Fourth-order Runge-Kutta Method Research Articles

New Analytical and Numerical Solutions for Squeezing Flow between Parallel Plates under Slip

In this article, the effects of physical flow parameters on squeezed fluid between parallel plates are explored through the Darcy porous channel when fluid is moving as a result of the upper plate being squeezed towards the stretchable lower plate, such as velocity slip, thermal slip, solutal slip, thermal stratification parameter, solutal stratification parameter, squeezing number, Darcy number, Prandtl number, and Schmidt number. The governing equations are transformed into a nonlinear ordinary differential equation using the appropriate similarity transformations. The resulting equations are solved by using the perturbation iteration method (PIT) to produce a convergent analytical solution with high accuracy. The phenomena of the squeezing fluid as the plates are moving apart and when they are coming together are illustrated using the resulting analytical solutions. Plots are used to discuss the significant effects of physical parameters on velocity, temperature, and fluid concentration profiles. The skin friction coefficient and Nusselt Sherwood values have graphical interpretations that are listed. For strong velocity slip parameters, the results demonstrate the existence of a minimum velocity profile close to the plate and a growing velocity profile distant from the plate. Additionally, as the slip effects rise, the fluid temperature and concentration both considerably drop. The results of the fourth-order Runge-Kutta method (RK4M) and the presented analytical solutions provided are in excellent agreement.

Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function. It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.

Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation

This study presents families of the fourth-order Runge–Kutta methods for solving a quadratic Riccati differential equation. From these families, the England version is more efficient than other fourth-order Runge–Kutta methods and practically well-suited for solving initial value problems in general and quadratic Riccati differential equation in particular. The stability analysis of the present method is well-established. In order to verify the accuracy, we compared the numerical solutions obtained using the England version of fourth-order Runge–Kutta method with the recently published works reported in the literature. Several counter examples are solved using the present methods to demonstrate their reliability and efficiency.

Research on NURBS Curve Interpolation Algorithm Based on Mline-Gear

Aiming at the problem of speed fluctuation caused by the large error between the ideal interpolation step and the actual interpolation step in the traditional non-uniform rational B-spline ( NURBS ) curve interpolation, a NURBS curve interpolation algorithm based on Mline-Gear is proposed. The differential and derivative definitions are derived to transform the Mline and Gear formulas respectively, and the prediction and correction formulas of the interpolation parameters are obtained. Then, it is judged whether the interpolation parameter values obtained by the correction formula meet the speed fluctuation rate requirements, and the interpolation can be continued. Otherwise, the Steffensen descent formula is used to iteratively correct the interpolation parameter values. In order to ensure the calculation accuracy and avoid solving high-order derivatives, the fourth-order Runge-Kutta method is used to solve the initial two interpolation parameters. Finally, the algorithm is simulated. The simulation results show that the algorithm reduces the feed speed fluctuation rate while ensuring a small number of iterations.

Iterated Crank–Nicolson Method for Peridynamic Models

In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, we apply the ICN algorithm for temporal discretization and the midpoint quadrature method for spatial integration. Several numerical tests are carried out to evaluate the performance of the ICN method. In general, the ICN method demonstrates second-order accuracy, consistent with the Störmer–Verlet (SV) method. When the weight is 1/3, the ICN method behaves as a third-order Runge–Kutta method and maintains strong stability-preserving (SSP) properties for linear problems. Regarding energy conservation, the ICN algorithm maintains at least second-order accuracy, making it superior to the SV method, which converges linearly. Furthermore, selecting a weight of 0.25 results in fourth-order superconvergent energy variation for the ICN method. In this case, the ICN method exhibits energy variation similar to that of the fourth-order Runge–Kutta method but operates approximately 20% faster. Higher-order convergence for energy can also be achieved by increasing the number of iterations in the ICN method.

