First-order Ordinary Differential Equations Research Articles

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Analysis of hole wall stability in saturated clay using undrained solution for cylindrical cavity contraction

To investigate the stress state of soil during the instability of the hole wall (hole shrinkage) in saturated clay, the theory of cylindrical cavity contraction in an infinite medium is applied, utilizing the modified Cam-Clay (MCC) model. By employing intermediate variables, the issue of cylindrical cavity contraction is reformulated into a first-order ordinary differential equation in Lagrangian form, with three effective stress components and pore pressure as the unknowns. In MATLAB, the stress conditions at the elastoplastic boundary serve as the initial values for solving the initial value problem of the differential equation system. After determining the correlation between the stress distribution around the cavity and the diameter reduction ratio, the analysis proceeds to evaluate the internal support pressure required for stabilizing the hole wall according to various stability criteria. This pressure is then integrated into the hole wall stability coefficient formula within the soil pressure model, facilitating the evaluation of hole wall stability. The effectiveness and accuracy of the proposed semi-analytical elastic–plastic solutions were subsequently confirmed through comparison with finite element analysis results, underscoring its relevance for addressing hole wall stability challenges.

Combined Effect of Non-Linear Mixed Convection, and Non-Uniform Heat Source/Sink on Casson Ternary Hybrid Nanofluid Flow Across a Stretched Rotatory Disk

This present study emphasizes the importance of a specific type of fluid called a Casson ternary hybrid nanofluid. Our research explores a novel approach using the combination of several complex phenomena, including magnetohydrodynamic radiative flow, non-linear mixed convection, a non-uniform heat source or sink, a chemical reaction described by the Arrhenius model, and multiple slip effects. The researchers transformed the system of governing equations into a set of first-order ordinary differential equations using appropriate mathematical transformations and then solved them numerically using the bvp4c solver. The study investigated the behavior of velocity profiles, thermal dispersion, concentration dispersion, and heat and mass transfer for different values of the parameters involved. Our results indicated that the rising values of the heat transmission rate escalated by 2.98% for Casson ternary hybrid nanofluid when compared to ternary hybrid nanofluid. Further, ternary hybrid nanofluid had a 7.49% and 6.89% higher heat transmission rate compared to the hybrid nanofluid and conventional nanofluid, respectively. Besides, the heat transmission rate is enhanced by 17.5% and 3.11% respectively under the existence of the mixed convective and non-linear thermal convection parameters. Also, the presence of chemical reaction parameter shows a positive impact on the rate of mass transmission

Phase change in a one-dimensional functionally graded material

Functionally graded materials (FGMs) are a class of materials with engineered microstructure to achieve spatial variation in properties so as to serve a desired function. While FGMs have been well-researched for their structural properties, there is a lack of investigation of FGMs in solid-liquid phase change heat transfer processes. A well-designed functionally graded phase change material (FG-PCM) may overcome some of the well-known limitations of the phase change process. In order to investigate this possibility, this work presents a theoretical model to predict phase change propagation in a one-dimensional FG-PCM containing an arbitrary spatial variation in thermal conductivity, thermal diffusivity and latent heat. The problem is solved using the method of eigenfunction expansion to determine the temperature distribution, wherein, spatial variation in thermophysical properties listed above is accounted for by deriving a first-order matrix ordinary differential equation in the coefficients of the series solution. Results are found to correctly reduce to the well-known Stefan solution under special conditions. The impact of variation in thermophysical properties is investigated in detail. It is found that while spatial variation in thermal conductivity and latent heat has a significant effect on the rate of phase change, the impact of the thermal diffusivity function is negligible for realistic values of the Stefan number. The effect of the Stefan number governing this problem on the rate of melting is investigated. The nature of phase change under linear (1+aξ), polynomial (1+aξn) and exponential (eaξ) functional forms of thermophysical properties is investigated. It is shown that an appropriate choice of these spatial functions may help overcome the well-known self-limiting nature of phase change and produce a nearly-linear phase change propagation curve. For example, results show 33% increase in phase change propagation for a linear (1+5ξ) variation in thermal conductivity compared to baseline. The theoretical tools developed here may help design and optimize functionally graded PCMs towards improved phase change heat transfer in applications of practical interest.

