Form Of Differential Equations Research Articles

Formation of the Spectral Composition of the Output Voltage of Converters for Nuclear Magnetic Resonance

In this paper, we consider the construction of an asynchronous control system for semiconductor converters for nuclear magnetic resonance. The relevance of research in this direction, main problems that arise during the construction of these systems are shown. Derived the mathematical model of the inverter, on the basis of which a control system with asynchronous pulse-width modulation is constructed. Proposed an equivalent converter circuit with a constant structure, constant parameters, and an equivalent EMF generator. Established correspondence between the equivalent circuit of the converter and its mathematical model in the form of differential equations. On the basis of mathematical equations, a structural diagram of the converter with a control system was developed and the principle of operation of the device was described according to it. To develop an algorithm for the control system, the dependence of the frequency change relative to the resonant frequency on the phase shift between the current through the filter and the voltage on the antenna circuit is determined. For which the model was built in Simulink and the corresponding simulation was carried out. Numerical dependences of reference signal frequency change and phase shift were obtained. The increase in efficiency of device for nuclear magnetic resonance is considered due to the use of multilevel inverters with tuning the frequency of operation to the frequency of the resonant circuit. Simulation of a three-level diode-clamped inverter showed that when the capacitance of the resonant circuit changes and, therefore, the resonant frequency of the circuit, the input current increases. The work obtained specific numerical dependencies. Time diagrams of voltages and currents on the main elements of the converter are given, which illustrate the implementation asynchronous pulse-width modulation in the control system. After working out the automatic frequency control algorithm, the increase in current consumption can be leveled off. The simulation results also show that it is possible to reduce the amplitude of the third harmonic. The disadvantages of the proposed system include the fact that the frequency of the converter is adjusted at each subsequent period of its operation. At the same time, the parameters of the filter can change again, which leads us to believe that adjusting the frequency will never give a 100% result, but will only allow to get as close as possible to the set parameters of the sounding signal. The work also indicates that it is possible to improve the spectral composition of the probing voltage generated by the converter by using more levels of a diode-clamped multi-level inverter. However, increasing the number of levels reduces the action speed of the system and complicates the control system itself. Therefore, the need to maintain a balance between the number of levels of the inverter and the complexity of the system is indicated.

СТОХАСТИЧНЕ МОДЕЛЮВАННЯ ОПТИМАЛЬНИХ ЗВ’ЯЗКІВ МІЖ ТІНЬОВОЮ ТА ЛЕГАЛЬНОЮ ЕКОНОМІКАМИ

The article is devoted to the stochastic modelling of optimal links between the shadow and legal economies using Wiener and Poisson processes. The study focuses on identifying the best management strategies that will help reduce the size of the shadow economy. The paper first considers a deterministic economic and mathematical model of the relationship between the shadow and legal economies, which is presented in the form of differential equations with initial conditions. These differential equations describe the rate of change of real sector workers' savings, the dynamics of owners' savings, the rate of change of the price of aggregate output, etc. The assumptions about the dynamics of cash savings and the price of aggregate output are supplemented by initial conditions. Restrictions on the consumer shares of savings used for consumption of legal and shadow products for employees and entrepreneurs are formed. To build an optimal process, the author formalizes an objective function (objective criterion) that maximizes the average total (integral) savings of employees and entrepreneurs over a certain period of time and uses stochastic sufficient conditions for optimality, where the controls are the shares of savings used to purchase aggregate products of the legal and shadow economies of employees and business owners, and the phase trajectories are the monetary savings of employees and business owners of legal and shadow economies, the price of a unit of aggregate products, and the price of a unit of aggregate products. Thus, the main idea of the process of solving the optimal management problem takes into account both the management process itself (strategies, management methods) and the paths along which the system moves from the initial state to the final result (phase trajectories). The structure of the optimal process of the proposed stochastic model is described. It is established that the optimal controls are deterministic values and do not depend on the coefficients of the Wiener processes in the proposed stochastic dynamic model. It is also worth noting that the specific formulation of this problem of optimal control of the interaction between the shadow and legal economies depends on the context of the study and may vary depending on the tasks and assumptions set.

