Fourth-order Runge-Kutta Method Research Articles

Paving the way for future advancements in superconductivity research through gold ormus studies

Abstract Background Gold ormus is a type of superconductor that can exhibit superconductivity at temperatures below 1 Kelvin, allowing it to conduct electricity without resistance. While not as widely used as other materials like niobium or lead, gold ormus is valuable for research in superconductivity. Limited studies have been conducted on gold ormus. Numerical simulations of the Ginzburg–Landau theory have yielded important results for both gold ormus. Results Class-I and class-II superconducting gold ormus, have been successfully simulated using the Runge–Kutta fourth-order method. Our analysis shows the convergence of our simulation outcomes and emphasizes the importance of considering truncation error and selecting appropriate step sizes for accurate results. The periodic factor of penetration (PFP) for each superconductor has been determined, with class-I superconducting gold ormus having a PFP of 250 nm, class-II superconducting gold ormus having a PFP of 566.2 nm. The relationship between the PFP and the length of the penetration depth has also been revealed. Conclusions Our study confirms the accuracy of the Runge–Kutta fourth-order method in simulating superconductors. By analyzing the PFP for different superconducting materials, we have identified trends in penetration depth that contribute to understanding superconductivity. Our simulations provide valuable insights for further research in the field of superconductivity. Adjusting parameters carefully ensures reliable simulations and advances progress in superconductivity research.

A Mathematical Model of Poverty and Cybercrime Dynamics in South-South, Nigeria

Poverty and cybercrime are two topical concepts that have attracted a lot of interest in Nigeria and in the global community today. This study used the compartmental mathematical modelling approach to provide a comprehensive framework for analyzing the dynamics of poverty and its relationship with cybercrime in the South-South region of Nigeria. The model was derived from a five-compartment representation from which a set of five ordinary differential equations (ODEs) was developed, which captures its dynamism. The stability analysis and determination of the conditions for stability at steady state of the system were done using the Routh Array criterion. The ODEs were solved numerically using the fourth-order Runge Kutta method. Findings reveals that intervention programmes at different stages can help decrease the population in both impoverished and cybercrime compartments in the region.

Thermal instability of three-dimensional boundary layer stagnation point flow towards a rotating disc

In this paper, the thermal instability of a three-dimensional boundary layer axisymmetric stagnation point flow towards a heated horizontal rotating disk is considered. A large number of works have been done on stability analysis. However, they did not check the thermal stability of the non-parallel-flow in the face of small disturbances that occur in the vicinity of the heated rotating disk. The governing equations of the basic flow are reduced to three coupled nonlinear partial differ-ential equations, and solved numerically with the fourth-order Runge-Kutta method. Thermal stability is examined by making use of linear stability theory based on the decomposition of the normal mode of Görtler-Hammerlin. The resulting eigenvalue problem is solved numerically using a pseudo-spectral method based on the expansion of Laguerre’s polyno-mials. The obtained results are discussed in detail through multiple configurations. As the main result, for large Prandtl numbers (Pr), the rotation disk parameter (Ω) has a destabilizing effect while for small Pr (around the unity) it tends to stabilize the basic flow. It was found that as the disk radius r→0, the flow is linearly stable, and the disturbances grow rapidly away from the stagnation point. For low values of Pr, the flow becomes more stable, and strong thermal gradients are necessary to destabilize it. However, an increase in Pr leads to a significant expansion of the instability region.

