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During worldwide epidemic of COVID-1, people started to depend on social media apps to cure their boredom. It was beneficial to provide information easily to a wide audience. The increased time people spent on their phones led to many, including teenagers and children, becoming addicted to social media, particularly TikTok. As a result, this usage of TikTok became an epidemic itself. Therefore, in this work, a fractional mathematical model is being developed to analyze the influence of TikTok on human population of different ages. In this paper, two equilibrium points for this model will be discussed and the theoretical stability will be proven. For numerical analysis, fractional Euler’s method will be used and the results will be discussed graphically and which group of population will be affected the most by frequent use of TikTok. [ABSTRACT FROM AUTHOR]

The spectral mapping theorem, in conjunction with the concept of a spectral set, is utilized to construct reducing pairs of subspaces for a given operator. By leveraging the theory of functional calculus, we revisit the well-established spectral mapping theorem for various spectra, with a particular focus on the point spectrum. Concluding this discussion, we present an application wherein we collect and establish properties of the point spectrum of a compact exponentially m-isometry. Also, we examine the behavior of solutions of a system of differential equations associated with exponentially m-isometric matrices. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find the approximate series solution of IVPs on nonlinear differential equations (NDEs) and on systems of NDEs via an iterative algorithm that can be easily exhibited through MATLAB code. We propose a modified iterative scheme based on the DGJM method of solving functional equations of the form u = f + N(u), and we observe that the method is far better than those existing in the literature. For the sake of illustration, series solutions to a few physical equations proposed by Klien-Gordon, Allen- Cahn and Benjamin-Bona-Mahony are provided along with the MATLAB code used for computation. Moreover, graphs are used to validate the efficacy of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of differential equations describing a mathematical model of a phase-locked loop with a delay. We introduce the concept of the curvature of a cycle, which is applied to analyzing the proximity of cycles of nonphase and phase systems. [ABSTRACT FROM AUTHOR]

We study the dynamics of a system of differential equations with the diffusion interaction and an additional internal coupling. Such systems are interesting because a slight variation in the coefficient at the additional coupling allows obtaining intricate scenarios of phase rearrangements. For the system under study, we find the critical dependence of the parameters such that zero equilibrium loses stability as two spatially inhomogeneous states appear in one case and a cycle in another case. With the parameter values close to the critical ones, asymptotic formulas are obtained for the regimes that branch off from the zero solution. [ABSTRACT FROM AUTHOR]

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider n-type Markov branching–immigration processes. The uniqueness criterion is first established. Then, we construct a related system of differential equations based on the branching property. Furthermore, the explicit expression of extinction probability and the mean extinction time are successfully obtained in the absorbing case by using the unique solution of the related system of differential equations and Kolmogorov forward equations. Finally, the recurrence and ergodicity criteria are given if the zero state 0 is not absorbing.

In the following paper, we have shown the existence and localization of solutions for a system of n third order differential equations under Sturm-Liouville type boundary conditions. Such systems appear in many physical problems, one of which is the jerk equations to locate the trajectory of a material point in space. [ABSTRACT FROM AUTHOR]

In this paper, we study the direct scattering problem on the half-line for the Dirac system of differential equations in the case of finite density with the boundary condition . The spectrum is studied, the resolvent is constructed, and the spectral expansion in the eigenfunctions of the Dirac operator is obtained. [ABSTRACT FROM AUTHOR]

In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving electric circuits, population growth, vibrational beams, and heat conduction. The Mohanad transform is defined and introduced in this study, along with its fundamental qualities, including linearity and convolution. It is also discussed in connection with other integral transforms and how it is used in derivatives. Additionally, we use the Mohanad transform to solve a few systems of ordinary differential equations (ODEs) and review its properties in this paper. Determining the concentration of a chemical reactant (material) in a series is a physical chemistry problem that we use in the application part. We achieve this by developing a model based on ordinary differential equations (ODEs) and then solving them using the Mohanad transform. This research proves that, with little computational effort, we can get the exact solutions of ordinary differential equations (ODEs) via the Mohanad transform. We used graphs and tables to show our answer. [ABSTRACT FROM AUTHOR]

Methods for constructing systems of second-order ordinary differential equations whose solutions satisfy the constraint stabilization conditions are outlined. Modified Helmholtz conditions are used to bring the system to the form of the Lagrange equations. The expression of the central force is determined, which ensures the stabilization of the points movement trajectory. [ABSTRACT FROM AUTHOR]