Study on dynamic characteristics of acceleration process of spherical hybrid sliding bearings rotor system

Purpose The transient loads on the spherical hybrid sliding bearings (SHSBs) rotor system during the process of accelerating to stable speed are related to time, which exhibits a complex transient response of the rotor dynamics. The current study of the shaft center trajectory of the SHSBs rotor system is based on the assumption that the rotational speed is constant, which cannot truly reflect the trajectory of the rotor during operation. The purpose of this paper truly reflects the trajectory of the rotor and further investigates the stability of the rotor system during acceleration of SHSBs. Design/methodology/approach The model for accelerated rotor dynamics of SHSBs is established. The model is efficiently solved based on the fourth-order Runge–Kutta method and then to obtain the shaft center trajectory of the rotor during acceleration. Findings Results show that the bearing should choose larger angular acceleration in the acceleration process from startup to the working speed; rotor system is more stable. With the target rotational speed increasing, the changes in the shaft trajectory of the acceleration process are becoming more complex, resulting in more time required for the bearing stability. When considering the stability of the rotor system during acceleration, the rotor equations of motion provide a feasible solution for the simulation of bearing rotor system. Originality/value The study can simulate the running stability of the shaft system from startup to the working speed in this process, which provides theoretical guidance for the stability of the rotor system of the SHSBs in the acceleration process.

Structural design and dynamic characteristics analysis of braided composite two-stage gear transmission system

In order to realize the lightweight design of the transmission system, the braided composite material is applied to the two-stage gear transmission system. According to the structural characteristics of the two-stage gear reducer box, the whole box is designed to be assembled with the braided base and the box wall. Woven composite materials are applied to the web parts of mixed metal composite gears to realize the design goal of lightweight gears. Then, under the assumption of ignoring the influence of friction, bearings and other factors on the system, the dynamic model of the two-stage gear transmission system considering the box is established. By normalizing and dimensionless processing of the equations, the dimensionless differential equations of motion are obtained. The fourth-order Runge–Kutta method is used to analyze the relationship between the connection parameters and the dynamic characteristics of the system under the two working conditions of rigid and flexible connection between the composite base and the box wall.Through the analytical analysis of vibration displacement of two-stage gear reducer and box, the theoretical basis is found for the numerical analysis results. Finally, the dynamic characteristics of the transmission system are studied by vibration resonance analysis through high and low frequency interference. It is found that in a certain frequency range, with the decrease of the mass and moment of inertia of the transmission parts corresponding to the mixed metal composite gear, the amplitude-frequency characteristic Q of the lightweight gear and gearbox transmission system is slightly lower than that of the common gear and gearbox system, and the stability of the system is increased, and the dynamic characteristics of the system are improved.

SPREADING PATTERN OF INFECTIOUS DISEASES: SUSCEPTIBLE INFECTED RECOVERED MODEL WITH VACCINATION AND DRUG-RESISTANT CASES (APPLICATION ON TB DATA IN INDONESIA)

Mycobacterium tuberculosis is the causative agent of the infectious illness tuberculosis (TB). Indonesia is the world's third-highest TB burden country. TB transmission is prevented by the BCG vaccination. A directly observed treatment, short-course (DOTS) treatment approach can cure TB illness. Recurrent TB may occur due to either relapse or reinfection with drug-resistant bacteria. The goals of this article are formulating the SVITR model with relapse and drug-resistant cases, applying the model to the TB data in Indonesia, determining the model accuracy, determining the spreading pattern and interpreting the result, and simulating the parameters. Literature study and application methods are used in this research. The SVITR model with relapse and drug-resistant cases is a first-order nonlinear differential equation system. The model is applied to TB in Indonesia based on annual data from Indonesian Health Profile, World Bank, and WHO. The model is solved by the fourth-order Runge-Kutta method. The model is accurate enough to explain the spread of TB in Indonesia with a MAPE value of 15,5%. The spreading pattern of tuberculosis infection is upward from 2010 to 2050. In 2050, there are still 8.115.976 TB cases in Indonesia. Hence in 2050, Indonesia's free of TB target has yet to be achieved. Simulation is conducted by increasing BCG vaccination to 95%, reducing contact with TB patients to 5%, increasing treatment to 95%, and lowering relapses and drug-resistant cases to 0.00005%, so the Indonesia free of TB target in 2050 can be achieved from 2042.

