International Journal of Mathematics and Physics

THE CONTROL PROBLEM ASSOCIATED WITH HEATING PROCESS OF A ROD

2025-06-05T13:41:02+00:00

In this paper, the control problem for a heat conduction equation with periodic boundary condition is considered. The solution with the control function on the border of the rod is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The considered control problem is reduced to the Volterra integral equation, which is the first type, using the Fourier method. The proof of the existence of admissible control is related to the existence of a solution of the integral equation. The existence of the solution of the integral equation is shown using the Laplace transform method and the admissibility of the solution is proved. In the last section, the control function needed to achieve the average temperature given certain values of the parameters is found.

Keywords: heat conduction equation, average temperature, periodic boundary condition, control problem, Volterra integral equation, Laplace transform.

MATHEMATICAL MODELING OF STEADY FLOW PAST CIRCULAR CYLINDER WITH SPLITTER PLATES BY R-FUNCTIONS METHOD

2025-03-05T14:07:10+00:00

In the paper the steady flow of a viscous incompressible fluid past a circular cylinder with attached splitter plates is considered. The mathematical representation of the problem takes the form of an external boundary value problem for the stream function. To solve the problem, a numerical method combining the R-functions and the nonlinear Galerkin method is used. The R-functions method is employed to construct a problem solution structure that exactly satisfies all the boundary conditions of the problem and has the required behavior at infinity. The Galerkin method is then applied to approximate the undetermined components of this structure, ensuring accuracy and efficiency in the solution process. A series of computational experiments was conducted to investigate the flow past a single circular cylinder and past a circular cylinder with triangular and rectangular splitter plates at various Reynolds numbers. For the case of a single cylinder, a quantitative error analysis confirms the convergence of the numerical method, with relative errors dropping below 1% when using a moderate number of basis functions. The computational cost remains practical, with each solution obtained in approximately 11 minutes on a standard workstation. Drag and lift coefficients are computed for the single-cylinder case, allowing for quantitative assessment of aerodynamic performance and validation of the numerical model against known reference data. The influence of splitter plate geometry on the flow structure is explored through visualizations, highlighting changes in vortex patterns and symmetry. The proposed approach demonstrates strong numerical accuracy and computational robustness for the single-cylinder case and offers a flexible framework for studying external viscous flows with complex boundary configurations.

Keywords: Viscous incompressible fluid, External boundary value problem, Stream function, R-functions method, Nonlinear Galerkin method.

INVERSE SOURCE PROBLEM FOR ADVECTION-DIFFUSION EQUATION FROM BOUNDARY MEASURED DATA

2025-04-17T06:58:28+00:00

The inverse problem for the advection-diffusion equation is considered in this paper. The study focuses on reconstructing a space-dependent source of a variable-coefficient advection-diffusion equation with separable sources from time-dependent temperature measurements at the right boundary of the domain. The Tikhonov regularization method is used to determine the space-dependent source function. These problems arise in various fields of science and engineering. The source term takes the form of separated variables, where one function describes the time evolution and the other represents the spatial distribution of some contaminant source. Such source terms also arise as control terms in the context of heat equations. Numerical experiments were conducted to demonstrate the accuracy and robustness of the proposed method. A non-iterative inversion algorithm is developed and numerically implemented for identifying the unknown space-dependent source. Consequently, identifying space-dependent or time-dependent sources is crucial in addressing environmental issues, which served as the motivation for proposing this problem.

Key words: Inverse source problem, advection-diffusion equation, space-dependent heat source, Dirichlet data

PURE LINEAR ORDERINGS OF MORLEY O-RANK 1

2025-04-05T06:21:22+00:00

The study of pure linear orderings—sets equipped solely with a linear (total) order—has deep historical roots in mathematical logic and order theory. Initial investigations trace back to Cantor’s work on ordinal numbers, which laid the foundation for understanding different sizes of ordered sets. Early 20th-century research by Hausdorff and others explored order types and their classification. Later, developments in model theory and set theory refined the structural properties of pure linear orderings, including their rigidity, embeddability, and definability in various logical frameworks. So, pure linear ordering and its classification are one of the classical mathematical questions. Descriptions of o-minimal and weakly o-minimal pure linear orderings are known, as well as we know that any pure linear ordering has an o-superstable elementary theory. The aim of this paper is to start investigation of pure linear ordering that have o-ω-stable elementary theory. So, we give the complete description of pure linear ordering of Morley o-rank 1.