Dynamical systems are often being described with the help of the ordinary differential equations. But only a few problems can be solved analytically. Basically, the quantity theory of ordinary differential equations or numerical methods are used to analyze the properties of the solution. But most of the difference schemes can lead to the numerical instability with the regard for the constraints. To avoid the accumulation of errors, J. Baumgarte suggested to use the method of constraint stabilization. In this paper, the methods of solving inverse dynamical problems with regard for constraint stabilization are investigated. [ABSTRACT FROM AUTHOR]

This article presents a model for organizing railway freight transportation between two node stations based on the interaction of neighboring stations depending on their technical capabilities and the demand for freight transportation. The main objective of such interaction is to synchronize the inbound and outbound flow at the stations, specifically reducing the degree of inconsistency between the receipt and dispatch of goods at the stations. This characteristic represents the imbalance between the volume of incoming and outgoing goods at the stations per unit of time and is described by a nonnegative function bounded above by unity. Its dynamics are described by a system of differential equations containing a set of parameters characterizing the infrastructure of the stations, the mode of distribution of goods from the final node station, and the demand for freight transportation. The range of parameter variations for which the specified system has a solution has been determined, as well as such a set of these parameters for which the task of synchronizing the inbound and outbound flow at the stations is best solved. It has been found that regardless of the initial value, starting from a certain point in time, the degree of inconsistency between the receipt and dispatch of goods at all stations except the initial node station becomes zero. [ABSTRACT FROM AUTHOR]

Currently, systems of neural ordinary differential equations (ODEs) have become widespread for modeling various dynamic processes. However, in forecasting tasks, priority remains with the classical neural network approach to building a model. This is due to the fact that by choosing the neural network architecture, a more accurate approximation of the trajectories of a dynamic system can be achieved. It is known that the accuracy of the mentioned approximation significantly depends on the settings of the neural network parameters and their initial values. In this regard, the main idea of the article is that the initial values of the neural network parameters are taken to be equal to the parameters of the neural ODE system obtained by modeling the same process, which will then be simulated using a neural network. Subsequently, the singular ODE system was used to adjust the parameters of the LSTM (Long Short Term Memory) neural network. The results obtained were used to model the process of epilepsy.

In this paper, we investigate the convergence of solutions of certain nonlinear system of two nonlinear differential equations using a suitable Lyapunov functional with sufficient conditions to establish our new result. An example is given to demonstrate the effectiveness of the result obtained and geometric argument to show that the solutions of the system are better rapidly converging under the criteria obtained. [ABSTRACT FROM AUTHOR]

This article analyses the exothermic reaction of flame ignition in the combustion chamber of a gas turbine unit, which is characteristic of combustion chambers operating on traditional hydrocarbon fuels. The combustion of gases as an explosive process in confined and semi-enclosed areas remains a poorly understood section of thermal physics. Without a detailed review of the physical and chemical processes taking place in the combustion chamber, it cannot be said whether the gas turbine unit will run sustainably. It is also important to know what combustion modes are in principle possible after a loss of stability in the combustion chamber in order to take action against this in advance. To describe flame ignition and quenching in the flow of the fuel–air mixture through a combustion chamber, a system of differential conservation equations of energy and reactive species supplemented with the equation of state is used. Nonstationary combustion processes in gas-turbine engines were studied, and flame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes. Calculation methodologies for various operating modes of the combustion chamber of a gas turbine unit are developed and realized. The results of the calculations that were carried out are presented with graphical interpretation and with their analysis provided in sufficient detail. Based on this analysis, recommendations are then provided. From the graphs, it can be observed that the combustion chamber of a gas turbine unit reaches its maximum limit of stable operation at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and the mixing zones of the flame tube (λOC)opt = 0.22 when the fuel–air mixture is at maximum depletion, ensuring that combustion does not stop and flame failure does not occur. The topic of this article relates to the intensification of hydrocarbon fuel combustion and the technological improvement of combustion chambers in gas turbine units. This topic is of exceptional importance and relevance, emphasizing its significance. The purpose of this work is to develop and implement a methodology for calculating various modes of operation of the combustion chamber of a gas turbine unit. [ABSTRACT FROM AUTHOR]

