Form Of Differential Equations Research Articles

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.

Stagnation point on MHD boundary layer flow of heat and mass transfer over a non-linear stretching sheet with effect of Casson nanofluid

ABSTRACT The present study analyzes the effects of the magnetohydrodynamic stagnation point and Casson fluid flow considered over the stretching surface. Due to the growing need for non-Newtonian nanofluid flows in industry and engineering areas, the present work focuses on the Casson nanofluid flow over a non-linear stretching surface with effects. The Casson nanofluid is more helpful for cooling and friction-reducing agents compared to Newtonian-based nanofluid flow. The mathematical explanation of the problem is elaborated with the help of partial differential equations. The coupled nonlinear form of ordinary differential equations (ODEs) is solved numerically by the Keller-Box method using a suitable MATLAB program. On the effect of the Casson parameter on the velocity parameter, it is observed that for large values of the Casson parameter, the velocity profile decreases. The reason behind this behavior is that increasing the values of the Casson parameter increases the fluid viscosity, i.e. reduces the yield stress. Therefore, the momentum boundary layer thickness reduces and the thermal boundary layer thickness increases.

OVERVIEW OF PHYSICS-INFORMED NEURAL NETWORKS APPLICATIONS

When simulating various physical phenomena, the law of the phenomenon is often known in advance, in the form of a partial differential equation, that needs to be solved. Numerical methods, such as the finite element method, have been developed over decades, and these methods approximate the solution to the partial differential equation. However, these methods can be computationally demanding. On the oth- er hand, neural networks, can provide predictions that approximate the given partial differential equation. Neural networks are computationally more efficient than numerical methods, but they often face issues of generalization and consequently problems with solution accuracy. Insufficient generalization, among other things, can result from data collected from numerical simulations. In the last few years, physics-informed neural networks are being developed, for which it’s not necessary to gather data from simulations. These networks use automatic differentiation and during training, they minimize the residuals of the partial differ- ential equation, its initial, and boundary conditions. After training, these neural networks can be used as a replacement for traditional numerical solvers.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

In the recent decade, the reliability analysis of a stochastic system coupled with the uncertainty related to the system’s parameter has attracted much attention. Probability density evolution method (PDEM) is one of the viable options that estimates the probability density function of the structural response by solving generalized density evolution equations (GDEEs). The advantage of PDEM is that it is derived based on the principle of probability conservation, where GDEEs are decoupled from the physical system. In general, GDEEs in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, leading to computationally inefficient for high-fidelity models. With this in view, this study proposes a physics-informed neural network (PINN), a novel deep learning method, based PDEM, for solving the GDEEs. PINN utilizes physical information in the form of differential equations to enhance the performance of the neural networks. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Three numerical examples are presented in this study to illustrate the proposed PINN-based PDEM, including a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resisting frame coupled with a nonlinear energy sink with negative stiffness and sliding friction, and a high-rise timber building coupled with shape memory alloy-based outriggers. The first example is utilized to show the accuracy of the proposed method by comparing results with the Fokker–Planck–Kolmogorov equation and Monte Carlo simulation. The rest two examples are investigated for estimating time-dependent probability of failure. Numerical results show that the proposed PINN-based PDEM can estimate the probability of failure efficiently.

The continuous stochastic gradient method: part I–convergence theory

In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.

Darcy resistant of Soret and Dufour impact of radiative induced magnetic field sutterby fluid flow over stretching cylinder

The incompressible two-dimensional steady flow of Sutterby fluid over a stretching cylinder is taken into account. The magnetic Reynolds number is not deliberated low in the present analysis. Radiation and variable thermal conductivity are considered to debate the impact on the cylindrical surface. The Dufour and Soret impacts are considered on the cylinder. The mathematical model is settled by employing boundary layer approximations in the form of differential equations. The system of differential equations becomes dimensionless using suitable transformations. The dimensionless nonlinear differential equations are solved through a numerical scheme(bvp4c technique). The flow parameters of physical effects on the velocity, temperature, heat transfer rate, and friction between surface and liquid are presented in tabular as well as graphical form. The velocity function declined by improving the values of the Sponginess parameter. The fluid temperature is reduced by increment in curvature parameter.

