Three-dimensional Differential Equations Research Articles

Analyzing epidemics and designing policies with a modified SIR model

This research paper aims to study the impact of COVID-19 on the economy, where we have extended the standard epidemiological model, the Susceptible–Infected–Recovered (SIR) model. In the economy’s dynamics, we have incorporated aggregate demand and aggregate supply and further set up a system of three-dimensional nonlinear differential equations, which is more likely the Labor–Infected–Quarantine (LIQ) model, an extension of the SIR model. Here, we have derived the steady-state condition and analyzed the model’s stability. In addition, we have done a comparative static analysis of parameters with some policy implications. We aim to achieve the twin objective, i.e., controlling the pandemic and reviving the economy under the set of the policy mix.

Stability Properties of Multi-Order Fractional Differential Systems in 3D

This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions for the asymptotic stability of such systems are obtained.

GLOBAL STABILITY OF DISEASE-FREE EQUILIBRIA IN COVID-19 SPREAD THROUGH LIVING AND INANIMATE OBJECTS MATHEMATICAL MODEL

Covid-19 is a dangerous disease that is easily transmitted, both through living media in the form of interactions with infected human, as well as through inanimate objects in the form of surfaces contaminated with the Coronavirus. Various preventive and repressive efforts have been made to prevent the spread of this disease, such as isolating and recovering the infected human. In this study, the authors construct and analyze a new mathematical model in the form of a three-dimensional differential equations system that represent the interactions between subpopulations of coronavirus living on inanimate objects, susceptible human, and infected human within a population. The purpose of this study is to investigate the criteria that must be met in order to create a population free from Covid-19 by considering inanimate objects as a medium for its spread besides living objects. The model solution that represents the number of each subpopulation is non-negative and bounded, so it is in accordance with the biological condition that the number of subpopulations cannot be negative and there is always a limit for its value. The eradication rate of Coronavirus living on inanimate objects, the recovery rate of infected human, and the interaction rate between susceptible human and infected human such that the population is free from Covid-19 for any initial conditions of each subpopulation were investigated in this study through global stability analysis of the disease-free equilibrium point of the model.

SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.
 
 The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.
 
 Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.
 
 The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.
 
 Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

Математическое моделирование натекания водорода в гермзоне реакторного здания АЭС с ВВЭР-1200

Introduction. The patterns of hydrogen-air mixtures formation during the flow of hydrogen into the upper part of a large volume room in the initial moments of time are insufficiently studied. Therefore, the determination of regularities of formation of local explosive and fire-hazardous hydrogen-air mixtures when hydrogen flows into the lower part of the dome space of the reactor building is important. Goals and objectives. The purpose of the article is a theoretical study of occurrence of local fire and explosive zones of the hydrogen-air mixture generated during hydrogen leakage in the containment area of the reactor building in order to substantiate parameters of concentration sensors of the hydrogen concentration control system. To achieve it, a zone mathematical model of hydrogen concentration calculation in a pressurized room has been developed. Numerical experiments on regularities of hydrogen-air mixtures formation have been carried out. Theoretical basis. The generalized three-dimensional non-stationary differential equation of the laws of conservation of mass, momentum and energy is used to calculate local hydrogen concentration fields. The developed zone model makes it possible to determine hydrogen concentrations in the convective column and in the ceiling layer. Results and discussion. The characteristic fields of hydrogen mass concentrations in the volume of the dome space are obtained. It has been shown that at the initial stage of hydrogen leakage, a zone of the ceiling layer is formed under the ceiling of the dome space, which confirms the validity of the usage of the zone model. The hydrogen concentrations obtained by the field and zone models are compared. The distributions of hydrogen mass concentration along the convective column height at different Reynolds numbers in the hydrogen leakage hole have been obtained. It has been shown that hydrogen concentration sensors can detect a hydrogen leakage mode in the dome space only in a narrow area of Reynolds Re = 900–5,000. There is a leakage mode (Re = 3,358), in which the maximum hydrogen volume concentration is generated at the location of the concentration sensors at the maximum size of the fire and explosive hydrogen-air mixture zones in the room. Conclusions. The hydrogen concentration sensors used in the dome space of the NPP reactor building with water-­water reactors may not detect hydrogen at the top point of the dome at a sensitivity threshold of 2 % vol. In this case, in terms of the height of the convective column, hydrogen-air mixtures are formed within the fire and explosive concentration limits.