Application of the evolutionary approach to structural and parametric identification of dynamic objects

The paper examines the evolutionary approach to structural and parametric identification of dynamic systems in the form of differential equations. The approach is based on a genetic programming algorithm to determine a structure of the equation and differential evolution method for parameters selection. The author proposed approach based on such evolutionary algorithms as genetic programming and differential evolution. The search for the structure is carried out by genetic programming. The selection of numerical parameters and initial conditions is implemented by a method of differential evolution. The problem of finding a model that describes changes in the efficiency of a hydraulic system is solved with the help of this approach. The proposed approach is compared with a recurrent neural network and a nonparametric kernel regression estimation.

Research on the distribution of heat fluxes during infrared drying of grain material

Purpose. To develop mathematical models of unsteady heat and mass transfer during the infrared (IR) drying process, and to create an algorithm for calculating the variable power of IR energy input to ensure quality drying of thermolabile material. Methods. The methods of mathematical modeling used to describe the unsteady drying process of thermolabile material in a dryer are based on the formalized description in the form of nonlinear inhomogeneous differential equations, taking into account the change in the power of IR energy input during the drying process. Results. A dependency for calculating the perceivable power of the IR source by the grain, based on the given process mode parameters, has been obtained. The drying kinetics of the grain, derived from formula (10), more accurately characterizes the process from a physical point of view than the drying equation (8) with a constant coefficient kck. It was established that three characteristic periods of the drying process can be traced in the graphical dependencies: I – heating; II – constant drying rate; III – falling drying rate. The calculation of heat fluxes for downward cross-ventilation of the grain layer, where air washes over IR heaters and reflectors and heats up to 35 °C, is presented. It was found that in the implementation of the process with a downward airflow, it is necessary to supply 20–25 % less radiant energy to the material at the same drying exposures according to the graphical dependencies. Conclusions. A mathematical description of the unsteady drying process of thermolabile material in a dryer, considering the change in the power of IR energy input during drying, has been formulated. An algorithm for calculating parameters that can be used for process identification and optimization of the operating parameters of existing grain dryers has been developed. Keywords: drying, grain material, mathematical model, diffusive transfer, moisture content, temperature, speed.

NUMERICAL STUDY OF CARREAU FLUID FLOW ALONG AN EXPONENTIAL CURVED STRETCHING SURFACE

A two-dimensional incompressible boundary layer Carneau fluid flow with heat-transfer analysis over a curved stretching surface is analyzed. The energy equation with the inclusion of thermal radiation and viscous dissipation effects is considered. The governing partial differential equations which govern such flow phenomena are transformed into suitable form of ordinary differential equations for integration by using stream function formulation. The developed non-linear problem has been solved by computational approach based on shooting technique using sixth-order Runge-Kutta method and Matlab built-in function bvp4c program. The effects of non-dimensional controlling parameters on temperature and velocity profile are analyzed with the aid of tables and figures. The surface drag force and Nusselt numbers are studied for the different values of the governing parameters. It is predicted that velocity of the fluid and boundary layer thickness is decreased when radius of curvature parameter δ is increased. Furthermore, the temperature profile dwindles for the growing values of δ. Other important information is that for shear-thinning fluid the velocity profile shows its increasing nature, whereas for shear-thickening fluid the opposite trend has been observed. For increasing values of curvature parameter δ from 2 to 1000, the temperature distribution and velocity profile is decreased. The radiative heat flux is included to enhance the temperature of the system, so, for the increasing values of radiation parameter <i>R<sub>d</sub></i> from 0.2-0.5 the temperature distribution is increased. Further, as the Biot number and Eckert number are increased from 0.2-2 and 0.1-1, respectively, the temperature distribution is increased.