Numerical Computation and Statistical Interpretations of Heat Transfer of Tween-20/ethyl Acetate Nanofluid Flow with Melting Rheological Quality and Activation Energy

Tween-20 plays a significant role in the biological, food, and pharmaceutical industries. Additionally, it plays a vital role in improving the quality of reverse mechanisms of multi-drug resistance. This paper uses artificial neural computing to examine Tween-20 nanoparticles’ behaviors in a base fluid called ethyl acetate to enhance the applications in nanomedicine, drug delivery and biotechnology. The study focuses on the nonlinear model of a viscous fluid at the stagnation point, where mixed convection processes and activation energy are present. The study incorporates slip velocity and melting boundary conditions to examine heat and mass transfer, taking into account thermal and solutal stratification. Various fields of research and technology, specially in industrial engineering that include hydrodynamics, panto-graph systems, and biomedical mathematics, extensively utilize artificial computing. The datasets are based on velocity, temperature, and concentration outlines. The fourth-order Runge-Kutta method is used to generate the datasets. To validate the LMM-ABNNs, we have compared them with a numerical solution, showing a high level of agreement. We utilize the error histogram and mean square error results to validate the performance, scrutinize the training and testing methods, and explore the validity of the approximate answer. Furthermore, the quality characteristics such as skin friction, rate of heat, and mass movement (Nusselt and Sherwood numbers) are statistically analyzed to forecast the model’s durability. This article is the best example of investigating different fluid parameters with the latest artificial neural computing.

The Performance of Fixed Step Size and Adaptive Step Size Numerical Methods for Solving Deterministic Cell-Growth Models

Deterministic cell-growth models describe the growth of cell populations using fixed mathematical rules, assuming no randomness in the system. These models are often based on differential equations that account for the rates of cell division, death, and other biological processes. The solution to the system is obtained via numerical methods. Most of the developed approaches are based on fixed step sizes. However, fixed step size implementation failed to offer the optimal solutions when dealing with stiff challenges. Fixed step size methods can be unstable for stiff equations, where some components of the solution change much more rapidly than others. The step size, required to maintain stability can become impractically small. Thus, the adaptive step size method is required. Adaptive step size methods adjust the step size dynamically based on the behavior of the solution, aiming to maintain a desired level of accuracy while optimizing computational efficiency. These methods are particularly useful for solving ordinary differential equations (ODEs) where the solution can vary rapidly in some regions and slowly in others. This study is devoted to comparing the implementation of fixed step size and adaptive step size in solving ordinary differential equations (ODEs). The fixed step size and adaptive step size numerical method are solved in this study via the fourth order Runge-Kutta method (RK4) and Runge-Kutta Fehlberg 45 method (RKF45). The performance of both numerical methods used will be analyzed by comparing the numerical results approximated with the actual data. Subsequently, the absolute error, relative error, and rounding-off error will be calculated to compare both approaches. Based on the more precise findings, this work has shown that adaptive step size is predicted to be the optimal representation for solving ODEs. As a result, this may help mathematicians to choose the most effective numerical approach for solving ODEs.

An efficient Volterra series approach for identifying nonlinear system vibration responses to an excitation

This paper focuses on the issue of nonlinear vibration responses identification of nonlinear systems. An efficient algorithm is presented, in which the nonlinear vibration system under studied is decomposed into a multiple-input/single-output (MISO) linear system with a series of power characterized inputs based on Volterra series, and the nonlinear output responses of different orders are identified by taking power spectra operation to the input and output data, revealing the contributions of each order nonlinearity to the output of the system. Compared to the existing method, the input signal of the system required in the presented approach need not remain unchanged in waveform and adjustable in magnitude. To verify the approach, a classic Duffing–Van der Pol oscillator was simulated, obtaining results very close to the fourth order Runge–Kutta method. Finally, an experiment analysis was carried out, in which the vibration transmission properties of a bolt connection were tested when the bolt was tight and loose, respectively. The results proved that the approach is effective in identifying nonlinear vibration frequency responses.