The smallest bimolecular mass-action system with a vertical Andronov–Hopf bifurcation

We present a three-dimensional differential equation, which robustly displays a degenerate Andronov–Hopf bifurcation of infinite codimension, leading to a center, i.e., an invariant two-dimensional surface that is filled with periodic orbits surrounding an equilibrium. The system arises from a three-species bimolecular chemical reaction network consisting of four reactions. In fact, it is, up to a natural equivalence, the only such mass-action system that admits a center via an Andronov–Hopf bifurcation.

Experiment on local convective heat transfer for successive droplets impacting on heated cylindrical surface

An experiment was conducted to investigate the underlying heat transfer mechanism when successive droplets impacting on the heated cylindrical surface. The hydrodynamics of droplets were captured by visual method and the local convective heat transfer characteristics were obtained by combining direct measurement and numerical solution of three-dimensional differential equation of heat conduction. The influences of impact frequency, impact velocity and input heat flux were experimentally discussed. It is found that, based on the evolution of local wall temperature, there are three zones along the circumference, i.e., impingement zone, heat diffusion zone and tail detachment zone. Increasing the impact frequency and input heat flux enhances the convective heat transfer performance. However, when the impact frequency is high enough, continuously increasing the impact frequency, the degree of convective heat transfer improvement gradually decreases. The increase of impact velocity mainly enhances the convective heat transfer on the impingement zone and heat diffusion zone, but has little effect on the tail detachment zone. The continuous increase of impact velocity results in the splashing phenomenon and the decrease of convective heat transfer performance. Furthermore, some dimensionless correlations were proposed for predicting the droplet splashing threshold and the local convective heat transfer performance along circumference, respectively.

An improved dynamic model and numerical simulation of yarn in the ring spinning system

The nonlinear dynamic behavior of ballooning yarn in the ring spinning system is investigated. An improved dynamic model of ballooning yarn with axial conveying speed is established. Based on Hamilton’s law of variation principle, the three-dimensional dynamic differential equations are derived. Then, the shooting method is employed to solve the two-point boundary value problem of ordinary differential equations. The numerical results of balloon profiles and distribution tension curves of ballooning yarns are discussed. The numerical results imply that with the increase of Ω, the yarn conveying speed increases, the minimum tension at the guide eye increases, and the helical loops of the balloon profile decrease. It is also found that as Ω increases, the spatial area of the guide eye tension [Formula: see text] that forms the dynamic balloon decreases. When the yarn conveying speed is medium or high, under different tensions at the guide eye, the yarn tensions along the height direction of the balloon change in the same trend, and the tensions of the ballooning yarn at the same height are very close. In addition, with the increase of Ω, the distribution tension of the ballooning yarn changes significantly under the same guide eye tension.

Дифференциальные уравнения равновесия в перемещениях трехмерной геометрически и физически нелинейной теории упругости для разрывных замыкающих уравнений в декартовой системе координат

We consider the construction of coefficients of equilibrium differential equations in displacements for a three-dimensional physically and geometrically nonlinear theory of elasticity. Differential equations of equilibrium are done in a rectangular Cartesian coordinate system. Closing equations are variable modules of volumetric and shear deformation; physical relations are approximated by bilinear functions. Physical dependencies are recorded for four possible rules of continuum deformation in accordance with bilinear graphs of volume and shear deformation. The rules of deformation on each linear section of bilinear diagrams of volumetric and shear deformation are determined by the secant modules of bilinear graphs of volumetric and shear deformation diagrams. The analysis shows that the coefficients of differential equations of equilibrium in displacements, which are second-order partial differential equations of displacements along spatial coordinates, are quadratic functions of the first derivatives of displacements along spatial coordinates. The constructed three-dimensional differential equations of equilibrium in displacements can be applied in the calculation of structures using three-dimensional equilibrium equations of physically and geometrically nonlinear theory of elasticity in displacements, the closing equations of physical relations for which are approximated by bilinear functions.

Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories

Modified gravity theories can accommodate exact solutions, for which the metric has the same form as the one in general relativity, i.e., stealth solutions. One problem with these stealth solutions is that perturbations around them exhibit strong coupling when the solutions are realized in degenerate higher-order scalar-tensor theories. We show that the strong coupling problem can be circumvented in the framework of the so-called U-DHOST theories, in which the degeneracy is partially broken in such a way that higher-derivative terms are degenerate only in the unitary gauge. In this sense, the scordatura effect is built-in in U-DHOST theories in general. There is an apparent Ostrogradsky mode in U-DHOST theories, but it does not propagate as it satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface. We also clarify how this nonpropagating mode, i.e., the "shadowy" mode shows up at the nonlinear level.

3D Differential Equation Model for Patients' Choice of Hospital in China.

