Form Of Differential Equations Research Articles

Thermal transport characteristics of fluid-particle suspension in a vertical uniform tube: A computational study on Rabinowitsch fluid with uniform heat source

Problem statement The thermal transport of fluid-particle suspension in Rabinowitsch fluid is discussed theoretically in a vertical uniform tube under the consideration of a uniform heat source. Methodology The fluid and particle phase models are used to develop the mathematical model in the form of partial differential equations which are solved through the symbolic software MATHEMATICA version 12.0 and present the exact solution of temperature distribution, velocity of fluid and particle phases, wall shear stress, pressure gradient, pressure rise, and stream function, etc. Computational results and findings The volume fraction density of the particles slows down the motion of the fluid particles. The velocity of Newtonian fluid is maximum as compared to pseudo-plastic and Dilatant fluids. The fluid phase velocity is superior to particle phase velocity distribution for all pertinent parameters of the analysis. The pressure rise increases in the case of Newtonian and pseudo plastic fluid and decreases in Dilatant fluid. The trapping phenomena are strongly associated with the volume fraction density of the particle. The shear stress shows a decreasing trend against the volume fraction density of the particles, Grashoff number, and heat source parameter. Special case The current fluid model (Rabinowitsch fluid) can be reduced in Newtonian, pseudo plastic, and Dilatant fluid for taking κ → 0 , κ < 0 , and κ > 0 , respectively. Applications The current analysis can be useful in biomedical engineering to observe the blood supply through the arteries, the removal of kidney stones, and also the passage of urine to the bladder from the ureter. Originality The present research is original and has not been published before.

Pell Collocation Approach for the Nonlinear Pantograph Differential Equations

Pantograph equations, which we encounter in the branches of pure and applied mathematics such as electrodynamics, control systems and quantum mechanics, are essentially a particular form of the functional differential equations characterized with proportional delays. This study focuses on exploring the approximate solution to the Pantograph differential equation. Since there is no analytic solutions for this equation class, only the approximate solutions can be obtain. For this purpose, Pell Collocation Method which is one of the numerical solution methods is chosen. As the result of applying the method to the equation, an algebraic equation system has been gained and then the approximate solution has been found by using MATHEMATICA via the given initial conditions. The method is applied to the some test examples and then the results are presented by both graphically and by table. The error estimations show that the method works accurately and efficiently.

Creation of a Mathematical Model of a Stationary Rail Circuit in the Form of a Finite Discrete Automaton

Purpose. Ensuring the safety of train traffic is a mandatory task in the development of technical equipment of railway transport in Ukraine. To diagnose and verify the performance of such systems, simulation models of overhead devices, in particular, the rail circle, are used. The most commonly used models are in the form of differential equations and in operator form. Unfortunately, they are not fully suitable for solving this problem. In this regard, there is a need to create a mathematical model that is easier to integrate for checking both relay electrical interlocking and microprocessor-based interlocking systems. Methodology. To achieve this goal, the authors proposed to create a mathematical model in the form of a finite discrete automaton. This paper considers the creation of a model of a station rail circuit as a directed graph. During the creation of the model, the input and output values of the model and the states are determined. The tables of inputs and outputs of the automaton are constructed, sequential expressions for the abstract model of the automaton are created, and their minimization is performed. The states of the automaton are coded using trigger circuits. Findings. In the course of the research, a mathematical model of the rail circle in the form of a Moore model finite automaton was created, and its performance was tested in the Proteus software environment. The developed model allows to simulate the operation of a stationary rail circuit at the level of abstraction, which operates with binary signals. This makes it possible to simplify the coordination of the model with microprocessor-based centralization software. In general, it is now possible to more effectively check the performance of microprocessor-based interlocking systems at the design and commissioning stages. Originality. The developed mathematical model makes it possible to determine the response of the microprocessor-based centralization software to the behavior of the rail circuit in various, in particular atypical, operating modes, as well as to determine the response of the station electrical centralization system to individual failures and to the occurrence of several failures simultaneously. Practical value. The proposed mathematical model can be used both to check the operation of microprocessor-based centralization systems at the design and implementation stages and for relay centralization systems when developing diagnostic complexes for monitoring their performance.