An exact solution for unsteady three-dimensional magnetohydrodynamic Casson flow of dusty nanofluid over a porous stretching sheet

The unsteady three-dimensional (3D) Casson flow of a nanofluid containing dust particles over a porous, linearly stretching sheet in the presence of an external magnetic field is studied. It is assumed that the sheet is stretched in both directions along the xy plane. The governing equations of the two-phase model are partial differential equations that are transformed into ordinary equations using similarity transforms. The nanofluid is a suspension of water-based nanoparticles. In this study, we look at how nanoparticle size affect the properties of dusty nanofluid flow. The mathematical model contains the basic equations for the fluid and dust phases in the form of three-dimensional partial differential equations, which are transformed into dimensionless ordinary-dimensional equations using an appropriate similarity transformation. An exact analytical solution to this boundary value problem is obtained. The effects of various physical values on dust and nanofluid velocities are discussed in detail, including the Casson parameter, magnetic parameter, porosity parameter, fluid-particle interaction parameter, mass concentration of dust particles, and nanoparticle size. In a few specific instances, the current analytical solution demonstrates a good agreement with previously published numerical investigations.

КИНЕТИЧЕСКИЙ АНАЛИЗ ТЕХНОЛОГИЧЕСКИХ КРИТЕРИЕВ ЭФФЕКТИВНОСТИ ПРОЦЕССА ОКИСЛЕНИЯ ЭТИЛБЕНЗОЛА В ПРИСУТСТВИИ КОМПЛЕКСОВ ДИБЕНЗО-18-КРАУН-6 ЭФИРА С ХЛОРИДАМИ CA, SR, BA

A kinetic scheme is proposed that is common to the processes of ethylbenzene oxidation and ethylbenzene hydroperoxide decomposition in the presence of dibenzo-18-crown-6 ether complexes with Ca, Sr, and Ba chlorides. The scheme includes reactions of the formation of intermediate adducts from components of the reaction mixture and catalyst molecules, classical reactions of initiation, propagation and termination of the chain, reactions of initiation and propagation of the chain with the participation of intermediate adducts, and molecular reactions (includingthose with the participation of intermediate adducts).Basedonexperimentaldataandtheproposedscheme, akineticmodelwasmade.The kinetic model, written in the form of nonlinear differential equations according to the mass action law describes in time the rate of change of the concentrations of all components of reaction mixtures for ethylbenzene oxidation and ethylbenzene hydroperoxide decomposition in the presence of dibenzo-18-crown-6 ether complexes with Ca, Sr, Ba chlorides. As a result of solving the inverse kinetic problem using the zero-order direct search method, the values of reaction rate coefficients were found where the kinetic model reproduces the experimental values of the concentrations of all components of the reaction mixtures of the processes under consideration within the average relative error of 25%.As a result of the analysis based on the kinetic model, it was established that the conversion of ethylbenzene in the process of its oxidation is most affected by the reaction of ethylbenzene with the ethylbenzene peroxyl radical (target reaction).Using the kinetic model, the following was shown: 1) the values of the conversion of ethylbenzene, depending on the nature of the catalyst, line up in a row that corresponds to a number of values of the reaction rates of ethylbenzene with ethylbenzene peroxyl radical; 2) the model qualitatively reproduces the trend of selectivity changes when changing the catalyst.