Enhancing accuracy and efficiency: A novel implicit–explicit approach for fluid dynamics simulation

This study presents an innovative implicit–explicit time-stepping algorithm based on a first-order temporal accuracy method, addressing challenges in simulating all-regimes of fluid flows. The algorithm's primary focus is on mitigating stiffness inherent in the density-based “Roe” method, pivotal in finite volume approaches employing unstructured meshes. The objective is to comprehensively evaluate the method's efficiency and robustness, contrasting it with the explicit fourth-order Runge–Kutta method. This evaluation encompasses simulations across a broad spectrum of Mach numbers, including scenarios of incompressible and compressible flow. The scenarios investigated include the Sod Riemann problem to simulate compressible Euler equations, revealing the algorithm's versatility, and the low Mach number Riemann problem to analyze system stiffness in incompressible flow. Additionally, Navier–Stokes equations are employed to study viscous and unsteady flow patterns around stationary cylinders. The study scrutinizes two time-stepping algorithms, emphasizing accuracy, stability, and computational efficiency. The results demonstrate the implicit–explicit Runge–Kutta algorithm's superior accuracy in predicting flow discontinuities in compressible flow. This advantage arises from the semi-implicit nature of the equations, reducing numerical errors. The algorithm significantly enhances accuracy and stability for low Mach number Riemann problems, addressing increasing stiffness as Mach numbers decrease. Notably, the algorithm optimizes computational efficiency for both low Mach number Riemann problems and viscous flows around cylinders, reducing computational costs by 38%–68%. The investigation extends to a two dimensional hypersonic inviscid flow over cylinder and double Mach reflection case, showcasing the method's proficiency in capturing complex and hypersonic flow behavior. Overall, this research advances the understanding of time discretization techniques in computational fluid dynamics, offering an effective approach for handling a wide range of Mach numbers while improving accuracy and efficiency.

Illustration of thermal buoyancy on the flow of MHD Casson CNT–water nanofluids over a vertical stretching surface

This study addresses the applications of non-uniform heat source/sink and viscous dissipation in the magnetohydrodynamic (MHD) flow of Casson nanoparticles toward a porous stretchable sheet. The free convective flow for the inclusion of thermal buoyancy along with dissipative heat encourages the flow phenomena. The governing equations of the flow model are presented in terms of partial differential equations and then transformed into a system of ordinary differential equations using similarity transformations. These systems of equations are solved numerically by using the Runge–Kutta fourth-order method with a very efficient shooting technique. The effects of various parameters such as the Casson parameter, elastic parameter, porosity parameter, Prandtl number, non-uniform heat source/sink constant, Eckert number, skin fraction and Nusselt number on the flow area are computed and represented graphically. The present results reflect that the velocity of nanoparticles declined effectively with the porosity parameter and nanoparticle volume fraction. The temperature profile is increased with the elastic parameter and heat source parameter while decreasing with the Eckert number and Casson fluid parameter. Moreover, it is observed that when the Hartmann number is maximum, a retardation in wall shear force against nanoparticles volume fraction is marked.