Heat and mass transmission through the nanofluids flow subject to exponential heat source/sink and thermal convective condition across Riga plates

The analysis of the squeezing nanofluid flow across bounded domains received great attention from researchers and engineers due to its tremendous application in automobiles, energy exchangers, and aerodynamics. In the current analysis, we are using double Riga plates parallel to each other, in order to induce the squeezing nanofluids flow with the significances of chemical reaction, thermal radiation, and heat source/sink. The nanoliquid has been prepared by the dispersion of Copper (Cu) nanoparticles (NPs) in kerosene oil (C12H26C15H32) and water (H2O). The nanofluid flow has been modeled in form of nonlinear partial differential equations, which are further convert into the dimensionless form of ordinary differential equations. The obtained set of non-dimensional equations is numerically resolved by using the Matlab built-in package bvp4c. The results are compared to another numerical technique for accuracy purposes. The detailed results are presented in Figures. It has been detected that the energy field of nanofluid is enriched with the effect of heat source/sink component. Moreover, the velocity curve magnifies with the variation of the squeezing parameter of Riga plates, whereas declines with the accumulation of Cu-NPs in both types of base fluid (C12H26C15H32 and H2O).

A parameter estimation method for chromatographic separation process based on physics-informed neural network

Chromatographic separation processes are most often modeled in the form of partial differential equations (PDEs) to describe the complex adsorption equilibria and kinetics. However, identifying parameters in such a model requires substantial computational effort. In this work, a novel parameter estimation approach using a Physics-informed Neural Network (PINN) model is developed and tested for a binary component system. Numerical accuracy of our PINN model is confirmed by validating its simulations against those of the finite element method (FEM). Furthermore, model parameters in the kinetic model are estimated by the PINN model with sufficient accuracy from the observed data at the column outlet, where parameter fitting error can be reduced by up to 35.0 % from the conventional method. In a comparison with the conventional numerical method, our approach can reduce the computational time by up to 95 %. The robustness of the PINN model has also been demonstrated by estimating model parameters from noisy artificial experimental data.

The transition from equations presented in matrix form for the perturbed Earth’s rotation to excitation equations

A convenient method for studying the influence of disturbing factors on Earth’s rotation, such as influence of oceans, atmosphere and hydrology, is representation of the Earth’s rotation equations in the excitation equations form, which was introduced by W. Munc and G. Macdonald. Traditionally, these factors are taken into account in two stages: first, an exact solution is found for an equation of the unperturbed rotation for given Earth’s model, and then the influence of small perturbation factors is taken into account in the form of corrections to the exact solution of unperturbed Earth’s rotation. The equations of unperturbed rotation for most modern theories of Earth’s rotation (and, possible, futures too) can be written in the form of matrix linear differential equations. The purpose of this study is to take into account the influence of small perturbations such as oceans, atmosphere and hydrology on Earth’s rotation in newly created theories of Earth’s rotation. A fairly general algorithm of converting perturbed equations in matrix form to the form of the excitation equations has been developed and presented. As an example, this method was applied to the relatively simple equations of the Sasao, Okubo and Saito theory based on the original Earth’s model of Molodenskii. The equations for finding corrections to the original solution are reduced to the form of excitation equations. The developed algorithm is proposed to be used to transition to equations in the form of excitation in newly created theories of the Earth’s rotation.

MATHEMATICAL MODELING OF THE AUTONOMOUS HEAT SUPPLY SYSTEM WITH TUBULAR GAS HEATERS IN BUILDING STRUCTURES WHEN OPERATING IN THE CONDENSATION MODE

Problem statement. The application of modern autonomous heat supply systems is one of the directions of reducing the consumption of natural energy resources. There are many different autonomous systems of heat supply of objects. One of the variants of such systems is the system with tubе gas heaters. A tube gas heater consists of a gas burner, a radiating tube and a fan. One of the technical solutions for such heat supply systems is a tubular heater located inside the building structure. Increase of the efficiency of gas equipment can be achieved by operating this equipment in the condensation mode. Therefore, the operation of tubular gas heaters in building structures in the condensation mode is quite interesting in terms of increasing the efficiency of utilisation of the thermal potential of gaseous fuel and ensuring its saving. For research and practical design of autonomous heat supply systems with gas tube heaters in building structures with regard to the condensation mode of operation it is essential to develop a mathematical model for calculating the thermal and hydraulic modes of the system. The purpose of the article is to develop a mathematical model of an autonomous heat supply system with tubular gas heaters in building structures when operating in the condensation mode. Conclusion. The mathematical model of hydraulic and thermal modes of autonomous heat supply system with tubular gas heaters in building structures when operating in the condensation mode was developed. It is presented in the form of differential equations. The model is based on the equations of conservation of mass, motion and energy for the gas-air mixture inside the canal in two-phase flow, the equation of heat transfer inside the building structure, the equation of heat transfer from the external surface of the heater to the environment. The mathematical model of hydraulic and thermal modes of autonomous heat supply system with tubular gas heaters in building structures when operating in the condensation mode will be used to calculate and design such systems.