Stability and bifurcation analysis of a 2DOF dynamical system with piezoelectric device and feedback control

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Dynamic Modeling and Configuration Transformation of Origami with Soft Creases

Origami has attracted much attention from scientists and engineers in recent years. Classic origami only permits rotations around creases, while origami-like structures in nature can also undergo a certain degree of extension at creases. In this paper, an earwig-inspired origami with rotational symmetry is proposed and analyzed by introducing soft creases. Such a soft crease is equivalent to a combination of a rotational spring and an extensional spring. The motion of the earwig-inspired origami is described using spherical triangles. Based on Lagrange's Equation, the nonlinear dynamic equation of the system is formulated. It is then solved by using the fourth-order Runge-Kutta method. The results of theoretical calculations are consistent with those of simulations in ADAMS. With the established framework, the bifurcation behaviors of the equilibria of the proposed system, including supercritical pitchfork and saddle-node bifurcations, are investigated. Such origami can realize both mono-stable and bi-stable mechanisms, while the corresponding design parameters are demonstrated in the design map. In particular, the properties of the bi-stable origami can vary with different equilibria. The configuration transformation could be achieved using a continuous excitation. However, it is sometimes sensitive to the initial conditions. Inspired by the working mechanism of earwig wings, a simple control approach for origami configuration transformation is proposed. The key is to stop exciting the origami when its deformation crosses an energy barrier. This work lays a foundation for the design and study of novel multi-stable and morphing structures and provides an efficient approach for their configuration transformation.

Comparison of Runge-Kutta Methods for Solving Nonlinear Equations

This research explores the numerical solutions of nonlinear ordinary differential equations using four distinct methods: Heun method, the Midpoint method, Ralston's method, and the fourth-order Runge-Kutta method (RK4). Given the complexity and prevalence of nonlinear equations in various scientific fields, effective numerical techniques are essential for obtaining accurate solutions. This research contributes to the understanding of numerical methods in solving complex differential equations, aiding practitioners in selecting the most appropriate approach for their needs.

Global Analysis of Meningitis Disease With Optimal Control

The meningitis epidemic has impacted lives negatively, especially in Sub-Sahara Africa, dubbed the ‘Meningitis Belt’. The epidemic has been a public health concern due to an improper understanding of the disease’s dynamics. To implement a control measure that will help minimize the epidemic, we introduce a non-linear Meningitis model that describes the dynamic behaviour of the disease and explains the transmission trend. The model explores the condition that leads to local or global asymptomatic stability of the equilibria. The model is subjected to a sensitivity analysis to find the parameters that influence the R0. The model is modified into an optimal control by adding timedependent controls. The control model is solved qualitatively using Pontryagin’s maximum principle and numerically using MATLAB and the fourth-order Runge-Kutta method. We provide a control strategy that can be relied on for management decision-making based on the results.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

Enhancing energy capacity of the car’s hood door under excitation frequency via functionally graded triply periodic minimal surface material: Application of machine learning and mathematical simulation

This study investigates the nonlinear dynamic response and vibration characteristics of a car’s hood door, focusing on its energy capacity under various excitation frequencies. The hood door is modeled using a functionally graded triply periodic minimal surface (FG-TPMS) material, which offers superior mechanical properties and lightweight design. The analysis is conducted using a higher-order shear deformation theory (HSDT), which accounts for shear deformations more accurately than classical theories. The incorporation of von Karman nonlinear terms captures the geometric nonlinearity due to large deformations, providing a realistic simulation of the hood door’s behavior under dynamic loads. To solve the complex equations of motion derived from the HSDT and von Karman terms, a fourth-order Runge–Kutta method is employed. This numerical method ensures accurate time integration and stability of the solution. The dynamic response is further analyzed to determine the energy absorption capacity of the hood door material, crucial for safety and durability in automotive applications. Validation of the numerical model is performed using previous published articles and a hybrid machine learning algorithm, which combines data-driven approaches with traditional physics-based models. This hybrid validation enhances the accuracy and reliability of the predicted responses, ensuring that the proposed model can be effectively used in the design and optimization of car components. The results demonstrate the potential of FG-TPMS materials in automotive applications, offering improved energy absorption and vibration control, which are essential for enhancing vehicle safety and performance.