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

Mathematical modelling of the emission characteristics of a field electron cathode in a scanning electron microscope under biosampling conditions

Currently, the use of electron microscopes in medicine is developing intensively, including scanning electron microscopes (SEM), which are designed to solve a huge number of problems in various fields with a wide range of electron accelerating voltages and electron beam energies. The development of an SEM with certain emission characteristics, with a range of lower beam energies for the study of biological samples, is an urgent task because modifying the SEM to solve problems in medicine, for example, would make it possible to obtain higher-quality images of biospecimens for diagnostics and monitoring the effectiveness of therapy. To develop new SEMs with certain characteristics, it is proposed to conduct less expensive research using numerical methods based on mathematical models of processes in electron-optical SEM systems. In this regard, this work sets the task of determining the size and shape of the beam, the main emission characteristics of the field electron cathode (FEC) of the SEM, which is under the influence of the electric field that excites electron emission and the external longitudinal magnetic field by studying the movement of the outermost electron of the beam, taking into account the influence of space charge beam electrons, external magnetic field. In the model, the FEC is approximated by a paraboloid of rotation, and the concept of a boundary “outermost” electron is introduced, the trajectory of which determines the shape and size of the beam. The problem of calculating the emission characteristics along the trajectory of the outermost electron of a FEC is solved using a mathematical model that includes the following equations: motion of the “outermost” electron, Maxwell outside and inside the beam, continuity of the current density, Fowler-Nordheim equation. As a result, a system of 18 first-order ordinary differential equations was obtained, the numerical calculation of which using the 4th order Runge-Kutta method allows us to obtain the emission characteristics of the FEC. As a result, it is suggested that it would be feasible to modify SEMs for more effective use in the medical field, taking into account their increasing use in disease diagnosis and the possible improvement of image quality through the development of FEC SEMs with more suitable characteristics.

Mathematical model with sensitivity analysis and control strategies for marijuana consumption

The current study focuses on reducing the use of marijuana in the population. Marijuana seems to represent a serious threat to public health in developing countries since it is an illegal narcotic with various harmful health effects. The novelty of this manuscript is to modify the non-user, experimental users, recreational users, and addict's (NERA) model for marijuana consumption by incorporating a new class of addicted users under treatment, called the hospitalized class. This real-world issue is expressed in mathematical terms using first-order non-liner ordinary differential equations, and as a result, a mathematical model for marijuana consumption is established. The general population is divided into two main groups: marijuana users and non-users. Subsequently, four subgroups of abusers are created, each reflecting a different stage or degree of substance misuse. The invariant region and basic reproduction number R0 are those parts that are used for the validation of the proposed modified model and to find out the initial transmission rate of marijuana smoking in the general population, respectively. The essential factors influencing cannabis growth were identified based on the results of sensitivity analysis. Additionally, strategies were constructed in the form of prevention objectives to prevent individuals from experiencing increasing effects of marijuana. The effectiveness of these control strategies was subsequently confirmed through numerical simulations, and the numerical simulations were carried out with the help of MATLAB using the RK-4 method.

Boundary Layer Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Carbon Nanotubes: Numerical Analysis and Response Surface Methodology under the Influence of Magnetohydrodynamics

The present study aims to offer new numerical solutions and optimisation strategies for the fluid flow and heat transfer behaviour at a stagnation point through a nonlinear sheet that is expanding or contracting in water-based hybrid nanofluids. Most hybrid nanofluids typically use metallic nanoparticles. However, we deliver a new approach by combining single- and multi-walled carbon nanotubes (SWCNTs-MWCNTs). The flow is presumptively steady, laminar, and surrounded by a constant temperature of the ambient and body walls. By using similarity variables, a model of partial differential equations (PDEs) with the magnetohydrodynamics (MHD) effect on the momentum equation is converted into a model of non-dimensional ordinary differential equations (ODEs). Then, the dimensionless first-order ODEs are solved numerically using the MATLAB R2022b bvp4C program. In order to explore the range of computational solutions and physical quantities, several dimensionless variables are manipulated, including the magnetic parameter, the stretching/shrinking parameter, and the volume fraction parameters of hybrid and mono carbon nanotubes. To enhance the originality and effectiveness of this study for practical applications, we optimise the heat transfer coefficient via the response surface methodology (RSM). We apply a face-centred central composite design (CCF) and perform the CCF using Minitab. All of our findings are presented and illustrated in tabular and graphic form. We have made notable contributions in the disciplines of mathematical analysis and fluid dynamics. From our observations, we find that multiple solutions appear when the magnetic parameter is less than 1. We also detect double solutions in the shrinking region. Furthermore, the increase in the magnetic parameter and SWCNTs-MWCNTs volume fraction parameter increases both the skin friction coefficient and the local Nusselt number. To compare the performance of hybrid nanofluids and mono nanofluids, we note that hybrid nanofluids work better than single nanofluids both in skin friction and heat transfer coefficients.

