Order Runge-Kutta Method Research Articles

Breakdown of quantization in nonlinear Thouless pumping

The dynamics of solitons driven in a nonlinear Thouless pump and its connection with the system’s topology were recently explored for both weak and strong nonlinear strength. Using both a self-consistent algorithm and 4th order Runge Kutta method, this work uncovers the fate of nonlinear Thouless pumping in the regime of intermediate nonlinearity, thus establishing a fascinating crossover from the observation of nonzero and quantized pumping at weak nonlinearity to zero pumping at strong nonlinearity. We identify the presence of critical nonlinearity strength at which quantized pumping of solitons breaks down regardless of the protocol time scale. Such an obstruction to pumping quantization is attributed to the presence of self-crossing in nonlinear topological bands. By considering another type of pumping involving Bloch states, we further show how the presence of self-crossing bands also leads to breakdown of quantization, but in a completely different manner from that in the case of soliton pumping. Our results not only unveil a missing piece of physics in nonlinear Thouless pumping, but also provide a means to detect loop structures of nonlinear systems investigated in real space and momentum space.

Inverse potentials for all ℓ channels of neutron-proton scattering using reference potential approach

Reference potential approach (RPA) has been successful in obtaining inverse potentials for weakly bound diatomic molecules using the Morse function. In this work, our goal is to construct inverse potentials for all the available ℓ-channels of np-scattering using RPA. Riccati-type phase equations for various ℓ-channels are solved using 5th order Runge-Kutta method to obtain the scattering phase shifts (SPS) in tandem with an optimisation procedure to minimize mean squared error (MSE). Interaction potentials for a total of 18 states have been constructed using only a three parameter Morse interaction model. The obtained MSE is <1% for 1 S 0, 3 P 1 and 3 D 1 channels and <2% for 1 P 1 channel and <0.1% for rest of the 14 channels. Total scattering cross-sections at various lab energies obtained using RPA are found to match well with experimental ones. A complete study of the np-scattering using RPA has been undertaken using Morse function as zeroth reference for the first time.

Thermal Analysis of a Micro-Satellite in a Low Earth Orbit

The thermal design of a micro-satellite is presented in the paper. The theoretical studies are carried out to describe briefly the relationship among various design orbital parameters and are depicted graphically. Solar heating on a micro-satellite is considered as a thin walled circular cylinder rotating with uniform speed about its geometric axis is analyzed for a situation in which heat transfer by convection and heat exchange within the cylinder is negligible. The nonlinear ordinary differential equations for the thin walled cylinder are solved using fourth order Runge-Kutta method to compute temperature distribution. The results of the analysis are presented at various revolution per minute RPM of the micro-satellite. Radiation heat transfer inside the package is analyzed with networking method.

THE APPLICATION OF THE EXTENDED CELLS METHOD TO SIMULATE THE FLOW OF COMBUSTION GASES IN THE LPRE CHAMBER

&#x0D; &#x0D; &#x0D; Abstract. The numerical modeling of the process of two-dimensional axisymmetric flow of combustion gases in the chamber of a liquid rocket engine is considered in this study. In general, when solving such problems, meshes are used which lines coincide with the boundaries of the computational domain. However, an alternative solution is proposed here, which is to apply the extended cells method. It allows using rectangular Cartesian grids, which lines do not coincide with the boundaries of the computational domain, without reducing the stability of the numerical solution due to the fractional finite volumes. This also simplifies the setting of boundary conditions in such volumes. The advantage of the proposed approach over the generally accepted one is the absence of the global geometric transformations during the entire modelling process, which leads to a reduction in its duration. To perform the numerical modelling, an inviscid ideal compressible gas of constant chemical composition was chosen as a basic model of a continuum. It is described by a system of the unsteady Euler equations in integral form, which was closed by the Mendeleev-Clapeyron equation of state. For the numerical solution of this system, the finite volume method was used with the reconstruction of the flow parameters by the WENO algorithm of the third order of accuracy. The solution of the Riemann problem was carried out using the Lax-Friedrichs relations. Time integration of the system of equations was performed using the explicit Runge-Kutta method of the third order of accuracy. All calculations were carried out on a uniform rectangular Cartesian mesh, which lines did not coincide with the boundaries of the computational domain. The results were compared with the solution of the same problem using the ANSYS Fluent on an unstructured mesh coinciding with the boundaries of the computational domain. The value of the relative error obtained as a result of comparing both solutions did not exceed 0.05.&#x0D; &#x0D; &#x0D;

Drought adaptation of plants based on improved Lotka-Volterra model

In arid environments, plant communities often cooperate to fight the harsh conditions. In order to study the interaction between plant populations and explore their survival under drought conditions, an improved Lotka-Volterra model was established by means of mechanism analysis to consider the interaction between different plant species, and the species correlation degree was added. Light, water, air competition among plants and other important parameters affecting plant growth, a relatively complete model of plant drought adaptability was fitted. Then, the fourth order Runge Kutta method was used to solve the model numerically, and the relationship between the amount of various plants in the community with time was obtained, and the relationship between plant populations under drought conditions was explored.

