategies. The results of the study quantitively identify tuning of controllers and sizing of heat exchangers in residential and commercial solar installations.

This paper considers a system of singularly perturbed equations consisting of four first-order equations in a certain domain of the complex plane. The matrix function of the leading approximation has four distinct eigenvalues forming two pairs of complex conjugates. One pair has simple poles, while the other has double poles. The problem of establishing the validity of the limiting transition in a certain subdomain of the region under consideration is formulated. This problem is solved using the method of level lines and successive approximations.

The article examines the essence of modular learning as one of the modern pedagogical technologies. The main principles, structure, and features of the organization of modular learning are revealed. Special attention is paid to the issues of individualization of the educational process, the development of students’ independence, and the improvement of learning effectiveness. The advantages and challenges of implementing the modular approach, as well as the conditions for its effective application, are analyzed. The role of the teacher in modular learning is highlighted, along with the possibilities of using this technology in various educational contexts, including distance and e-learning. The study concludes that modular learning is an effective and promising direction for the development of the modern education system.

Examines the asymptotic behavior of solutions to a system of singularly perturbed differential equations with logarithmic poles. The system under consideration possesses an equilibrium point whose stability is lost at a certain value of the slow variable. The main focus is on the phenomenon of solution delay after the loss of stability, in which the trajectory of the solution corresponding to the fast variables remains in the vicinity of the unstable equilibrium point for a finite time interval. For the analysis, the system is transformed into a complex form and reduced to a special type of integral equation. An appropriate domain is constructed in the complex plane, and the existence of a solution is proved using the methods of level curves and successive approximations. An asymptotic estimate of the solution on an interval, part of which corresponds to the unstable equilibrium, is obtained. The results clarify the influence of logarithmic poles on the system dynamics and describe the region in which the phenomenon of solution delay is observed.

The phenomenon of delay of solutions in autonomous singularly perturbed systems of differential equations is investigated. A system involving fast and slow variables is considered, in which the system of fast variables has an equilibrium state whose stability depends on the value of the slow variable. As the slow variable passes through a critical value, the equilibrium state loses stability. However, the solution of the system may continue to remain in a neighborhood of this unstable equilibrium state for a finite interval of variation of the slow variable. This phenomenon is referred to as the delay of solutions near an unstable equilibrium state. The delay phenomenon has been studied for broad classes of singularly perturbed systems. In existing results, sufficient conditions for the occurrence of delay are generally established. The problem of determining necessary and sufficient conditions for the considered class of problems requires further investigation. In this paper, a criterion for the delay of solutions is established in terms of the level lines of the corresponding harmonic functions. It is shown that the existence of level lines connecting the stable and unstable intervals is a necessary and sufficient condition for the occurrence of delay. The obtained criterion makes it possible to characterize the asymptotic behavior of solutions in a neighborhood of an equilibrium state undergoing a loss of stability.