Covid-19 has become a pandemic in various countries since 2019. The spread of Covid-19 is easy and fast through droplets that come out when coughing, sneezing, or talking. Several ways to overcome the spread are conducting home quarantine, medical quarantine, carrying out health protocols, and vaccinating. This study proposed a deterministic model called SVEQ1IQ2R, which considers home and medical quarantine and vaccination. The model divides the population into seven compartments, namely susceptible (S), exposed (E), vaccinated (V), home quarantine (Q1), infected (I), medical quarantine (Q2), and recovered (R). This study aims to analyze the stability of equilibrium points and the sensitivity of the proposed model. The stability analysis results show that diseasefree and endemic equilibrium points are locally asymptotically stable using Routh-Hurwitz criteria and CastilloChavez and Song theorems, respectively. Moreover, the sensitivity analysis shows that reducing the contact rate and increasing vaccine effectiveness are the most appropriate steps to take to reduce the spread of Covid-19. Based on numerical simulations, increasing the vaccine effectiveness and the vaccination rate as well as reducing the contact rate can make the transmission of Covid-19 resolved more quickly [ABSTRACT FROM AUTHOR]

The normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases.

Researchers and practitioners in the fields of science and engineering encounter significant challenges when it comes to mitigating the proliferation of computer worms, owing to their rapid spread within computer and communication networks. This study delves into a comprehensive analysis of the mathematical model governing the hazard of worm propagation in such networks. Specifically, the mathematical framework employed herein encompasses a system of ordinary differential equations. In numerous instances, mathematical models have been employed to quantitatively investigate the propagation patterns of worms across computer networks. In this scholarly article, we present an enhanced Susceptible-Exposed-Infected-Quarantined-Vaccinated (SEIQV) model, denoted as Susceptible-Exposed-Infected-Quarantined-Patched (SEIQP), which effectively captures the dissemination dynamics of an insider threat within a network featuring air gaps. To facilitate the study, we leverage the power of feedforward neural networks that are trained using the backpropagated Levenberg-Marquardt optimization algorithm. These neural networks serve as surrogate tools, providing solutions to the SEIQP model. To evaluate the efficacy of our approach, we meticulously assess their performance across three distinct scenarios. Additionally, the stability of the mathematical model is examined by manipulating the probability of an insider threat removing a patch from the host, denoted as $\eta $ . Our empirical findings conclusively establish the effectiveness of the proposed approach in addressing the intricate challenges associated with insider threats within network environments.

The positioning problem considered in this paper is an attempt to find system’s of a few points position using accelerations. Today’s industry provides with various types of accelerometers suitable for acceleration measure. That data may be used to trace object movement. Final results showed that such movement trace is possible and may be applicable. The results also showed origins for errors which according to the results, are quite high globally. The final conclusion follows: the model created must be analized further in a deeper level, and error origins must be eliminated or at least minimized by either chosing different solving or modeling methods, either by revising used ones.

In this paper, we are interested in modeling the production of mucus by diatoms under the constraint of a nutrient limitation and carbon overabundance. The initial questioning comes from the observation of the so-called "liga" on the Acquitaine coast. The biological origin of the phenomenon is presented and discussed based on the existing literature. An original mathematical model incorporating mucus secretion is proposed and its theoretical properties are analized: well-posedness and differentiability with respect to the model parameters. Finally, numerical experiments are provided, investigating the possibility of parameter identification for the model using chemostat-type experiments. [ABSTRACT FROM AUTHOR]

This article analyses the exothermic reaction of flame ignition in the combustion chamber of a gas turbine unit, which is characteristic of combustion chambers operating on traditional hydrocarbon fuels. The combustion of gases as an explosive process in confined and semi-enclosed areas remains a poorly understood section of thermal physics. Without a detailed review of the physical and chemical processes taking place in the combustion chamber, it cannot be said whether the gas turbine unit will run sustainably. It is also important to know what combustion modes are in principle possible after a loss of stability in the combustion chamber in order to take action against this in advance. To describe flame ignition and quenching in the flow of the fuel–air mixture through a combustion chamber, a system of differential conservation equations of energy and reactive species supplemented with the equation of state is used. Nonstationary combustion processes in gas-turbine engines were studied, and flame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes. Calculation methodologies for various operating modes of the combustion chamber of a gas turbine unit are developed and realized. The results of the calculations that were carried out are presented with graphical interpretation and with their analysis provided in sufficient detail. Based on this analysis, recommendations are then provided. From the graphs, it can be observed that the combustion chamber of a gas turbine unit reaches its maximum limit of stable operation at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and the mixing zones of the flame tube (λOC)opt = 0.22 when the fuel–air mixture is at maximum depletion, ensuring that combustion does not stop and flame failure does not occur. The topic of this article relates to the intensification of hydrocarbon fuel combustion and the technological improvement of combustion chambers in gas turbine units. This topic is of exceptional importance and relevance, emphasizing its significance. The purpose of this work is to develop and implement a methodology for calculating various modes of operation of the combustion chamber of a gas turbine unit.