Elastoplastic solution for cylindrical cavity contraction in unsaturated soils

This paper develops an elastoplastic solution for cylindrical cavity contraction in unsaturated soils under constant suction conditions. The elastoplastic cavity contraction problem is formulated into a set of first-order ordinary differential equations by introducing a new auxiliary variable, which is solved as an initial value problem. The new solution is validated by comparison with numerical simulation results. Finally, parametric studies show that, as soil suction increases, the internal support pressure decreases faster with cavity contraction, the unloading-induced plastic zone becomes narrower, and the changes in effective stresses are smaller for a given tunnel convergence.

Theoretical investigation of the free fall of a spherical particle in a viscous fluid using continuous piecewise linearization method

This article presents a theoretical investigation of the problem of free fall of a spherical particle in a viscous fluid. The classic Boussinesq-Basset-Oseen (BBO) model for particle motion in laminar flow was modified for generalized flow by using a drag law that is applicable for 0<Re≤2.0×105. By assuming that the acceleration in the Basset force integral is constant, the Basset force effect was approximated to form an integrated added mass coefficient. Consequently, the integro-differential equation of the BBO model was transformed to a first-order nonlinear ordinary differential equation that accounts for the Basset force effect and was solved using the continuous piecewise linearization method (CPLM). The CPLM algorithm was developed based on the jerk-velocity relationship and is applicable to zero and non-zero initial conditions, steady motion, increasing or decreasing velocities and the corresponding acceleration and jerk responses. The CPLM algorithm was shown to predict published experimental results accurately and compared very well with numerical solutions and existing analytical solutions. Examination of the fall response under varying parameters showed that the fall distance, fall time and terminal velocity depend strongly on the sphere diameter, sphere density, and the density and viscosity of the fluid medium. Also, an analytical solution for the power dissipated in the fluid medium as the sphere falls to reach its terminal velocity was derived. The power dissipated was found to increase exponentially as the initial velocity deviates positively from the terminal velocity.

Prescaling Relaxation to Nonthermal Attractors.

We study how isotropic and homogeneous far-from-equilibrium quantum systems relax to nonthermal attractors, which are of interest for cold atoms and nuclear collisions. We demonstrate that a first-order ordinary differential equation governs the self-similar approach to nonthermal attractors, i.e., the prescaling. We also show that certain natural scaling-breaking terms induce logarithmically slow corrections that prevent the scaling exponents from reaching the constant values during the system's lifetime. We propose that, analogously to hydrodynamic attractors, the appropriate mathematical structure to describe such dynamics is the transseries. We verify our analytic predictions with state-of-the-art 2PI simulations of the large-N vector model and QCD kinetic theory.

Analytical analysis of spherical cavity expansion considering particle breakage effect of sand and its application for CPT tests

Particle breakage in sand significantly influences the stress and deformation predictions of the spherical cavity expansion theories. However, existing theories often overlook or lack a quantitative description of particle breakage, leading to significant errors in the actual results. Based on the Simple Critical State Sand (SIMSAND) model, considering the particle breakage effect, the traditional Euler’s problem of the drainage spherical cavity expansion is converted into a set of first-order ordinary differential equations described by Lagrangian. Fontainebleau sand is taken as an example, and the influence of the initial stress of the sand on the stress state and void ratio is studied. An analysis of the relative breakage surrounding the cavity reveals that the primary impact zone extends to a distance thrice the expansion radius. An axisymmetric model of the cone penetration test is established, the analytical and numerical solutions of expansion stress and cone tip resistance are analysed and the correctness of the spherical cavity expansion theory is verified. Furthermore, the cone tip resistance results, derived from the theory of spherical cavity expansion, are consistent with the results of numerical calculations. Notably, when the particle breakage effect is not considered, the calculated results of cone tip resistance significantly increase.

One-step three-parameter optimized hybrid block method for solving first order initial value problems of ordinary differential equations

A one-step three-parameter optimized hybrid block method and second derivative hybrid block method with optimized points were proposed to solve first-order ordinary differential equations. The techniques of interpolation and collocation were adopted for the derivation of the methods using a three-parameter approximation. The hybrid points were obtained by optimizing the local truncation error of the method. The schemes obtained were reformulated to reduce the number of occurrences of the source term. The hybrid points were used in the derivation of the second derivative hybrid block method. The discrete schemeswere produced as a by-product of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The resulting schemes are self-starting, do not require the creation of individual predictors, and are consistent, zero-stable, and convergent. The accuracy and efficiency of the methods were ascertained using several numerical test problems. The numerical results were favourably compared to some techniques from the cited literature.

Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems

This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM” for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods' capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables, demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.