Key words: Pure linear ordering, o-minimal structure, o-stable theory, Dedekind’s cut, ordered structure, Morley o-rank.

DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH BOUNDARY TERMS

2025-05-16T04:19:29+00:00

In the referenced study, a differential equation (DE) exhibiting a hybrid structure is examined. The principal objective of this manuscript is to determine the feasibility of substituting the given supplementary boundary conditions with alternative equivalent conditions. This is achieved through the establishment and proof of four theorems, providing a rigorous foundation for the proposed substitutions. Incipiently, the existing (3) conditions are considered in a nonhomogeneous context. Subsequently, new conditions, denoted as (7), are introduced. These newly formulated conditions are demonstrated to be equivalent to the original ones, ensuring the unique solvability of the hybrid-structured system labeled as (1). The system under consideration is characterized as hybrid due to the presence of both unknown and algebraic components. This dual nature necessitates a nuanced approach to boundary condition formulation and analysis. The methodology employed in this study underscores the importance of flexibility in boundary condition specification, particularly in complex or hybrid configurations. By establishing the equivalence of different boundary conditions, article provides valuable insights into the solvability and analysis of such frameworks. Furthermore, the study meticulously details prior research in this domain, delineating the specific conditions and configurations previously explored. This comprehensive review situates the current manuscript within the broader context of hybrid DE analysis.

Key words: boundary, hybrid, dissipative, DAE, BVP.

BIANCHI TYPE-II RENYI HOLOGRAPHIC DARK ENERGY MODEL IN SAEZ-BALLESTER THEORY OF GRAVITY

2024-10-19T08:18:33+00:00

In Saez-Ballester's (Physics Letters A: 113, 467, 1986) theory of gravitation, the paper presents the study of Bianchi type- interacting Rényi holographic dark energy determined. We determine interacting dark energy models by considering a correlation between the metric potentials to solve the field equations of the model. This results in a dynamical deceleration parameter which demonstrates a shift in the cosmic rate of acceleration from deceleration to acceleration, with a redshift change that is compatible with observations. Despite assuming several values to parameters close to at (the present epoch) and is in agreement with the most recent observations. Next, we discovered that the squared sound speed, , is negative at the initial epoch and positive at the present epoch, implying instability against perturbations at the initial epoch and stable at the present epoch. The - plane is constructed to investigate the evolution of the models’ EoS parameter turned out to be in the freezing zone. As should be the case in an expanding universe, the strong energy conditions of the models are violated. Statefinders plane confirms that our model includes the Chaplygin gas, phantom and limit.

Keywords: LRS Bianchi type- metric, Saez-Ballester theory, Rényi holographic dark energy, scalar-tensor theory, cosmology.

ENERGETIC EFFICIENCY OF TURBULENT BIOFUEL COMBUSTION FOR ADVANCED BIOENERGY TECHNOLOGIES

2025-05-19T15:42:38+00:00

The combustion processes of biodiesel droplets in a combustion chamber using numerical modeling of two-phase reacting flows and complex turbulent currents are investigated in this study. The focus is on analyzing temperature fields, aerodynamic characteristics, and soot particle distribution at varying Reynolds numbers. The results show that optimal combustion conditions for biodiesel are achieved at Reynolds numbers between 20,000 and 25,000, where maximum combustion temperatures reach up to 2700 K, indicating high thermal efficiency and intense combustion processes. Additionally, this range of Reynolds numbers leads to a significant reduction in soot particle concentration (50–75 g/g), suggesting more complete fuel combustion and improved oxidation conditions. The findings confirm that increasing Reynolds numbers not only enhances combustion temperature but also improves the environmental performance of biodiesel by reducing emissions of solid combustion products. This research demonstrates the potential of biodiesel as an environmentally friendly alternative to traditional hydrocarbon fuels, offering optimal combustion characteristics under high turbulence conditions.

Keywords: biodiesel, combustion, numerical modeling, soot emissions, turbulent flow.

