International Journal of Mathematics and Physics

HISTORICAL EVOLUTION OF MATHEMATICS EDUCATION IN KAZAKHSTAN: FROM THE MIDDLE AGES TO INDEPENDENCE

Alma Abylkassymova — 2025-12-19

Abstract. Over the past three centuries, Kazakhstan has been part of various states. It was initially a part of the Russian Empire; from 1920 to 1991, it was among the republics that formed the Soviet Union; and since 1992, it has existed as an independent country. In this context, it is important for the public of Kazakhstan to understand how mathematics was taught to the population at different stages of historical development.

The subject of this study is the historical evolution of mathematics education in Kazakhstan from the Middle Ages to the present day. The main focus is on the teaching of mathematics in national schools. This study represents the first research conducted in this direction; its scientific and methodological significance is undeniable, and it also has practical value for specialists in the field of mathematics education.

The primary attention is devoted to the study of literary sources and materials from state archives. Since many authors were subjected to repression in the 1940s, their works were removed from libraries and remained inaccessible for a long period. An in-depth methodological and mathematical analysis of these materials was conducted.

During the period of the Russian Empire, the first mathematics textbooks in the Kazakh language were written by S. Gramenitsky (1897) and M. Dulatuly (1914). In the first fifteen years of Soviet rule, mathematics textbooks were authored by mathematicians associated with the national movement “Alash.” However, in 1935, the country transitioned to instruction in the Russian language across schools nationwide, and national schools began using translated Russian textbooks. This policy remained in effect until the collapse of the Soviet Union.

Following the independence of Kazakhstan, all mathematics textbooks in the country began to be published in the Kazakh language. The results of the study, based on authentic archival materials, reveal the actual state of mathematics teaching in the Kazakh language. These materials can be incorporated into modern mathematics textbooks and teaching methodologies, taking into account the importance of the historical evolution of mathematics education among the Kazakh population.

Keywords: mathematics education, historical evolution, textbooks, school, pedagogical university, Bologna process.

On the inverse problem of identifying the source term in a pseudoparabolic equation with a final time overdetermination condition

Aidos Shakir — 2025-12-19

In this paper, we consider the inverse problem for a linear pseudoparabolic equation describing the temperature distribution taking into account external forces that depend only on the spatial variable. The classical solution to the inverse problem under consideration satisfies the usual pseudoparabolic equation, initial and nonlocal boundary conditions, and a final additional condition. The issues of existence and uniqueness of the solution to the inverse problem are the subject of study in the work presented by the author. As the main result, theorems on the existence and uniqueness of the classical solution to the problem under study are formulated and rigorously proven. These theorems are completely proven in a mathematically rigorous language using the method of separation of variables. In the course of the proof, a system of orthogonal and biorthogonal basis functions of a special type was chosen in accordance with the nonlocal boundary conditions. First of all, to prove the theorem on the existence of a solution, an analytical formula for the solution was derived in the form of a series in the system of these functions, their uniform convergence was analyzed according to the Weierstrass theorem, and the convergence to the classical solution of the inverse problem under consideration was investigated. The proof of the theorem on uniqueness was carried out by the method of the opposite assumption.

Keywords: inverse problem, pseudoparabolic equation, existence of solution, uniqueness of solution, nonlocal boundary condition.

Numerical solution of the inverse problem of magnetotelluric sounding

Syrym Kasenov — 2025-12-19

This study focuses on the coefficient inverse problem arising in magnetotelluric (MT) sounding, which plays a crucial role in geophysical exploration and subsurface characterization. The main objective is twofold: first, to construct a reliable forward numerical model based on the Helmholtz equation with a complex-valued conductivity coefficient, and second, to develop a stable inversion procedure for reconstructing the conductivity distribution from boundary measurements. The forward problem is discretized using a finite-difference approximation, ensuring numerical stability and accuracy for both the direct and adjoint formulations. To address the ill-posed nature of the inverse problem, a misfit functional is introduced, measuring the discrepancy between simulated and observed boundary data. This functional is minimized using the iterative Landweber method, which provides a simple yet robust tool for stabilizing reconstructions. Numerical experiments are carried out for a synthetic conductivity model consisting of a smooth background medium with an embedded localized anomaly. The obtained results demonstrate the ability of the proposed method to recover key structural features of the anomaly. The presented framework offers a promising foundation for the development of practical inversion algorithms applicable to real geophysical MT data.