A self-excited bistable oscillator with a light-powered liquid crystal elastomer

The nonlinearity in the dynamics of a liquid crystal elastomer-based systems plays an important role when designing and controlling a responsive LCE-based smart system. In this work, we proposed an LCE-based bistable oscillator and investigated its self-excited oscillation behavior. The oscillator consists of a lumped mass, connected by two lateral helical springs, and one liquid crystal elastomer fiber spring which is subjected to a steady light illumination. After inserting the dynamic model of the light-powered LCE fiber into that of the oscillator, the nonlinear vibration response is obtained by using a fourth-order Runge-Kutta method. Different from the reported mono-stable limit circle modes, the self-vibration has a snap-through oscillation mode owing to an opto-mechanical coupling where the light energy can overcome the potential well of the mechanical part. Since the light illumination energy compensates the dissipative energy loss in each nonlinear vibration period, the nonlinear vibration of the LCE-based bistable oscillator is sustainable. We discuss the influence of both the linear and nonlinear parameters on the self-excited vibration response of the bistable oscillator. The nonlinear parameters of the lateral spring, such as the stiffness ratio and the length ratio, as well as the linear parameters of the LCE fiber, including the damping ratio, spring stiffness, light intensity, light contraction coefficient, and gravity, are all taken into consideration in our analysis. Our results reveal that the self-excited vibration can be controlled by adequately tuning these linear and/or nonlinear parameters. The investigation may pave a way for promising applications such as intelligent structural design, energy harvesting, sensing, actuating, and bio-mimic/bio-inspired system using the LCE fiber.

New particle tracking method for 2D steady-state groundwater flow around wells: A modified algorithm using the stream function

Particle tracking is an essential process in analyzing groundwater flow and solute transport in aquifers. A drawback of existing particle tracking methods, which are based on approximate estimations of the velocity in a time step, is the accumulative errors of numerous time steps would lead to deviation of particles away from the exact streamlines, especially when pumping and injecting wells exist. In this study, to improve the accuracy of particle tracking for the two-dimensional (2D) steady-state flow around wells, we propose a stream-function-based (SFB) particle tracking method in which a particle is traced along the exact streamline with a given value of the stream function, and the branch cut effect caused by the multivalued stream function around a well is eliminated. Three synthetic cases are used to demonstrate the advantages of the SFB method over the existing Pollock method and the fourth-order Runge-Kutta method. In the cases of a single pumping well and an injecting-pumping well pair, the Pollock and Runge-Kutta methods yield pathways that may be far away from the exact streamline, but the SFB method performs much better. In the case with a pumping well and two injecting wells, the SFB method successfully delineated a narrow capture zone of the pumping well, which requires a high accuracy particle tracking process. Moreover, the accuracy of particle positions obtained by the SFB method is insensitive to the step-travel distance. The study would be useful for accurate investigation of flow fields in aquifers disturbed by wells.

Modeling of self-oscillating flexible circuits based on liquid crystal elastomers

Deformable and movable flexible circuits are characterized by simple construction, precise control, flexible and versatile morphology, and various functional forms, which can be used to build dynamic circuit systems, achieve the integration of drivers and controllers, and automated control of circuits, etc. Limited to conventional materials, challenges remain in designing novel self-oscillating flexible circuit. In this paper, a concept for a self-oscillating flexible circuit based on electrothermally responsive liquid crystal elastomers (LCEs) is presented, which is capable of generating periodic continuous oscillation under steady applied voltage. The governing equation of the self-oscillating LCE-based flexible circuit is formulated, and numerically calculated by fourth-order Runge-Kutta method with Matlab software. The numerical calculations reveal that the self-oscillating flexible circuit involves two motion patterns, i.e., static pattern and self-oscillation pattern. The self-oscillation arises from the coupling between the electrothermally driven deformation of the LCE film and its motion, and the electrothermal input compensates for the damping dissipation. Moreover, it is found that system parameters affect the self-oscillation of flexible circuit and that there are critical values of these parameters that trigger self-oscillation. Based on the self-oscillating flexible circuit, we conceptualized an automatic control device and a disturbance monitoring device, realizing automatic control of electrical appliances without human intervention and automatic monitoring of external disturbances. The proposed self-oscillating LCE-based flexible circuit features convenient operation, simple construction, flexibility, lightweight, and noiseless operation, rendering it suitable for constructing dynamic circuit systems, achieving actuator-controller integration, and facilitating circuit automation applications.