The number of patients in a hospital is a direct indicator of patients' choice of hospital, which is a complex process affected by many factors. Based on the national medical system and patients' preference for high-grade hospitals in China, this study establishes a three-dimensional differential equation model for calculating the time variation of the number of visits to three grades of hospitals. We performed a qualitative analysis of the system. We carried out a subsequent numerical simulation to analyze the impact on the system when the rate of leapfrog treatment and the maximum capacity of doctors and treatments changed. The results show that the sustainability of China's three levels of hospitals mainly depends on the level of hospital development. The strength of comprehensive health improvement at specific levels is the key to increasing the service efficiency of medical resources.

Solutions to Three-Phase-Field Model for Solidification

Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied.

The evaluation of actual fire resistance limits of steel structures exposed to real fire loading

Introduction. The fire resistance limits of load-bearing and enclosing structures can be identified using the method of heat-and-mass transfer calculation in case of fire. The multifactorial nature and nonlinearity of the problem makes the application of this method complicated. If necessary, the temperature regime, demonstrating the conditions of real fire, can be applied. In this work, actual fire resistance limits of metal structures of a thermal power station are calculated.Goals and objectives. The co-authors attempt to identify the actual fire resistance limits of bearing metal structures in case of the most dangerous fire development scenario.Methods. Taking into account the complexity of space-planning solutions of a building, the field-focused calculation method was selected. This method is applicable to premises, featuring complex geometric shape, where one geometric dimension is much larger than the others. Non-stationary three-dimensional differential equations of mass, momentum and energy conservation are solved for the gas medium inside a room (the Reynolds type of Navier–Stokes equations), as well as the components of the gas medium and the optical density of smoke. A heat transfer equation is solved to determine the temperature distribution inside the building structure for a one-dimensional case. The fire resistance limit of the building structure is identified as the moment in time following the start of fire, when the temperature in, at least, one point of the structure reaches a critical value.Results and discussion. Calculation results show that in case of the most dangerous fire development scenario, within 15 minutes as of the start of fire, maximum temperatures of load-bearing metal structures are far below the critical temperature of 500 °C.Conclusions. Load-bearing metal structures in an engine room, that has steam turbines, don’t need fire protection.

MATHEMATICAL MODELING OF THERMAL EFFICIENCY OF BUILDING EN-VELOPE OF MULTI-STOREY BUILDINGS WITH ACCOUNT OF INDIVIDUAL INSULATION

At present, the problem of general thermal modernization of building envelopes is given much attention both at the level of scientists and consumers. This is one of the effective ways to reduce natural gas consumption, reduce the negative impact on the environment, maintain and improve comfortable indoor conditions. Over the last decade, the population has rapidly begun to insulate their homes in order to raise the indoor air temperature to a comfortable level in the multi-storey residential sector. Due to insufficient attention of the authorities in the housing and communal sector, the lack of scientific research and widespread public awareness, there is a massive thermal insulation of building by residents of multi-store buildings within their own apartments. But the study of thermal processes that occur in individual thermal insulation of enclosing structures is currently not fully completed. Therefore, in the context of significant increases in gas and electricity prices, this problem is relevant.
 In the study was carried out mathematical modeling of a fragment of a partially insulated wall of an enclosing structure with determination of heat flux by solving a three-dimensional differential equation of thermal conductivity with boundary conditions of II, III and IV kind and distribution of characteristics of building structures and insulation. These results can be used in the analysis of the efficiency of insulation of the building taking into account the fragmentary insulation and of comparison with systemic thermal modernization.
 As a result of modeling, the three-dimensional temperature fields of wall surfaces, are determined, there are additional heat fluxes (thermal bridges), which are not considered in the simplified one-dimensional calculation. In one-dimensional calculation, the heat flux from the wall is reduced by 2.43 times during insulation. Taking into account the total heat flow from the side surfaces near the window (thermal bridges) and system insulation - by 1.75 times. With fragmentary insulation and considering the total heat flux from the side surfaces near the window - by 1.6 times.
 The next stage of calculations is the determination of the actual air temperatures in the premises of a multi-storey building considering the actual condition of enclosing structures and heating systems, heaters, mode parameters of the coolant and outdoor air parameters. The methods and means of this analysis can take into account the final data of heat loss adjustment after the mathematical modeling presented in this paper. In consequence, the results will be taken into account in the projects of thermal modernization of buildings, reconstruction of heating systems, rational placement of sources, selection of equipment and regulation of devices.

Application of Triple Shehu Transformation for solving Volterra Integro-Partial Differential Equation

This paper depends on some concepts related to the triple Shehu transform; and how its various properties can be used to solve three-dimensional linear Volterra Integro-Partial Differential Equations; in terms of obtaining the exact solutions. Some examples will be studied to support the effectiveness of this transform.

Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories

We study U-DHOST theories, i.e., higher-order scalar-tensor theories which are degenerate only in the unitary gauge and yield an apparently unstable extra mode in a generic coordinate system.We show that the extra mode satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface, and hence it does not propagate.We clarify how to treat this “shadowy” mode at both the linear and the nonlinear levels.

Similarity Criteria of Water Drive Physical Simulation of Pressure-Sensitive Fractured Reservoirs

A mathematical equation of water drive physical simulation of pressure-sensitive fractured reservoirs was established based on previous research results. In this study, the similarity criteria of water drive physical simulation of pressure-sensitive fractured reservoirs were derived according to the similarity theory. First of all, based on the three-dimensional differential equation of rock mechanics, a dimensionless analysis was conducted to determine the similarity relationship between the displacement of oil by water of pressure-sensitive fractured reservoirs, the similarity criterion was obtained, and the similarity criteria were formed. Secondly, according to the similarity criterion, the similar relationship between the stress-strain fields of the real object and the simulated object was worked out. Thirdly, the finite element software COMSOL Multiphysics was applied to model and calculate the multifield coupling process in the percolation of pressure-sensitive fractured reservoirs, verifying the correctness of the established similarity criteria and similarity relationship. The verifying results shows that the similarity between the physical model and the actual model can be realized by magnifying the geometric size N times in a certain direction and adjusting the load and boundary conditions according to the similarity principle, which can be used for the design of the pressure-sensitive fractured reservoir simulation model for a physical indoor test.

Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

Modeling of Environmental Pollution by Ammonia Emission from a Damaged Pipeline

Purpose. This work provides for the development of a hydraulic model for calculating the unsteady ammonia outflow from a damaged pipeline and the implementation of this model into a numerical model for predicting emergency air pollution. Methodology. To solve the problem, the calculated dependencies of the pressure flow hydraulics were used. An empirical model to calculate the evaporation of ammonia from a damaged pipeline was also used. To calculate the process of spreading ammonia in atmospheric air, a three-dimensional equation of convective-diffusion transfer of impurities was used. Mathematical modeling of the spread of ammonia from a damaged pipeline takes into account the change with height of the wind flow velocity, as well as the change with height of the vertical coefficient of atmospheric diffusion, the dynamics of changes over time in the intensity of ammonia leakage from the damaged pipeline. For the numerical solution of the three-dimensional differential equation for the transfer of ammonia in atmospheric air, its physical splitting is carried out: an equation that describes the transport of an impurity due to convection is singled out separately, an equation that describes the transport of an impurity due to atmospheric diffusion and separately an equation that describes a change in the ammonia concentration in air due to the action of the emission source. The McCormack method is used to numerically integrate the equation for the convective transfer of ammonia in air. The Richardson method is used to numerically integrate the equation of diffusion transfer of an impurity. The Euler method is used to numerically integrate the equation that describes the change in ammonia concentration under the influence of an emission source. Findings. Based on the developed model of unsteady ammonia outflow from the damaged pipeline and the created numerical model of ammonia propagation in the atmospheric air, a computational experiment was carried out to assess the level of atmospheric air and underlying surface pollution in the event of an emergency ammonia outflow in the section where the Tolyatti – Odessa ammonia pipeline crosses the Dnipro River. Data on non-stationary environmental pollution were obtained. Originality. A mathematical model that allows calculating the unsteady process of ammonia outflow from a damaged pipeline has been developed. A numerical model is proposed to determine the areas of contamination during an emergency ammonia outflow from the Tolyatti – Odessa ammonia pipeline. Practical value. Based on the developed model, a code has been created that makes it possible to promptly predict the environmental pollution dynamics during an emergency ammonia outflow. The proposed mathematical model can be used in the development of emergency response plan for chemically hazardous facilities.

Statements of the boundary value problems in mathematical simulation of a quasistationary electric field in the atmosphere and ionosphere

Boundary-value problems which have to be solved in the frame of the mathematical models of quasi-stationary electric fields and currents in the global conductor, that consists of the Earth’s ionosphere and the atmosphere are formulated. The approach based on the domain decomposition is used. A quadratic energy functional is constructed. It allows us to reduce the solution of the boundary value problem for a three-dimensional elliptic differential equation with asymmetric coefficients to minimizing of the functional. Estimates of the quadratic form in comparison with the Dirichlet principle for the Poisson equation are obtained. These estimates allow us to assess the condition number of the matrix of the system of linear algebraic equations, arising in the numerical solution to the problem.