Key words: Helmholtz equation, magnetotelluric sounding, inverse problem, Landweber method, numerical solution.

CFD Study of the Effect of Coarctation on the Distribution of Velocities, Pressures, and Shear Stresses in the Thoracic Aorta

Alibek Issakhov — 2025-12-19

This study focuses on numerical modeling of thoracic aortic hemodynamics in patients with coarctation, a congenital narrowing of the vessel lumen that impairs blood flow and increases hemodynamic load. The aim of the study was to identify the influence of geometric changes in the aorta on the distribution of velocity, pressure, and shear stress using computational fluid dynamics (CFD) methods. The mathematical model is based on the Navier-Stokes equations for an incompressible non-Newtonian fluid, which describes the rheological properties of blood. Calculations were performed using the finite volume method with an explicit time-dependent scheme for two geometric configurations—normal and pathological. Analysis of the resulting velocity and pressure fields revealed that with coarctation, the maximum velocity increases by approximately 1.6 times, and the pressure difference between the ascending and descending aorta reaches 0.6 kPa. The shear stress distribution revealed localized areas of extreme values that potentially contribute to endothelial dysfunction. This study contributes to the development of personalized blood flow modeling and demonstrates the potential of CFD methods for assessing hemodynamic disturbances in vascular pathologies, which has practical implications for preliminary diagnosis and treatment planning.

Keywords: hemodynamics, coarctation of the aorta, computational fluid dynamics, CFD, numerical modeling.

Solar Assisted Auto-Cascade Heat Pump for Water Heating in the Continental Climates

Zarina Abdulina — 2025-12-19

Heat pumps are widely recognized as energy-efficient technologies for domestic hot water production. However, conventional single-stage vapor compression heat pumps utilizing ambient air as the heat source are limited in delivering outlet water temperatures above 323 K under low ambient conditions. This limitation restricts their applicability in continental climates characterized by large diurnal and seasonal temperature variations. Two-stage cascade systems can achieve higher outlet temperatures exceeding 343 K, but they require two compressors, resulting in increased energy consumption and higher capital costs. To overcome these drawbacks, the present study proposes an auto-cascade compression heat pump system employing an environmentally friendly binary zeotropic refrigerant mixture to achieve water outlet temperatures above 343 K with improved efficiency and reduced system complexity. In addition, solar collectors are integrated to enhance low-grade heat extraction from the environment. A numerical simulation of the proposed auto-cascade system was conducted for binary zeotropic refrigerant mixtures including R32/R134a, R32/R1234yf, R32/R1234ze, and R32/R245fa within an ambient temperature range of 223–273 K. The results show that the coefficients of performance (COP) for R32/R134a, R32/R1234yf, and R32/R1234ze mixtures vary between 2.72 and 2.75, while that of R32/R600a reaches 2.55. Based on the comparative analysis, the R32/R134a mixture demonstrated the best performance and is recommended as a promising working fluid for auto-cascade heat pump systems designed for water heating applications in continental climate regions.