The focus on renewable energy is increasing globally to lessen reliance on conventional sources and fossil fuels. For renewable energy systems to work at their best and produce the desired results, precise feedback control is required. Microbial electrochemical cells (MEC) are a relatively new technology for renewable energy. In this study, we design and implement a model-based robust controller for a continuous MEC reactor. We compare its performance with those of traditional methods involving a proportional integral derivative (PID), H-infinity (H ∞ ) controller and PID controller tuned by intelligent genetic algorithms. Recently, a dynamic model of a MEC continuous reactor was proposed, which describes the complex dynamics of MEC through a set of nonlinear differential equations. Until now, no model-based control approaches for MEC have been proposed. For optimal and robust output control of a continuous-reactor MEC system, we linearize the model to state a linear time-invariant (LTI) state-space representation at the nominal operating point. The LTI model is used to design four different types of controllers. The designed controllers and systems are simulated, and their performances are evaluated and compared for various operating conditions. Our findings show that a structured linear fractional transformation (LFT)-based H ∞ control approach is much better than the other approaches against various performance parameters. The study provides numerous possibilities for control applications of continuous MEC reactor processes. [ABSTRACT FROM AUTHOR]

In this paper, a numerical study has been undertaken on the susceptible-infected-recovered (SIR) epidemic model that encompasses the mechanisms of the evolution of disease transmission; a prophylactic vaccination strategy in the susceptible populations, depending on the infective individuals. We furnish numerical and graphical simulation combined with explicit series solutions of the proposed model using the New Iterative Method (NIM) and Modified New Iterative Method (MNIM). The analytic-numeric New Iterative Method failed to deliver accurate solution for the large time domain. A new reliable algorithm based on NIM, the coupling of the Laplace transforms, and the New Iterative method is called Modified New Iterative Method (MNIM) which is presented to enhance the validity domain of NIM techniques. The convergence analysis of the MNIM has also been illustrated. The simulation results show that the vaccination strategy can slow down the spread of the epidemic rapidly. Numerical results illustrate the excellent performance of the MNIM and show that the modified method is much more accurate than the NIM. [ABSTRACT FROM AUTHOR]

Well-posed boundary value problems are constructed for calculating rotation shells of with a stiffness variable along the meridian in two directions, and also with variable bilateral with respect to the reference surface with the shell wall thickness. Algorithms for the numerical integration of systems of differential equations with variable coefficients are discussed.

As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse problem. The correct (vector) order can be found utilizing the available data. In this paper we offer an new method of solution of this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution U ^ (t , ξ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{U}(t, \xi)$$\end{document} evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of a system of governing pseudo-differential equations. [ABSTRACT FROM AUTHOR]

Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (t) (x , y ∈ I). This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which M f , g ; μ (x , y) = M F , G ; μ (x , y) (x , y ∈ I) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means. [ABSTRACT FROM AUTHOR]

We consider the problem of reconstructing an unbounded nonsmooth input of a system of ordinary differential equations that are nonlinear in the state variables and linear in control. The problem has two features. First, we assume that the state coordinates of the system are measured (with error) at discrete instants of time. Second, we assume that the unknown input is an element of the space of functions with square integrable Euclidean norm, i.e., it may be nonsmooth and unbounded. Taking into account this feature of the problem, we construct an algorithm for solving it that is stable to information noise and computational errors. The algorithm is based on a combination of constructions of the theory of ill-posed problems and the well-known extremal shift method from the theory of positional differential games. [ABSTRACT FROM AUTHOR]