Numerical investigation and thermal transport features of magnetic hybrid nanoparticles flow over a poignant tiny needle subject to Joule heating and slip boundary conditions

The primary purpose of this study is to provide more information on the stable and incompressible stream of a hybrid nanofluid over a poignant tiny needle in two dimensions under slip boundary conditions. In the hybrid nanofluid flow, Al2O3 and Fe3O4 are nanoparticles, water and ethylene glycol (50:50) are considered as the base fluids. Furthermore, the impacts of Joule heating and inclined magnetic fields are considered. The PDE’s governing equations are converted into ODEs by using similarity transformations and solved by a numerical technique based on Runge–Kutta fourth-order method. The results illustrate that the crucial parameters such as the magnetic parameter, Eckert number, nanoparticles of solid volume fractions, inclined angle parameter, and Prandtl numbers significantly affect the momentum and thermal profiles. The heat transfer rate and skin friction factors are used to calculate the numerical values of various parameters, which are displayed in a table. These analyses manifest that raising the magnetic parameter results in a decrease in the hybrid nanofluid velocity under slip and no-slip circumstances. The Nusselt number has also grown as a result of the volumetric fractions of nanoparticles and the intensification of the angle parameter. This analysis might include areas such as microfluidics, biomedical devices, heat exchangers, and other engineering applications where precise control over fluid behavior and temperature distribution is important.

Theoretical treatment and implementation of the SCM included Appell-Changhee polynomials for the fractional delayed carbon absorption-emission model

Some researchers have started to perform in-depth studies on carbon dioxide emissions phenomena because global warming has caused immense damage. In this work, as a generalization of these studies, we will investigate and describe the fractional delayed Carbon absorption-emission model which consists of two fractional (Caputo sense) differential equations. From this point, we study the behavior of the solution for this model by using an approximate technique based on Appell-type Changhee polynomials (ACPs). We present an approximation of the fractional-order derivative by using ACPs. We implement the spectral collocation method (SCM) and use the properties of the ACPs to convert the provided model into a set of algebraic equations. A special focus is placed on the examination of the existence and stability of the equilibrium point for the model according to the characteristic roots with the help of the Hopf bifurcation technique. We verify the effectiveness and efficiency of the presented numerical scheme by comparing its results to those of the fourth-order Runge–Kutta method (RKM4). The results confirm that the current method is a straightforward and efficient tool for simulating the solutions to such models.

Accelerate the numerical convergence of incompressible flows: Novel preconditioned characteristic boundary conditions

For the first time, the locally power-law preconditioning method (LPLPM) is used to formulate the preconditioned characteristic boundary conditions (CBCs). Then, it is implemented to solve the numerical modeling of unsteady and steady flows from viscous to turbulent regimes. The compatibility equations and Riemann invariants are mathematically derived and then utilized to the incompressible flow solvers as suitable boundary conditions. This method discretizes time derivative and governing equations' space terms by applying the four-stage, fourth-order Runge–Kutta method, and a finite volume, respectively. The preconditioning matrix in the LPLPM is automatically derived by local velocity sensors through a power-law formulation. The baseline k−ω is applied as an appropriate turbulence model. Several test cases are conducted around airfoils of Office National d'Etudes et de Recherches Aerospatiales, NACA0012 (National Advisory Committee for Aeronautics), and S809 at varied angles of attack of 0–20 and Reynolds numbers of 500 to 5.25 × 106 to examine the effectiveness and accuracy of the LPLPM employing preconditioned CBCs. A sensitivity analysis is also performed to examine how numerical parameters affect the simulation. The results show that using preconditioned CBCs in conjunction with LPLPM at the artificial boundary is precise, reliable, and computationally efficient in simulating viscous/turbulent flows. Furthermore, it is also concluded that the present approach considerably improves the convergence speed contrasted to the simplified boundary conditions.