Nanoparticle Shape Effects of Aligned Magnetohydrodynamics Mixed Convection Flow of Jeffrey Hybrid Nanofluid over a Stretching Vertical Plate

This study investigates the nanoparticle shape effects of aligned magnetohydrodynamics (MHD) mixed convection flow of Cu-Al2O3/water-EG Jeffrey hybrid nanofluid over a stretching vertical plate. Five different shapes of nanoparticles which are spherical, cylindrical, blades, bricks, and platelets are considered in this study. The governing equations in the form of Partial Differential Equations (PDEs) had been reduced to nonlinear Ordinary Differential Equations (ODEs) by using similarity transformation. The solver bvp4c in MATLAB software was applied to numerically solve the dimensionless governing equations towards the physical parameters which are aligned angle of magnetic field, interaction of magnetic field, mixed convection, Deborah number, volume fraction of nanoparticles, and nanoparticle shape factor. The effects of nanoparticle shapes and other relevant thermophysical parameters on fluid velocity, temperature, skin friction coefficient and Nusselt number are discussed in tables and figures. In comparison of all shapes, this study found that blade shaped nanoparticles have the highest values of skin friction coefficient and Nusselt number. The parameters blobid0.pngblobid1.pngblobid2.png and blobid3.png decrease the velocity profiles but increase the temperature profiles.

Optical solitons for the dispersive concatenation model with spatio-temporal dispersion having multiplicative white noise

This investigation of the current work presents the dispersive concatenation model that integrates the spatio-temporal dispersion and the effect of multiplicative white noise in the Itô sense. The study aims to secure stochastic optical soliton solutions within this framework. To achieve this purpose, we implemented the enhanced Kudryashov technique and the new projective Riccati equations strategy. Initially, the traveling wave transform hypothesis was applied to convert the stochastic model into its nonlinear ordinary differential equation form, encompassing real and imaginary components with emerging the related parametric restrictions. Subsequently, we introduced the enhanced Kudryashov and the new projective Riccati equations techniques successfully employed in analyzing the generated nonlinear ordinary differential equation. By utilizing the appropriate solution sets, which encompass variables with unknown values and necessary transformations, we have derived many stochastic optical soliton-type solutions for the model. Detailed discussions for the dynamics of some selected soliton solutions are presented with the supporting of the 3D and 2D graphs that illustrate the white noise effect on these solitons.

Formulation and Numerical Solution of Plane Problems of the Theory of Elasticity in Strains

This article is devoted to the formulation and numerical solution of boundary-value problems in the theory of elasticity with respect to deformations. Similar to the well-known Beltrami–Michell stress equations, the Saint-Venant compatibility conditions are written in the form of differential equations for strains. A new version of plane boundary-value problems in strains is formulated. It is shown that for the correctness of plane boundary value problems, in addition to the usual conditions, one more special boundary condition is required using the equilibrium equation. To discretize additional boundary conditions and differential equations, it is convenient to use the finite difference method. By resolving grid equations and additional boundary conditions with respect to the desired quantities at the diagonal nodal points, we obtained convergent iterative relations for the internal and boundary nodes. To solve grid equations, the elimination method was also used. By comparing with the Timoshenko–Goodyear solution on the tension of a rectangular plate with a parabolic load, the validity of the formulated boundary value problems in strains and the reliability of the numerical results are shown. The accuracy of the results has been increased by an average of 15%.

Managing the staffing process of the Samara region oil industry

The problem of constructing dynamic models in the form of differential equations or transfer functions linking the performance indicators of the oil industry of the Samara region with the performance indicators of a regional university with a large institute of oil and gas technologies, considered as the controlling factors of staffing the oil industry, is solved. The input factors of the model are the normative performance indicators of the Samara State Technical University (SamSTU). The effectiveness of the regional oil cluster of the Samara region is assessed by indicators established by the Federal State Statistics Service (Rosstat), in particular, the amount of oil received for processing. For the algorithm of personnel management of the oil industry of the region, the problem of structural and parametric identification of the personnel training process at the university based on statistical characteristics as experimental data is solved. Using the library of typical linear models of management objects by N.S. Raibman, mathematical models of the process of staffing the industry in the form of transfer functions are constructed. Verification of mathematical models is carried out by comparison with official statistical data. On the basis of the obtained transfer functions, the synthesis of algorithms for automatic management of the personnel management process of the regulatory industry in the management decision support system is carried out.