Well-Testing Model for Dual-Porosity Reservoir considering Stress-Sensitivity and Elastic Outer Boundary Condition

Stress sensitivity and the elastic outer boundary (EOB) condition have a great effect on the analysis of the characteristics of the fluid flow in a reservoir. When researchers analyzed the characteristics of the fluid flow, they have considered the stress sensitivity and the EOB condition separately but have not considered them simultaneously. Therefore, errors are inevitable during the analysis of well testing. The main object of this work is to present a well-testing model for stress-sensitivity dual-porosity reservoir (DPR) with EOB to improve the accuracy of the analysis of well-testing data. To this end, in this paper, we established a well-testing model for the DPR, considering the stress sensitivity and the EOB simultaneously, and presented its semianalytical solution. On the basis of the consideration of the EOB condition and stress sensitivity of permeability (SSP), a seepage model for the DPR with the EOB is built using the continuity equation, motion equation, state equation, and interporosity flow equation between matrix and fracture, which considers the stress sensitivity, wellbore storage, and skin. To solve this model, a nonlinear partial differential equation is changed into a linear form of a partial differential equation by introducing an effective well radius and applying Pedrosa’s transformation and perturbation transformation. Applying the Laplace transformation, an analytical solution in the Laplace space is obtained, and curves of pressure and pressure derivative (PPD) are drawn by numerically inverting them. The model is verified by comparing it with the EOB without consideration of SSP and using case data. The sensitivity of parameters on the curves of PPD is analyzed. This work may be significant for evaluating more accurately the parameters of wells and reservoirs using well testing.

Improved artificial gorilla troops optimizer with chaotic adaptive parameters - application to the parameter estimation problem of mixed additive and multiplicative random error models

For mixed additive and multiplicative random error models (MAM models), due to the complex correlation between the parameters and the model power array, derivative operations will be inevitable in the actual calculation. When the observation equation is in nonlinear form, the operations will be more complicated. The swarm intelligence optimization algorithm (SIO) can effectively solve the derivative problem when estimating the nonlinear model parameters using conventional iterative algorithms. However, for different problems, the conventional SIO cannot effectively balance the ability of global and local behavior, resulting in the algorithm falling into prematureness and failing to output effective parameter information. To address the above problems, the improved artificial gorilla troops optimizer (CAGTO) algorithm with chaotic adaptive behavior is proposed. To address the problem that the population generated by the algorithm using pseudo-random numbers in the initialization population phase has poor traversability in the feasible domain, the chaotic sequence is applied to initialize the population instead of pseudo-random number generation to ensure that the population can traverse the feasible domain as much as possible and improve the global search capability of the algorithm. Adaptive parameters that vary linearly and nonlinearly with the algorithm process are constructed to balance the global search and local search ability, while accelerating the convergence speed. Two CAGTO algorithms with different parameter settings are constructed for different problems, and the experimental results show that both CAGTO algorithms can effectively solve the parameter estimation problem of MAM models with different nonlinear forms of observation equations compared with several other comparative algorithms.

Magnetohydrodynamic Stagnation-point Flow towards a Permeable Stretching/Shrinking Sheet with Slip and Heat Generation/Absorption Effects

This study of the magnetohydrodynamic (MHD) stagnation-point flow towards a permeable stretching/shrinking sheet in the presence of slip and heat generation/absorption effects is considered. The governing equations in the form of partial differential equations are transformed into a system of ordinary differential equations by using similarity transformation, and then solved numerically using bvp4c function in Matlab software. The variations of the numerical solutions for the skin friction coefficient and the local Nusselt number as well as velocity and temperature profiles are obtained for several values of the governing parameters. It is found that the solution is unique for the stretching case whereas dual (first and second) solutions exist for the shrinking case in certain range of parameters.