A local differential quadrature method for the generalized nonlinear Schrödinger (GNLS) equation

A local differential quadrature method based on Fourier series expansion numerically solves the generalized nonlinear Schrodinger equation. For time integration, a Runge-Kutta fourth-order method is used. Matrix stability analysis is used to examine the method's stability. Three test problems involving the motion of a single solitary wave, the interaction of two solitary waves, and a solution that blows up in finite time, respectively, demonstrate the accuracy and efficiency of the provided method. Finally, the numerical results obtained from the presented method are compared with the exact solution and those obtained in earlier works available in the literature.

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic
wind loads, focusing on control strategies and multistability. We consider time varying wind speeds and analyze a specific case where hydrodynamic drag forces align
with the flow velocity. The stability of the model’s equilibrium points is analyzed
theoretically and numerically. Melnikov’s method is employed to identify conditions
for homoclinic bifurcation. Numerical simulations employing basin of attraction
confirm the analytical predictions. Our findings show a decrease in the threshold
for chaos with increasing amplitudes of external excitation, damping coefficient,
and parametric damping. The global dynamics are explored numerically using a
fourth-order Runge-Kutta method. When solely subjected to external excitation, the
system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode
oscillations, and chaos. Conversely, with self- and parametric drag forces, the system
displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity,
transient chaos, and chaos. Poincar´e maps analyze the geometric structure of chaotic
attractors, revealing a strong influence of dimensionless drag parameters. These
parameters can be manipulated to control or eliminate chaos. Furthermore, the system
exhibits multistability, with coexisting attractors. Beyond its application in protecting
infrastructure from wind damage, this research can contribute to ecological balance by
improving our understanding of tree wind resistance.

Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized [Formula: see text] distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced [Formula: see text], as well as waste [Formula: see text] and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor [Formula: see text] of regularized [Formula: see text]-distribution (RKD) distributed ei, the superthermality of ei particles [Formula: see text], the ratio of ion to electron temperature [Formula: see text], and the ratio of dust to electron density [Formula: see text] illustrated. It is noticed that the DAW shows promotion in width and amplitude as the frequency [Formula: see text] increases and it becomes rarefaction as the cut-off parameter [Formula: see text] increases. On the contrary, it becomes compressive under the impact of superthermality [Formula: see text] and [Formula: see text]. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized [Formula: see text] dispersed ei, as well as in the interstellar medium.

SACR EPIDEMIC MODEL FOR THE SPREAD OF HEPATITIS B DISEASE BY CONSIDERING VERTICAL TRANSMISSION

Hepatitis B is an infectious disease that causes inflammation of the liver due to infection with the Hepatitis B virus. Hepatitis B is divided into two phases: the acute phase and the chronic phase. Hepatitis B virus (HBV) can be prevented through vaccination and treatment of susceptible and infected individuals. The spread of the virus can be modeled using mathematical modeling of epidemics. In this study, the model used consists of four classes, namely vulnerable individuals (S), acute individuals (A), chronic individuals (C), and recovered individuals (R). The purpose of this study is to explain the formation of the Hepatitis B disease epidemic model, analyze the stability of the model, perform simulations, and conduct parameter sensitivity analysis on the basic reproductive number. The result of this study is the construction of an epidemic model of the spread of hepatitis B disease in the form of a SACR model. This model takes into account the transmission that occurs not only through interactions between susceptible individuals and chronic individuals but also through the birth process, which occurs in chronic subpopulations because babies born can be chronically infected (vertical transmission from mother to baby). The model produces two equilibrium points, the disease-free equilibrium and the endemic equilibrium. Both points were analyzed for stability using the linearization method and were found to be asymptotically stable. Furthermore, the model simulation was carried out using the fourth-order Runge-Kutta method and sensitivity analysis of the basic reproduction number. From the results obtained, it can be concluded that the spread of hepatitis B disease can be minimized by reducing contact between susceptible and chronic individuals, increasing treatment of chronic individuals, and increasing the number of vaccinated individuals in susceptible populations.