Impact of Darcy-Forchheimer for the flow analysis of radiated tangent hyperbolic fluid with viscous dissipation and activation energy

In the present analysis, tangent hyperbolic fluid is assumed to be flowing over a stretching cylinder. The thermophysical characteristics of fluids are assumed to be variable. The thermal and solutal rates are investigated by using viscous dissipation, activation energy, and thermal radiation. The governing equations occur in the form of partial differential equations, and then these equations are transformed into ordinary differential equations by using similarity transformations. Then, by using the BVP4C method, we obtain the numerical solution to the equations. The impacts of the involved parameters, such as Prandtl, Weissenberg, Schmidt, Eckert number, viscosity coefficient, and thermal radiation parameters, are presented through the graphs. The velocity profile decreases due to the power law index, Weissenberg number, and temperature-dependent viscosity. However, it increases due to the Darcy-Forchheimer number and curvature parameter. The temperature distribution rises with the Eckert number, curvature parameter, and radiation parameter. On the other hand, it declined for the Prandtl number and power law index. Similarly, the concentration profile increases for the activation energy, reaction rate, and temperature difference parameter, but it decreases for the curvature parameter and Schmidt number.

Magneto-Hydrodynamic Stagnation Point Flow of Casson Williamson Hybrid Nanofluid Incorporating Viscous Dissipation and Suction/Injection Effect Past an Exponentially Stretching Cylinder

The ongoing inquiry aims to analyze stagnation point flow characteristics of magneto-hydrodynamic (MHD) Cason-Williamson hybrid nanofluids over an exponentially stretched cylinder, incorporating phenomena like viscous dissipation and suction/injection effects, as no prior investigation has been conducted on it, which represents the distinctiveness of the flow model. To facilitate analysis, the original partial differential equation (PDE) formulation of the flow model is transformed into non-dimensional ordinary differential equations (ODEs) employing dimensionless quantities, a process facilitated by the MATLAB bvp4c approach. Various non-dimensional variables are examined for their impacts on velocity profiles, temperature distribution, shearing stress, and Nusselt number. Results conveyed through graphs and detailed tables show thermal profile enhancement with escalating Weissenberg, Eckert, and Biot numbers for Casson Williamson hybrid nanofluid. Increasing copper nanoparticle volume in this fluid raises friction drag compared to the Casson hybrid nanofluid, with a 9% enhancement in shear stress. Conversely, the heat transport rate is reduced by about 1.5% for Casson Williamson hybrid nanofluid compared to Casson hybrid nanofluid. These findings significantly advance fluid dynamics and nanofluid exploration, offering opportunities for improved heat and mass transmission in various industries.

Sensitive analysis of heat transfer enhancement in ternary Casson nanofluid flow between a conical surface and disk

This study explores the ternary nanofluid flow within the canonical gap between a cone and a disk with particle deposition and magnetic field effects. Reduced titanium dioxide, magnetite, and graphene oxide are used as nanoparticles in the base fluid ethylene glycol. The governing equations of the problem are in the form of partial differential equations, which are converted to nonlinear ordinary differential equations by using appropriate similarity transformations, and they are solved numerically by using Runge–Kutta–Fehlberg fourth fifth-order (RKF 45) technique. The main agenda of this work is to discuss the impacts of parameters on three cases. The effects of essential aspects on fluid flow, heat and mass transfer rates were studied and analyzed using a graphical representation. Additionally, the response surface methodology and sensitivity analysis are carried out to enhance the importance of the heat transfer rate. The results reveal that the flow field increases significantly with increased Reynolds numbers for both cone and disk rotations. It is observed that the sensitivity analysis of the Nusselt number toward the Eckert number is more for all the radiation parameter values and the Eckert number’s middle level.