An efficient non-uniform Haar wavelet method is proposed for the numerical solution of system of first order linear partially singularly perturbed initial value problem on piecewise uniform Shishkin mesh and ρ-mesh. Further, we apply same technique for solving system of second order linear partially singularly perturbed boundary value problems on piecewise uniform Shishkin mesh and q-mesh. Our method produces better results in comparison to uniform Haar wavelet, classical finite difference operator method and parameter uniform methods. We demonstrated two test problems to support the theory, accuracy and efficiency of the non-uniform Haar Wavelet method. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability, uniformly stability, asymptotically stability, boundedness, uniformly boundedness and square integrability of solutions of a system of differential equations of second order with variable delay by applying the direct method of Lyapunov- Krasovskii. By means of a new Lyapunov-Krasovskii functional, we simplify and extend some previous work that is found in the recent literature. Finally, the validity and applicability of the proceeded results are indicated by some numerical examples applying MATLAB-Simulink. By the results of this paper, we can obtain the results of Omeike et al. [Stability and boundedness of solutions of certain vector delay differential equations. J Nigerian Math Soc. 2018;37(2):77–87], Theorem 1.1 and Theorem 1.2, under weaker conditions. In addition, we establish two new results on the uniformly stability and integrabilty of solutions the considered equation. Finally, in particular cases, the applicability of the results of this paper can be shown by two new examples. These are the contribution of this paper to the subject and the relevant literature.

This paper deals with the existence of positive solutions for a system of nonlinear singular fourth-order differential equations with a parameter $\lambda$ subject two-point boundary conditions. Our analysis relies on the Krasnoselskii fixed point theorem and under suitable conditions, we derive explicit eigenvalue intervals of $\lambda$ for the existence of at least one positive solution for the system.

In this paper, a numerical study has been undertaken on the susceptible-infected-recovered (SIR) epidemic model that encompasses the mechanisms of the evolution of disease transmission; a prophylactic vaccination strategy in the susceptible populations, depending on the infective individuals. We furnish numerical and graphical simulation combined with explicit series solutions of the proposed model using the New Iterative Method (NIM) and Modified New Iterative Method (MNIM). The analytic-numeric New Iterative Method failed to deliver accurate solution for the large time domain. A new reliable algorithm based on NIM, the coupling of the Laplace transforms, and the New Iterative method is called Modified New Iterative Method (MNIM) which is presented to enhance the validity domain of NIM techniques. The convergence analysis of the MNIM has also been illustrated. The simulation results show that the vaccination strategy can slow down the spread of the epidemic rapidly. Numerical results illustrate the excellent performance of the MNIM and show that the modified method is much more accurate than the NIM.

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In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex.The solution, defined in this study, is compared with the Hukuhara- differentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, a class of rational methods of second to fourth order of accuracy are presented. The methods are developed by considering the concept of the closest points of approximation in its formulas. These methods require the application of a suitable method to calculate the starting approximation values as they are not self-starting, as well as higher derivatives of the problems. The illustrated region of absolute stability presented in the article shows that the rational schemes are all A-stable. The methods are tested on several initial value problems including systems of differential equations. From the numerical results, it appears that the proposed methods are comparable to some of the existing methods in terms of accuracy and efficiency. The capability of the schemes in solving problems whose solution possesses singularity and singularly perturbed problem can be obviously seen as they are compared to the existing rational multistep methods. The methods also show its reliability in solving stiff problems. Besides that, the schemes are tested on the Van der Pol's equation, which usually arises in physics and chemical reactions. The presented numerical solutions show that the proposed methods are comparably accurate to the existing methods. [ABSTRACT FROM AUTHOR]

Background. The work is devoted to the dynamics of a system of differential equations with diffusion interaction and additional internal constraint with cubic nonlinearity. The relevance of studies of such a system is due to the fact that a slight change in the additional coupling coefficient allows one to obtain complex scenarios of the behavior of stable equilibrium states. For the system under consideration, critical dependences were found at which the zero equilibrium state loses its stability with the appearance of two spatially inhomogeneous states in one case and a cycle in the other. For the values of the parameters close to the critical ones, asymptotic formulas were obtained for the modes branching from the zero solution. Materials and methods. For the problem in the complex, analytical and numerical solution methods were used. In the numerical study, special attention was paid to the values of the parameters at which the zero solution of the system of differential equations loses its stability. Results. Critical dependences of the parameters at which bifurcations of the zero equilibrium state occur. At the values of the parameters close to critical, a normal form was constructed and on its basis the conditions for the appearance of inhomogeneous equilibrium states in one case and a cycle in the other were determined. Conclusions. The results obtained can be used to solve problems of numerical modeling of some biophysical processes. It is also interesting to extend these results to other systems of differential equations with additional internal connection.