Response solutions for a kind of quasi-periodic forced neuron system

Based on the transcranial magnetic stimulation (TMS) therapy for major depression disorder and anxiety disorder, an abstract form of neuron differential equation with quasi-periodic coefficients is established. Considering the interactions among neurons, electrical effect induced by TMS therapy, and random effect, the probability density function with respect to the firing rates of neurons of the above equation is constructed, which satisfies a static Fokker–Planck equation with quasi-periodic coefficients. Hence, different clinical TMS therapies are discussed numerically by setting different parameters, such as the coupling strengths, the intensity, frequency, initial phase, phase difference of the electric field. In addition, a functional connectivity matrix related to the adjacency matrix with variable coupling strengths based on data from a real brain network is constructed. It is shown that the proper coupling strengths and the electric field parameters are crucial for TMS therapy. The methods include Kolmogorov–Arnold–Moser (KAM) theory, numerical analysis, and data analysis, having interdisciplinary characteristics. The result is original, which may be applied to TMS combined with drug therapy and the cognitive process of the brain.

Modeling and simulation analysis of current transformer based on magnetizing current as the solution variable

PurposeThis paper aims to develop a model that reflects the current transformer (CT) core materials nonlinearity. The model enables simulation and analysis of the CT excitation current that includes the inductive magnetizing current and the resistive excitation current.Design/methodology/approachA nonlinear CT model is established with the magnetizing current as the solution variable. This model presents the form of a nonlinear differential equation and can be solved discretely using the Runge–Kutta method.FindingsBy simulating variations in the excitation current for different primary currents, loads and core materials, the results demonstrate that enhancing the permeability of the B–H curve leads to a more significant improvement in the CT ratio error at low primary currents.Originality/valueThe proposed model has three obvious advantages over the previous models with the secondary current as the solution variable: (1) The differential equation is simpler and easier to solve. Previous models contain the time differential terms of the secondary current and excitation flux or the integral term of the flux, making the iterative solution complicated. The proposed model only contains the time differential of the magnetizing current. (2) The accuracy of the excitation current obtained by the proposed model is higher. Previous models calculate the excitation current by subtracting the secondary current from the converted primary current. Because these two currents are much greater than the excitation current, the error of calculating the small excitation current by subtracting two large numbers is greatly enlarged. (3) The proposed model can calculate the distorted waveform of the excitation current and error for any form of time-domain primary current, while previous models can only obtain the effective value.

Use of Reversion Method to Solve Mathematical Models on Series Electrical Circuits (Rl) with Nonlinear Inductors

Forming a mathematical model of a series electrical circuit with a nonlinear inductor obtained based on Kirchoff's laws one and two takes the form of a nonlinear differential equation. Approach methods are used to determine the electric current in the circuit, including the Reversion Method, namely by converting nonlinear differential equations into a system of linear differential equations.

Stochastic Modeling in Immunology Based On a Stage-Dependent Framework with Non-Markov Constraints for Individual Cell and Pathogen Dynamics

We present a systematic approach to modelling the responses of the immune system to virus infections. Two continuous-discrete stochastic models arising in mathematical immunology are developed and computationally implemented. The variables of the models are integer random variables that denote the quantity of individuals (cells and viral particles), and sets of unique types of individuals that take into account the current state and history of stay of individuals in some stages of their development. The distribution laws of the durations of the mentioned stages are different from exponential or geometric. A probabilistic description of a one-stage stochastic model of population dynamics is presented. A stochastic model of the development of HIV-1 infection in the lymph node in the initial period after infection of a healthy person is formulated. A computational algorithm based on the Monte Carlo method is given. Each of the stochastic models is complemented by a deterministic analogue in the form of integral and delay differential equations. The results of numerical simulation are presented.