Continuous-discrete extended Kalman filtering based on the neural ordinary differential equations method

In bearings-only tracking (BOT), the accuracy of state estimation significantly diminishes when confronted with an unknown target motion model. Therefore, this study proposed a novel continuous-discrete filtering algorithm, the neural ordinary differential equations-based continuous-discrete extended Kalman filtering (NODE-CD-EKF). The approach utilizes the neural ordinary differential equation (NODE) to build the target's system model, which is then used in the time update to precisely predict the target state at the next time point within the continuous-discrete Kalman framework. The NODE concisely and accurately describes the motion model in the form of ordinary differential equations in a data-driven way. This approach addresses the limitations of traditional Kalman filters, which suffer from degraded performance due to a lack of a precise mathematical model of target states. Our method is evaluated and compared to existing multiple-model and other data-driven algorithms in simulations of BOT and another scenario using the Accumulated Root Mean Square Error (ARMSE) of position. The ARMSE of our method is only 16.30 % of the multiple-model methods at least. Overall, this approach outperforms other multiple-model and data-driven algorithms and possesses the potential for solving other nonlinear tasks in real-world applications.

P-wave propagation in attenuative elliptical VTI media

Elliptical anisotropy is convenient to use as the reference medium in perturbation methods designed to study P-wave propagation for transverse isotropy (TI). We make the elliptically anisotropic TI model attenuative and discuss the corresponding P-wave dispersion relation and the wave equation. Our analysis leads to two conditions in terms of the Thomsen-type parameters, which guarantee that the P-wave slowness surface and the dispersion relation satisfy elliptical equations. We also obtain the viscoacoustic wave equation for such elliptically anisotropic media and solve it for point-source radiation using the correspondence principle. For the constant- Q TI model, we use the weighting function method to derive the viscoacoustic wave equation in differential form. Numerical examples validate the proposed elliptical conditions and illustrate the behavior of the P wavefield in attenuative elliptical TI models.

Does dynamical wormhole evolve from emergent scenario?

In the present work we analyze a dynamical wormhole solution with two fluids system (one isotropic and homogeneous and the other being inhomogeneous and anisotropic in nature) as the matter at the throat. We choose two different forms of Equation of State(EoS) and investigate two solutions of the wormhole geometry. The properties to ensure existence and traversability has been analyzed. Also, the model of the dynamic wormhole has been examined for a possibility of the Emergent Universe(EU) model in cosmological context. Finally, for the dynamical wormholes so obtained, Null Energy Condition(NEC) has been examined near the throat.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Elasto-plastic simulation of hot shape rolling of steel by a meshless method

This paper describes the development of meshless simulation of thermomechanics of steel in a continuous hot rolling process using a novel meshless solution procedure. During the process, steel billets at high temperatures are transferred through multiple roll passes, each with a specific groove shape and roll gap. The major outcome of this simulation system is to design the rolling schedule and confirm whether the final shape is inside the acceptable range. For this purpose, a perpendicular 2D slice model approach is used to obtain fast and reasonably accurate simulation results. A return mapping algorithm solves the elasto-plastic material model with linear hardening to obtain displacements, strains and stresses. They are calculated by the local radial basis function collocation method (LRBFCM), which uses the local interpolation of the strong form of partial differential equations. The predefined slice positions are entered into the rolling system, and for each position, the slice contact line and amount of reduction are predicted and used as boundary conditions. Thermal and mechanical models are run sequentially, which is repeated until the final slice position reaches the exit from the last roll. The simulation system was validated with multiple rolling schedules provided by the industrial partner and was successfully used in the steel production plant afterwards. The meshless LRBFCM successfully solves a complex elasto-plastic large deformation problem of hot rolling for the first time.

Nonlinear oscillation of microscale fiber-reinforced composite laminated beams under a thermal loading