A 1D model for the dynamic instability of magnetized beams in leakage flow energy harvesters

This paper presents a one-dimensional fluid–structure interaction model for investigating the dynamic stability of elastic mounted beams under the influence of magnetic forces in a leakage flow energy harvesting environment. The interaction between the beam and external magnetic fields is represented by a nonlinear long range force distribution that is non linearly dependent on the beam displacement. The formulation results in a two way coupled fluid–structure interaction system that is one way coupled with the electromagnetic system. A perturbation technique is utilized to linearize the second order equilibrium equations of the coupled system, which are then transformed into a set of first-order ordinary differential equations in the state space. The effect of the linearized magnetic force coefficient on the eigenvalues and dynamic stability of the system is analyzed through a parametric study. The results indicate the onset of oscillatory dynamics in the coupled magneto-fluid-structural system is a non monotonic function of the magnetic force.

Applications of First-Order Ordinary Differential Equations to Real-Life Systems

This article discussed applications of first-order ordinary differential equations to real-life systems, various types of differential equations with examples are presented. When a dead body is discovered somewhere, Police and detective security agencies are eager to identify the time of death and what caused the death. In this article, Newton’s Law of Cooling is used to estimate the time of death of a dead body discovered at midnight with the aid of information about the body’s surrounding temperature. It was discovered in this study that Newton’s law of cooling only works when the temperature of the body’s surroundings is kept constant. The study also considered other applications of first-order differential equations such as the population growth model and radioactive decay of radioactive isotopes and illustrative examples are given in each case.

Time-dependent simplified [formula omitted] approximation for neutron transport equation

Time-dependent transport is still too computationally expensive for various practical purposes. On the other hand, the diffusion approximation is unable to provide accurate results for many real problems. To establish a middle ground between the accuracy of the transport and the simplicity of the diffusion approximations, the simplified spherical harmonics (SPN) method was introduced. The present paper aims to improve the accuracy of the time-dependent SP3 method by employing the simplified double-P1 (SDP1) approximation, which is based on the one-dimensional double-spherical harmonics DP1 equations. Using the finite element method to treat the spatial dependence of the SDP1 equations leads to a set of first-order ordinary differential equations on the time variable. These equations are then converted into a set of linear algebraic equations by discretizing the time variable with the implicit backward finite difference scheme. Our numerical results indicate that the SDP1 approximation could compute more precise values than the widely used SP3 method in transient problems.

Design of a novel intelligent computing framework for predictive solutions of malaria propagation model.

The paper presents an innovative computational framework for predictive solutions for simulating the spread of malaria. The structure incorporates sophisticated computing methods to improve the reliability of predicting malaria outbreaks. The study strives to provide a strong and effective tool for forecasting the propagation of malaria via the use of an AI-based recurrent neural network (RNN). The model is classified into two groups, consisting of humans and mosquitoes. To develop the model, the traditional Ross-Macdonald model is expanded upon, allowing for a more comprehensive analysis of the intricate dynamics at play. To gain a deeper understanding of the extended Ross model, we employ RNN, treating it as an initial value problem involving a system of first-order ordinary differential equations, each representing one of the seven profiles. This method enables us to obtain valuable insights and elucidate the complexities inherent in the propagation of malaria. Mosquitoes and humans constitute the two cohorts encompassed within the exposition of the mathematical dynamical model. Human dynamics are comprised of individuals who are susceptible, exposed, infectious, and in recovery. The mosquito population, on the other hand, is divided into three categories: susceptible, exposed, and infected. For RNN, we used the input of 0 to 300 days with an interval length of 3 days. The evaluation of the precision and accuracy of the methodology is conducted by superimposing the estimated solution onto the numerical solution. In addition, the outcomes obtained from the RNN are examined, including regression analysis, assessment of error autocorrelation, examination of time series response plots, mean square error, error histogram, and absolute error. A reduced mean square error signifies that the model's estimates are more accurate. The result is consistent with acquiring an approximate absolute error close to zero, revealing the efficacy of the suggested strategy. This research presents a novel approach to solving the malaria propagation model using recurrent neural networks. Additionally, it examines the behavior of various profiles under varying initial conditions of the malaria propagation model, which consists of a system of ordinary differential equations.

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

BENTUK EKSPLISIT RUMUS BEDA MAJU DAN BEDA MUNDUR UNTUK TURUNAN KE-N DENGAN ORDE KETELITIAN KE-N BERDASARKAN DERET TAYLOR

In this paper, we aim to derive the explicit forms of forward and backward difference formulas based on the Taylor series. These formulas are employed to approximate the N-th derivative of a single-variable function with N-th order accuracy. The explicit formula is determined by utilizing the properties of the determinant of the Vandermonde matrix. Subsequently, we apply the obtained formulas to an example case involving a first-order ordinary differential equation