Keywords: Auto-cascade Heat Pump, Zeotropic Mixture, Continental Climate, Water Heating, COP

Exploring the Impact of Anisotropy Parameters on Stellar Structure

S. Toktarbay — 2025-12-16

In this study, we examine how variations in local pressure anisotropy affect the internal structure and equilibrium of white dwarfs. A generalized anisotropy model is developed, defined by three parameters: the amplitude coefficient α₀ and the shape exponents l and k. This formulation ensures that the anisotropic pressure continuously vanishes at both the center and surface of the star while reaching a single peak in the intermediate region. By applying appropriate boundary and regularity conditions, the model allows us to determine physically stable parameter domains consistent with realistic stellar configurations. Our analysis shows that even a small degree of anisotropy can have a measurable effect on the mass–radius relation and overall compactness of white dwarfs, which may help explain the origin of super-Chandrasekhar systems observed in astrophysical data. This modeling approach provides a clear and flexible way to describe compact stars with anisotropic pressures and can also be applied to neutron and quark stars.

Keywords: compact stars, white dwarfs, anisotropic pressure, stability analysis, generalized anisotropic factor

Response of Liver and Gastric Cancer Cells to Electron and X-ray Radiation

J. Batmyagmar — 2025-12-16

Radiotherapy remains a critical pillar of cancer treatment worldwide. This study evaluates the in vitro efficacy of high-energy ionizing radiation, specifically 6 MV electrons and 12 MV X-rays, generated by a Varian Clinac iX linear accelerator (linac), on human HepG2 (liver) and AGS (gastric) cell lines. Cell samples (1 ml) were irradiated with doses ranging from 0.5 Gy to 4 Gy. Cell viability was assessed using the WST assay 4-5 hours post-irradiation. The measured survival rates were critically compared with those predicted using the established linear-quadratic (LQ) model. The results revealed significant and consistent discrepancies between the experimental measurements and the theoretical predictions for both cell lines. For HepG2 cells, the measured survival rate at 4 Gy was higher than the predicted rate. Interestingly, AGS cells irradiated with 12 MV X-rays exhibited minimal cytotoxicity, with a viability rate of 99.0% at 3 Gy versus a predicted rate of 73.6%. These findings suggest a discrepancy between theoretical predictions and the short-term biological responses observed under the shallow in vitro irradiation conditions employed in this study. While the present study was not designed to isolate the underlying mechanisms, the results imply that factors inherent to high-energy beam delivery in thin in vitro geometries, together with the early (four to five hour) post-irradiation assessment window, may have contributed to the limited cytotoxicity observed in both cell lines. Further studies employing extended observation periods or complementary assays would be valuable in clarifying the temporal progression of MV-beam-induced cellular effects.

Keywords: Varian Clinac iX, high-energy radiation, HepG2, AGS, cell viability, WST assay, Monitor Unit (MU).

Selection of a Propellant Feed System for the LPRE of a Small Upper Stage

A. Dintayev — 2025-12-16

The design of small upper stages must meet many propulsion requirements: repeated restarts, stable operation at low mass flow rate, strict limits on mass and volume, and compatibility with limited ground infrastructure. The central engineering task is a justified selection of the liquid propellant rocket engine feed architecture for a given mission profile and propellant pair, since this choice affects specific impulse, service life, risk, and ground processing effort. This work systematizes propellant feed schemes for a small upper stage and proposes a unified analytical framework that links engine cycle and feed method with propellant selection and tank pressurization modes. Pressure fed and turbopump schemes are treated as mature solutions; pump cycles are used mainly with cryogenic propellants and in high energy demand cases that require high efficiency and compact hardware. In parallel, electro pump schemes are actively studied as a promising direction due to lower mechanical complexity, precise control, and straightforward integration with modern control systems. The analysis shows how chamber pressure, allowable throttling range, restart capability, and the mass and volume metrics of tanks and hardware bound the rational domain of each scheme. The outcome is a set of criteria for early design that maps mission requirements to feed system architecture and supports a technologically feasible choice for small upper stages.

Keywords: pressure-fed system, turbopump-fed supply system, small upper stages, rocket engines, liquid rocket engine.