Nonlinear vibration of porous truncated conical shell under unified boundary condition and mechanical load

The aim of this article is to research nonlinear vibration responses of a metal porous truncated conical shell undergoing mechanical harmonic loads in thickness direction. The artificial spring technique is employed to unify boundary condition which can arrive at arbitrary classical boundary condition in real-word by changing the stiffness of springs. Utilizing Hamilton's principle, theoretical formulations of the conical shell are developed in view of FSDT and von-Kármán geometric relation. Afterwards, by employing generalized differential quadrature method (GDQM), the governing equation of nonlinear vibration in the form of partial differential equation is truncated and discretized into an ordinary differential equation which is a high-dimensional nonlinear dynamic system described in physical space rather than the mode space. On this basis, the analysis of effect of exciting frequency on the amplitude is carried out for three different distributing types of porous truncated conical shells by numerical integration. The frequency-amplitude responses of the system are depicted utilizing the sweeping frequency. The bifurcation diagrams, time series and phase diagrams are utilized to analyze the nonlinear dynamics of the porous truncated conical shell under unified boundary condition. It is found that the multiple solutions high-incidence area is near the 1st primary resonance, and vibration amplitude is dominated by the 1st order primary resonance and the 3rd one. The hardening-spring nature decreases for Type3, Type2 and Type1 successively.

Modeling and analysis of the cutting temperature of titanium alloys for quasi-intermittent vibration assisted swing cutting

EVC has proven to be an effective means of machining difficult-to-process materials. QVASC inherits EVC's intermittent cutting and friction reversal advantages, eliminating EVC cutting residues and improving machining quality. In order to deeply understand the cutting mechanism of QVASC, it is necessary to understand the cutting heat generation and tool surface temperature distribution in the process of QVASC. In this paper, the kinematic and frictional characteristics of the QVASC machining process are analyzed, the heat generation and heat transfer model of QVASC machining process is established, and the cutting temperature prediction model considering the characteristics of QVASC machining is proposed. According to whether there is heat convection and frictional shear stress distribution between the tool and the environment, the heat flow distribution of the front surface of the tool is divided into four regions, and the heat flux of the tool is calculated respectively. On this basis, the tool temperature of QVASC is solved by using the heat balance equation in the form of the partial differential equation. Finally, a systematic comparison experiment was carried out, and the effects of cutting depth, cutting speed, tool swing frequency, and amplitude on cutting temperature were studied by single factor method. The results show that the average cutting temperature of QVASC is 10 % lower than that of OC, which verifies the advantage of QVASC in reducing cutting temperature. Under different cutting depth, cutting speed, tool swing frequency and amplitude, the change trend of theoretical and experimental cutting temperature is the same, and the average error between them is 9 %, 8 %, 10 % and 9 %, respectively, which is within the acceptable range, proving the validity of the theoretical model. This research contributes to a deep understanding of the cutting process of QVASC and effectively determines the optimal cutting parameters.

Hamilton Equations of Pendulum-Spring System

Abstract. The pendulum-spring system was studied by using Hamilton equations of motion. The total Hamiltonian of this system is complicated because of the complex mechanical system. The paper begins by introducing the physical system and defining a fixed coordinate system. Following this, the Lagrangian and Hamilton equations of motion are derived. Six equations of motion are obtained from the Hamilton equations in the form of ordinary differential equation. These equations can be employed for simulating the dynamical behavior of the system. The primary goal of this paper is to familiarize physics students with the Hamilton equations of motion as applied to the pendulum-spring system.Keywords: Hamiltonian, pendulum, equation of motion, ODE

HAM-based analysis of nonlinear convection of two-layer water and air-TiO2 flow model in a vertical channel

This article investigates the nonlinear convection transport of heat transfer in a nanoparticle laden two-layer flow in a vertical channel under the additional effects of magneto hydrodynamic (MHD) and radiation. Water and compressed air are chosen as the fluids in two-layer flow. This type of flow is commonly seen in the transportation of oil and gas in pipelines, therefore, correct understanding of flow dynamics including convective transport phenomenon, layers interaction, and graphical interpretation of flow variables is important for better designing of equipments. Titanium dioxide ( Ti O 2 ) nanoparticles are added into the air to magnify the transport mechanism. The mathematical model incorporating buoyancy, nonlinear convection, MHD, viscous dissipation, and radiation effects is presented in the form of nonlinear Ordinary Differential Equation (ODE’s). Then utilizing homotopy analysis method, these interconnected differential equations are resolved. A visual illustration shows how different physical factors affect fluid rate and temperature. Tabulated values of Nusselt number at both walls are presented to discuss the numerical data of convection and found that it decreases toward the left boundary but rises toward the right wall by increasing Eckert number and Prandtl number. The results of this two-layer flow will help in understanding this complex flow which has practical applications in various engineering processes and further enhance the scientific knowledge related to multilayer problems.