Numerical Solution Of The SIRV Model Using The Fourth-Order Runge-Kutta Method

This study aimed to predict the spread of the Covid-19 virus in Maluku Province using the fourth-order Runge-Kutta method. The mathematical model of the spread of the Covid-19 virus is a system of differential equations which includes Susceptible (S) variables, namely human subpopulations that are susceptible to Covid-19 virus infection, Infected (I), namely human subpopulations infected with the Covid-19 virus, Recovered (R) namely subpopulation of people who have recovered and Vaccination (V) namely a subpopulation that has been vaccinated and is immune to the Covid-19 virus, used as initial values. The values of are parameter values that are numerically solved by the fourth-order Runge-Kutta method performed for 24 literations with . Data were obtained from the Maluku Provincial Health Office from March 2022 - November 2022. Based on the data obtained, the average of the data is used as the initial value, where . The initial and parameter values were substituted into the numerical solution and simulated using Matlab. The rate value of each class for the next 24 months for the Susceptible (S), Infected (I), and Recovered (R) classes has decreased until it approaches zero equilibrium. It shows that the subpopulation of the three classes no longer exists, and the Vaccinated (V) class has increased significantly because almost all of the population has been vaccinated in the next 24 months. It shows that after an individual is vaccinated, he does not return to being vulnerable.

Optimal control and cost-effectiveness analysis for the human melioidosis model

In this work, we formulated and investigated an optimal control problem of the melioidosis epidemic to explain the effectiveness of time-dependent control functions in controlling the spread of the epidemic. The basic reproduction number (R0c) with control measures is obtained, using the next-generation matrix approach and the impact of the controls on R0c is illustrated numerically. The optimal control problem is analyzed using Pontryagin's maximum principle to derive the optimality system. The optimality system is simulated using the forward-backward sweep method based on the fourth-order Runge-Kutta method in the MATLAB program to illustrate the impact of all the possible combinations of the control interventions on the transmission dynamics of the disease. The numerical results indicate that among strategies considered, strategy C is shown to be the most effective in reducing the number of infectious classes compared to both strategy A and strategy B. Furthermore, we carried out a cost-effectiveness analysis to determine the most cost-effective strategy and the result indicated that the strategy B (treatment control strategy) should be recommended to mitigate the spread and impact of the disease regarding the costs of the strategies.

Multiscaled inviscid Taylor-Green vortex flow for examining energy conservation error in incompressible flows

This study investigates kinetic energy conservation errors in incompressible flows using multiplexed inviscid Taylor-Green vortices. Fourth-order precision Runge-Kutta methods, specifically five- and six-step methods, were employed and an inviscid two-dimensional periodic flow field was used as a test case. This method allows the energy conservation error to be suppressed to a very low level. Recent previous studies have dealt with turbulence generation using a multi-scale turbulence grid. In this study, the Taylor-Green vortex of the analytical solution is multiplexed to generate a flow field with a number of wavenumbers. Visualisation results of the initial field and the energy distribution over time were obtained. It was observed that the errors in the time evolution of the fluid energy were extremely small, and that the errors actually increased with time. However, the influence of the time step range was limited and the analytical and numerical solutions were in general agreement. As a result, it was confirmed that the inviscid Taylor-Green vortex is effective in examining the energy conservation error for incompressible flows.

Analysis of heat, mass transfer, and microorganism density rates in bioconvection nanofluid flow under the influence of thermal radiation using response surface methodology

This article presents innovative studies on bioconvection caused by microorganisms in a nanofluid flow with nonlinear radiation effects over a vertical plate. The set of governing partial differential equations is transformed by the similarity transformation into a set of nonlinearly coupled ordinary differential equations (ODEs). These are then solved numerically using the fourth-order Runge–Kutta method. The current findings are in remarkable agreement with the previous studies found in the literature. It is noted that the impact of bioconvection Lewis number and Peclet number on the microbe density rate profile gets reduced, and the increasing effect of the nonlinear thermal radiation parameter on the temperature profile is observed, which maintains the heat transmission distances inside the boundary layer. Additionally, the interactions of heat, mass transfer, and microbe density rates with the selected input parameters are discussed using the response surface methodology (RSM). This model is more applicable in a variety of fields, including synthetic biology, biofuel cells, and drug delivery devices.