A study of the 3-phase lag model to a two-dimensional isotropic micro-polar thermoelastic medium with memory-dependent properties

This article enlightens the two-dimensional (2D) isotropic micro-polar thermoelastic problem of the three-phase-lag (3PHL) model and heat conduction equation is formulated in the context of memory-dependent derivative (MDD). The equations are converted into the domain of the Laplace transform vector matrix differential equation form and solved by using the eigenvalue technique. To obtain an analytical solution of displacement, temperature, and stress components Silicon material properties are used. Inversion of the Laplace transform with Fourier series expansion technique is used to obtain the numerical solution. For obtaining graphical results Mathematica software is used. For the purpose of exhibiting the beauty of MDD in the present model comparisons are made between the time delay parameters and kernel functions (constant, linear, and non-linear kernels), respectively, over the micro-polar panel. The results obtained have a valuable impact on structural analysis, especially in the design of rotating machinery structures using accurate material properties.

Numerical Approach to Thermo-Convective Micro-Polar Fluid with Radiation in Permeable Medium

A steady, incompressible and thermo-convective flow of micro-polar fluid over a stretching permeable sheet with heat and mass transfer under effects of radiation, Soret, Schmidt and Dufour numbers has been analyzed. The modeled governing equations, of the classical Navier-Stokes are coupled with micro rotation, temperature and concentration equations, in the form of Partial Differential Equations (PDE’s), along with initial and boundary conditions, are transformed into a system of nonlinear coupled Ordinary Differential Equations (ODE’s) by using an appropriate transformation. The numerical solution is obtained by using the Parametric Continuation Method (PCM). For the validity of the scheme the results are compared with a numerical package bvp4c. It has been observed that both the results are in their best agreement with each other. The effects of associated parameters on the dimensionless velocity, micro-rotation, temperature and concentration profiles are discussed and depicted graphically. It has been detected that the permeability parameter give rise in micro-rotation profile.

Linearization in Financial Mathematics

We consider certain forms of nonlinear partial differential equations that arise in Financial Mathematics. Our central aim is to derive mappings that connect such equations with linear equations. We use point and contact transformations and also Hopf-Cole transformations. The concept of infinite-dimensional (Lie or contact) symmetries admitted by linearizable equations is discussed.

Modeling the Water Temperature Profile of Lakes: A Physics Informed Neural Network (PINN) Approach

Water temperature plays an important role in our environment and is applicable to nearly all limnology research, as the temperature of a body of water affects biological activity and growth of organisms such as algae and bacteria. Certain organisms have a preferred temperature range within which they can survive, while others become dormant or die when the water reaches extreme temperatures. The temperature of water also governs the maximum dissolved oxygen concentration of water. Dissolved oxygen in water is important for aquatic life because of its vital role in cellular respiration. Predicting water temperature is also an important factor in determining whether a body of water is acceptable for human use. Warm bodies of water may contain pathogens that can be dangerous to humans.
 
 Our research presents a computational method of determining lake water temperature through a novel technique known as physics-informed neural networks (PINNs). PINNs can be used to model and forecast the temperature of water over a specific time period by training a neural network using the data points derived from the discrete form of a partial differential equation and taking into account the boundary conditions. Several factors such as wind, precipitation, and solar energy effects on water temperature were investigated. By using a computer simulation in place of an analytical mathematical model, a tremendous increase in run time speed can be achieved. The results can be used to determine the patterns in water temperature throughout a year, demonstrating the advantages of a PINN over an analytical model.

Application of fixed point theorems in triple bipolar controlled metric space to solve cantilever beam problem

In this article, we introduce the notion of triple bipolar controlled metric space and prove some new fixed point theorems on this space, reformulate the concept of α-admissible mapping with respect to θ in triple bipolar controlled metric space, and introduce the concept of α-admissible crooked mapping with respect to θ in triple bipolar controlled metric space. We supported our results with suitable examples. We also present an application of the derived result to prove the existence and uniqueness solution of the cantilever beam problem, which is in fourth-order differential equation form.