Prediction of wellbore gas hydrate generation based on temperature-pressure coupling model in deepwater testing

With the advancement of oil and gas exploration and development technology, the development of the world's oil and gas resources is gradually advancing from land and shallow waters to deep waters. During the development of gas wells in deepwater regions, the temperature of the wellbore decreases from the bottom of the well to the wellhead, and therefore, common well conditions of high pressure, low temperature, and the presence of free water are encountered, which can lead to gas hydrate plugging of the pipeline and affect the transportation of natural gas. Simulation of temperature and pressure along the wellbore and identification of gas hydrate-generating zones in the wellbore can be used to take timely preventive measures and minimize losses. In this study, a wellbore gas hydrate generation and prediction model is developed by combining the mass, momentum, and energy conservation equations, and the model is solved by the fourth-order Runge-Kutta method, taking into account the temperature-pressure coupling in the wellbore. The effects of gas production, specific heat capacity, thermal conductivity, tubing size, test duration, and ground temperature gradient on the location of gas hydrate generation and the amount of inhibitor used were analyzed in a case study well in the South China Sea. Some suggestions for practical engineering operations, such as increasing the gas production in moderation and choosing smaller tubing sizes within the acceptable range, were given. The study's results can provide a theoretical basis for flow assurance for hydrocarbon production in the wellbore of deepwater gas wells.

Using Short Time Series of Monofractal Synthetic Fluctuations to Estimate the Foreign Exchange Rate: The Case of the US Dollar and the Chilean Peso (USD–CLP)

Short time series are fundamental in the foreign exchange market due to their ability to provide real-time information, allowing traders to react quickly to market movements, thus optimizing profits and mitigating risks. Economic transactions show a strong connection to foreign currencies, making exchange rate prediction challenging. In this study, the exchange rate estimation between the US dollar (USD) and the Chilean peso (CLP) for a short period, from 2 August 2021 to 31 August 2022, is modeled using the nonlinear Schrödinger equation (NLSE) and calculated with the fourth-order Runge–Kutta method, respectively. Additionally, the daily fluctuations of the current exchange rate are characterized using the Hurst exponent, H, and later used to generate short synthetic fluctuations to predict the USD–CLP exchange rate. The results show that the USD–CLP exchange rate can be estimated with an error of less than 5%, while when using short synthetic fluctuations, the exchange rate shows an error of less than 10%.

Stochastic Optimal Control Analysis for HBV Epidemic Model with Vaccination

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV.

Effect of side ratio on the two-degrees-of-freedom vortex-induced vibrations of the rectangular cross section model

The effect of the mutative side ratio (D/B) on the vortex-induced vibration (VIV) characteristic, aerodynamic characteristics, and the surrounding time-averaged and transient flow field of a rectangular cross section model were simulated numerically. Based on Fluent 19.0 platform, overset grid technology and the fourth-order Runge–Kutta method were used at Re 22 000. First, a rectangular cross section model with D/B = 0.25 was selected, and the simulation method and parameter settings were validated against previous literatures. The subsequent analysis compared and evaluated the effect of side ratio on the VIV response by focusing on statistical values of aerodynamic force coefficients, self-spectra, amplitude ratio, motion trajectory, and phase transition changes for stream-wise and cross-flow directions. Moreover, the study examined the influence of different models at different reduced velocities (Ur) on wake vortex-shedding. The findings suggest that, within a fixed cross-sectional area, a smaller side ratio leads to a weaker VIV characteristic and notably lower aerodynamic performance compared to a larger side ratio. The vortex-shedding mode of the rectangular cross section, particularly with a large side ratio, is less sensitive to changes in Ur compared to the standard square cylinder. An examination of the Reynolds number (Re) effect on the minimum and maximum side ratio models reveals that it has a more pronounced impact on the aerodynamic performance and VIV of the cross-flow when compared to in-line flow. In general, it is noted that larger side ratio model exhibits a stronger sensitivity to the variation of Re.