Comprehensive Structural and Stress Analysis Of Zno/Sic/Porous-Si/Si Multilayer Heterostructures Synthesized via Sequential Deposition Techniques

V. Kidalov — 2025-12-16

This research explores the fabrication and structural characteristics of ZnO/SiC/porous-Si/Si multilayer heterostructures synthesized through a controlled multi-step deposition process. The study combines electrochemical porosification of monocrystalline Si substrates, solid-phase epitaxial growth of silicon carbide films, and magnetron sputtering of ZnO layers under varied oxygen partial pressures. Two samples of ZnO films were synthesized under distinct oxygen atmospheres: 0.06 Pa and 0.1 Pa. Comparative XRD analysis reveals that films deposited at lower pressure (0.06 Pa) exhibit enhanced crystallinity, indicated by reduced peak broadening and distinct polycrystalline features. Residual stress analysis confirms compressive biaxial stress in both samples (−0.511 GPa and −0.287 GPa), indicating high crystalline quality and structural integrity of the ZnO films. These findings highlight the effectiveness of buffer layering and deposition control for optimizing ZnO film properties on complex silicon-based architectures.

Keywords: ZnO thin film, porous silicon, silicon carbide, heterostructure, residual stress, structural characterization, X-ray diffraction.

Quasi-2D Vortex Structures in Turbulent Flows: a Lagranjian Model with Fractal Effects

A. Imanbayeva — 2025-12-16

This paper presents a physically motivated model of quasi-two-dimensional vortex structures in turbulent flows. The theory of quasi-two-dimensional turbulence explains many phenomena in geophysical hydrodynamics, since due to the rapid rotation of the Earth, large-scale movements of the atmosphere and ocean almost two-dimensional. Quasi-2D turbulence is approximately two-dimensional and is described by equations containing additional terms. Such additions allow us to take into account weak three-dimensional effects that arise in real conditions, for example, in the atmosphere or ocean. We consider the basic equations for the velocity and pressure fields using the Lagrangian frame and incorporating centrifugal and Coriolis forces, as well as fractal disturbances on the vortex surface. Numerical simulations implemented in MatLab reproduce classical vortex behavior and reveal the influence of fractal corrections on field asymmetry. The model aligns well with existing experimental data and offers a foundation for analyzing energy transport and vortex interactions in stratified or thin-layered turbulent systems.

Keywords: quasi- two-dimensional turbulence, fractal boundary, Lagrangian frame, streamfunction, vortex elements, numerical simulations.

Elemental analysis and X-ray irradiation effect on the Mongolian dairy product – “Khorkhoi Aaruul”

B. Munkhbat — 2025-12-16

Aaruul is a traditional dairy product from Mongolia, recognized for its nutritional qualities and potential health benefits. Traditionally produced through manual milk processing at home, aaruul is increasingly mass-produced for commercial markets. This study aimed to investigate the effects of X-ray irradiation on Khorkhoi aaruul, a widely consumed small-curd variety. Samples were exposed to irradiation doses of 1 kGy, 3 kGy, and 5 kGy to evaluate reductions in bacterial contamination, extension of shelf life, and compliance with international food safety standards. The changes in vitamin C content and organoleptic properties (taste, texture, and appearance) were systematically assessed. Trace element composition was quantified using Energy-Dispersive X-ray Fluorescence (ED-XRF) spectrometry, focusing on phosphorus (P), calcium (Ca), zinc (Zn), iron (Fe), manganese (Mn), copper (Cu), and bromine (Br). Results indicated a substantial reduction in bacterial counts, achieving a 95.5% decrease at the 5 kGy dose (from 4.2×10⁶ to 2×10⁵ CFU). Critically, irradiation did not induce significant alterations in chemical composition, sensory quality, or vitamin C levels. These findings demonstrate the effectiveness of X-ray irradiation as a viable approach to enhancing microbial safety and prolonging the shelf life of traditional Mongolian dairy products without compromising their nutritional or sensory characteristics.

Keywords: Aaruul, irradiation, trace elements, ED-XRF, colony formation unit