Purpose. In order to study the dynamic phenomena arising when rolling stock moves along a rail track both in the straight and curved track sections, the article is aimed to construct a mathematical model of DPKr-2 diesel train car. It will be constructed on the basis of mechanical model of this car of Kryukiv Railway Car Building Works. Methodology. To construct a mathematical model a system of 38 differential equations of the diesel train movement is formed. When it is used a pneumatic spring in the core stage of spring suspension, its equivalent mechanical mo-del is presented as Kelvin-Voigt knot. It includes a parallel elastic element and an element of viscous friction. Rail track flexibility is taken into account by elastic and dissipative elements. During simulation it was assumed that the wheel pair and the track weight interacting with it were moving intact. Geometric inequalities of the left and right rails were accepted as disturbances when studying the forced vertical and horizontal oscillations. Findings. On the basis of the adopted mechanical model of the diesel train car we constructed the mathematical model consisting of 38 differential equations of motion . Originality. For the first time, for the DPKr-2 diesel train car we developed its spatial mathematical model taking into account the features of the interaction of individual elements of its construction and the possibilities of the rail track depression. When constructing the mathematical model, it was proposed to take into account the flexibility of the rail track by elastic and dissipative elements. Originality. The mathematical model of the diesel train car will be used for studying the dynamic phenomena and determining the dynamic loads of structural elements during operation. The study of these phenomena is necessary for optimal choice of the scheme and parameters of rolling stock equipment, in particular antivibration devices (spring suspension, horizontal, longitudinal and transverse joints of wheel pairs with the bogie frame, bogie with the body), as well as for reduction of dynamic forces acting on the elements of rolling stock construction and rail track.

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math.41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33]. [ABSTRACT FROM AUTHOR]

We consider a system of frequency-phase self-tuning whose mathematical model is a system of differential equations. In this paper, existence conditions for asynchronous modes of a phase system are obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider the following system of differential equations, y' = f(x, y); y(x0) = y1 and z' = g(x, z); z(x0) = z1, where f, g are bounded L¹ functions defined on a rectangle in R². We give sufficient conditions for the existence of two functions Τ and ψ on an interval I containing x0, such that |y1 + ∫xx0 f(t,ɸ(t))dt - ɸ(x)| ≤ |y1 - z1|; |z1 + ∫xx0 g(t,ψ(t))dt - (x)| ≤ |y1 - z1| for all x ∊ I. To establish the same, we introduce a notation of c-cyclic contractive mapping and prove the existence of best proximity pairs for such a mapping. [ABSTRACT FROM AUTHOR]

The article is devoted to a description of a mathematical model of a tethered high-altitude unmanned platform in which the power of propulsion systems and payload is supplied from a ground-based energy source via cable. The magnitudes and directions of the forces acting on the unmanned vehicle by the cable are determined from the derived system of differential equations which allows calculating the required power transmitted from the ground to the platform, depending on the lift height and the wind load. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the parameter-uniform overlapping numerical method for a coupled first-order system with prescribed initial conditions. An overlapping interval is used to form a numerical method for this problem. The problem is discretized using the mid-point difference scheme on the boundary and non-layer regions. The convergence analysis is given, and the method is proved to be second order uniformly convergent in maximum norm with respect to the singular perturbation parameter 𝜀 Numerical experiments are conducted to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and (E , F) ⊆ (A , B) be a nonempty and proximinal pair in X such that dist (E , F) = dist (A , B) . We prove that the pair (con ¯ (E) , con ¯ (F)) is also proximinal, where con ¯ (E) denotes the closed convex hull of the set E. Moreover, we introduce a new notion of cyclic (noncyclic) mappings involving measure of noncompactness and obtain some new existence results of best proximity points (pairs). As an application of our main conclusions, we study the existence of an optimal solution for a system of integrodifferential equations under new sufficient conditions. [ABSTRACT FROM AUTHOR]

To investigate the stressed-strained state of thermoelastic anisotropic cylindrical shells the original system of nineteen differential equations of eighth order in partial derivatives of the general theory of shells is transformed into a system of eight differential equations of first order. It is suggested to carry out numerical realization by the method of orthogonal marching. Numerical results are presented for the case of temperature drop over the shell thickness. [ABSTRACT FROM AUTHOR]