STATISTICAL ANALYSIS OF THE SITUATION IN GEOSTATIONARY ORBITS

2025-05-10T09:53:20+00:00

Statistical analyses of near-miss events involving geostationary zone objects from 2011 to 2024, derived from orbital propagation based on geostationary zone object orbital element catalogs and observational data of events, are presented. The data reveal a general increase in the number of near-miss events, attributable to the newly launched satellites and enhanced detection capabilities for space debris fragments. Notably, the average daily distribution of near-miss events exhibits longitudinal asymmetry, with a higher number of such events at 75°E longitude compared to 255°E. Conversely, potentially hazardous mutual approaches with distances as low as 10 km are more prevalent at 255°E. An investigation into the relationship between the average daily number of such events and the minimum object separation as a function of the catalog epoch indicates an increase in daily dangerous approaches within 1 km from 0.24 events/day in 2011 to 0.39 events/day in 2024. When considering a 20-meter collision threshold, the 2024 data suggest a potential collision event probability in geostationary orbit of one event per eight years. Projections based on the current trajectory of orbit occupancy suggest that, if prevailing trends persist, collision events in GEO could occur every six years by 2030 and every four years by 2040.

Keywords: Space debris, Near-Earth Space, Geostationary orbit, Near-Miss Events, Space situational awareness.

SYNTHESIS OF TWO-DIMENSIONAL NANOSTRUCTURES AT LOW TEMPERATURES

2025-05-26T04:26:02+00:00

The paper presents the results of experiments on synthesizing two-dimensional carbon nanostructures by thermal chemical vapor deposition using iron slugs as catalysts and substrates. The synthesis was conducted at a temperature of 250 ° C, a working pressure of 500 mbar, and a duration of 1 hour. The SEM results show that a two-dimensional structure is synthesized on the surface layer of the Fe slug with a vertical alignment. According to the detailed morphological characteristics of the results obtained by the scanning electron microscope, the nanostructures are two-dimensional carbon nanostructures, i.e. graphene-like. For a detailed determination of the structure of the obtained samples, a study was carried out by the method of Raman scattering and scanning electron microscopy. According to the Raman spectroscopy results, all spectra show the main carbon peaks D and G. In the high-frequency region of the spectra, second-order peaks D′, 2D and D + G are also observed, indicating the presence of graphene.

Keywords: two-dimensional carbon nanostructures, iron catalyst, thermal chemical vapor deposition method, scanning electron microscope, Raman spectroscopy.

STUDYING THE GROWTH MECHANISMS OF Si EPITAXIAL LAYERS FROM A LIMITED VOLUME OF Si-Sn SOLUTION USING THEORETICAL CALCULATIONS AND THE SEDIMENTATION METHOD

2025-02-28T10:45:57+00:00

Differences in the thicknesses of the layers formed during the growth of solid solution epitaxial layers such as Si_1-xGe_x, (Si₂)_1-x(GaP)_x, (Ge₂)_1-x(GaAs)_x and (Ge₂)_1-x(ZnSe)_x on horizontally placed top and bottom substrates from a confined volume of liquid solution based on programming cooling were determined. In order to study the cause of this situation, silicon crystal layers were grown from a limited amount of solution (Si+Sn) on horizontally placed Si substrates in the temperature range of 973-1323 K, and in this process, a thicker crystal layer was grown on the upper substrate. The quality of the crystal layer grown on the top substrate is relatively low. We have shown that the reason for this phenomenon is that large-sized silicon nanoparticles in the solution system settle on a high substrate due to their low density during crystallization. It was considered that these processes occur due to the formation of a large number of large-sized silicon nanoparticles at high temperatures. The placement of large-sized nanoparticles on the substrate located above causes some defects in the formation of single crystals in this part. Experiments have shown that when the distance between the upper and lower Si substrates is less than 1.2 mm, crystal layers of the same thickness can be obtained from the solution on both substrates.

Keywords: Silicon, molecule, nanoparticle, sedimentation.

SIMULATION OF RESONANT REACTIONS IN TTIK APPROACH BY C++ CODE

2025-05-07T14:23:44+00:00

We studied the interaction of ¹⁵N with hydrogen at low energies using the Thick Target Inverse Kinematics (TTIK) method combined with the Time of Flight (ToF) technique. This combination enables a clear identification of different light recoil particles without compromising energy resolution in the extended gas target. The use of a gas target ensures minimal background and allows precise measurement of energy loss and reaction kinematics across a broad excitation function. At the same time, the ToF technique provides particle separation without degrading energy resolution. In this paper, we presented the results of theoretical calculations of energy losses in the gas target, implemented using a custom C++ code. The obtained results show reasonable agreement with experimental data. The simulations agree well with the experimental results for α-particles and ¹²C nuclei, confirming the accuracy of the theoretical predictions for these reaction products. However, further improvements or theoretical refinements are required due to the insufficient fit observed in the proton-related calculations.

Keywords: TTIK, inverse kinematics, time of flight, resonant reaction, C++ simulation.