The work analyzes the nonlinear dynamic response of fiber-reinforced composite laminated microbeams in thermal environment, accounting for the dependence of the material properties on temperature. The governing equations of the nonlinear partial differential equations are based on the Euler-Bernoulli beam theory, and the modified couple stress theory involving the von Kármán geometrical nonlinearity. The governing nonlinear equations are reduced to a single equation by neglecting the axial inertia, which is discretized in the form of nonlinear ordinary differential equation, according to the Galerkin method, and it is solved analytically using the multiple time scales (MTS) method. A systematic investigation checks for the effect of the material length scale parameter, temperature increase, and damping parameter of the microscale beam on its time history and phase plane trajectory. Different stacking sequences are also considered within the parametric investigation. Based on results, it is noticed that the system is in a damped period-doubling bifurcation for four cycle oscillation states. It is also noticed that the stacking sequence affects significantly the nonlinear dynamic behavior of the system. It is found that the microscale composite laminated beam with [0/0/0] lay-ups has the highest dynamical deflection and the highest nonlinear frequency, followed by [0/90/0], [30/-30/30] (angle ply), [90/0/90], [45/-45/45] (angle ply) and [60/-60/60] (angle ply). However, the microscale composite laminated beam with [90/90/90] lay-ups has the lowest dynamical deflection and the lowest nonlinear frequency.

Dominance of polarization modes and absorption on self-focusing of laser beams in collisionless magnetized plasma

This work investigates the polarization modes i.e., Extraordinary (E-mode) and Ordinary (O-mode) on self-focusing of higher-order mode of elegant Hermite-cosh-Gaussian (EHChG) laser beams within collisionless magnetized plasma medium. In this study, the authors have taken transverse electromagnetic ([Formula: see text]) mode with mode indices (0, 4) and explored. The ponderomotive force is primarily responsible for the dielectric constant’s nonlinearity in this context. Also, WKB and paraxial approximations via the parabolic wave equation approach are used to establish the general form of differential equations for beam-width parameters in both transverse dimensions of the beam. In this paper, the authors have theoretically investigated the dominance of polarization modes and absorption coefficient on self-focusing. The final results show good enhancement of laser propagation through collisionless plasma medium.

Dynamics of non-Newtonian methanol conveying aluminium alloy over a rotating disc: consideration of variable nanoparticle radius and inter-particle spacing

The advancement of non-Newtonian nanofluid innovation is a crucial area of research for physicists, mathematicians, manufacturers, and materials scientists. In engineering and industries, the fluid velocity caused by rotating device and nanofluid has a lot of applications such as refrigerators, chips, heat ex-changers, hybrid mechanical motors, food development, and so on. Due to the tremendous usage of the non-Newtonian nanofluid, the originality of the current study is to explore the influence of nanoparticle radii and inter-particle spacing effects on the flow characteristics of Casson methanol-based aluminium alloy (AA7072) nanofluid through a rotating disc with Joule heating and magnetic dipole. The present problem is modeled in the form of partial differential equations (PDEs), and these PDEs are converted into ordinary differential equations with the help of suitable similarity transformations. The analytical solution to the current modeled problem has been obtained by using the homotopy analysis method (HAM) and numerical solutions are obtained by employing Runge–Kutta–Fehlberg method along with shooting technique. The main purpose of the present research work is to analyze the behavior of the velocity and temperature of the nanofluid for small and large radius of the aluminium alloy (AA7072) nanoparticles and inter-particle spacing. The radial and tangential velocities are enhanced due to rising ferro-hydrodynamic interaction parameter and the skin friction force for radial and tangential directions are enhanced 10.51% and 2.16% when h = 0.5. Also, the heat transfer rate is reduced 18.71% and 16.70% when h = 0.5% and R p = 1.5. In fact, the present results are compared with the published results and they met good agreement.

Static analysis of two-side supported 2-ply laminated glass panes through physics-informed neural networks

This paper utilizes an emerging machine learning technique, namely physics-informed neural networks (PINNs), to analyze the static mechanical behavior of two-side supported 2-ply laminated glass panes. The PINN model utilizes prior physical information in the form of partial differential equations during model training, eliminating the need for tremendous amounts of data required by traditional data-driven machine learning methods. The governing equations for the structural behavior of two-side supported 2-ply laminated glass panes are derived, based on which a PINN framework encompassing the neural network architecture and the loss functions is constructed. The training stability is evidently improved after the linearly inducing process, with the relative error maintained within a reasonable range. Extensive validation examples and case studies are provided to demonstrate the accuracy and effectiveness of the proposed method. Composite action analysis results reveal that below 70 % in the design is a conservative estimate for thinner laminated glass. Furthermore, the PINN method achieves exceptional computational efficiency, reducing computational times from six hours to just a few minutes for the uncertainty evaluation of laminated glass, indicating its potential in engineering practice.

On hierarchical competition through reduction of individual growth

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.