MODEL EPIDEMIK CAMPAK DENGAN ADANYA VAKSIN PADA POPULASI RENTAN DAN SUPPORT PADA POPULASI TEREKSPOSE

Measles is a highly contagious disease and often occurs in children due to malnutrition, especially children with vitamin A deficiency and a weakened immune system. In addition to vaccination, the role of parents is needed in the form of support to control the development of the virus in the body. This measles disease can be modeled through a mathematical model, especially epidemic model. This study aims to explain the formation of a mathematical model of measles, determine the equilibrium point, basic reproduction number, stability analysis, and to perform numerical simulations on the model. The research procedure begins with construct a model using a system of nonlinear differential equations. The basic reproduction number can be determined using the next generation matrix method and analysis of model stability using the linearization method. While numerical simulation has been carried out using the fourth order Runge Kutta method. The result of this study is the formation of a mathematical model of measles with a population consisting of four compartments, namely Susceptible, Exposed, Infected and Recovered. Disease control is carried out in the model, namely vaccines in the Susceptible population and support measures in the Exposed population. From the model formed, two equilibrium points are obtained, namely the disease-free equilibrium point and the endemic equilibrium point. Furthermore, the basic reproduction number formula and analysis of the stability of the model at the disease-free equilibrium point and endemic equilibrium point are also obtained. Finally, a simulation model is presented to support stability analysis and comparison of solutions for the Infected population before being given control support and after being given control support with variations in vaccine percentages.

Numerical Study in Effect of Thermal Slip on Two Fluid Flow in a Vertical Channel

The present study investigates the effect of thermal slip on an immiscible flow of micropolar and viscous fluids in a vertical channel. The left boundary is subjected to thermal slip with appropriate boundary and interface conditions, resulting in a linked system of nonlinear partial differential equations. The ND Solve technique in Mathematica software is used to implement the Runge-Kutta method of the sixth order. The velocity, temperature, and concentration equations are then calculated. The mass, heat, and velocity transmission rates at the boundaries were recorded for all the variations in the governing parameters. In addition, the impact of relevant parameters on various physical properties of micropolar and viscous fluids is analyzed through graphical means. The results are then discussed in detail. Thermal slip, Grashof number, molecular number, magnetic parameter, and Reynolds number are crucial factors that significantly affect heat and mass transfer in fluid flow. The effect of the increased thermal slip is noted to result in a decrease in both the velocity profile and temperature. It was also observed that higher values of Grashof and molecular Grashof numbers led to increased velocity and angular velocity.

SOLUTION OF ORDINARY DIFFERENTIAL EQUATION vvi (u)=f(u,v,v',v'',v''') USING EIGHTH AND NINTH ORDER RUNGE-KUTTA TYPE METHOD

The present paper presents the numerical conclusion to solve sixth order initial value ordinary differential equation (ODE). The concept of order conditions for three stage eighth order (RKSD8) &amp; four stage ninth order Runge-Kutta methods (RKSD9) has been derived for finding global truncation error of differential equation The global and local truncated errors norms, zero stability of extended Runge-Kutta method (RK) is well defined and demonstrated with the help of an example.

Soret and Dufour effects on MHD mixed convection flow of Casson hybrid nanofluid over a permeable stretching sheet

The use of hybrid nanofluids can improve the engine operation and fuel efficiency in automotive engineering, as well as the efficiency of cooling systems used in food processing and preservation. This study provides insight for the development of efficient heat exchangers, thermal energy storage systems and cooling systems for various industrial applications, which can improve product quality and safety. The study aims to investigate the Soret and Dufour effects in hybrid nanofluid flow, with the goal of understanding their behaviour and potential applications in various industries. The flow is over a vertically stretched permeable sheet with porous material under convective boundary conditions and a non-uniform heat source/sink. The reduced governing equations are represented by non-linear-coupled ordinary differential equations solved by means of the shooting method based on the fourth-order Runge–Kutta method. Enhancing the Casson parameter reduces the fluid velocity while enhancing the mass and thermal Grashof numbers increases the flow velocity. The Soret and Dufour effects improve temperature and concentration, and a higher Biot number increases the temperature. The novelty of this investigation lies in its incorporation of Soret and Dufour effects, porosity, heat source/sink and a more refined definition of nanoflow, which is compared to previous results.

Launching Process Analysis of Aircraft Gun with Closed Type Gas Reflection Device

Recoil reduction is one of the most important issues for the design of aircraft gun. Traditional approaches for this purpose will produce back-blast, which constitutes a hazard to fighter planes or attack helicopters. This paper presents a closed type gas reflection device for the aircraft gun aiming to reduce the recoil force without ejecting the propellant gas rearward. The launching process is modeled by coupling the classical interior ballistic model and the flow equations for the closed type gas reflection device. The fourth order Runge-Kutta method is adopted to solve the modified interior ballistic model to obtain the recoil efficiency of the closed type gas reflection device. On this basis, effects of various parameters on the recoil efficiency and muzzle velocity are studied systematically. Finally, shooting experiments are carried out upon a 30mm caliber aircraft gun. The simulated results agree well with the experiment ones. The results show that, by using the closed type gas reflection device, the recoil efficiency of 26.01% is attained without ejecting the propellant gas rearward, nor decreasing the muzzle velocity dramatically.

A Numerical Investigation of the Effects of Groove Texture on the Dynamics of a Water-Lubricated Bearing–Rotor System

This paper aims to investigate the combined effects of working condition and structural parameters of groove texture on the dynamic characteristics, stability and unbalance response of a water-lubricated hydrodynamic bearing–rotor system to avoid instability and excessive vibration of the rotor. The Navier–Stokes equation, standard K-ε model with enhanced wall treatment and Zwart–Gerber–Belamri cavitation model are considered using the commercial software Fluent to calculate the stiffness and damping coefficients of a groove-textured, water-lubricated bearing based on the dynamic mesh method; the critical mass to express the stability and the unbalance response solved by the fourth order Runge–Kutta method of the rotor are calculated based on dynamic equations. The results indicate that shallower and longer groove textures can improve the direct stiffness along the load direction kyy, weaken the stiffness in the orthogonal direction kxx, improve stability and decrease the unbalance response amplitude of the water-lubricated bearing–rotor system at a greater rotational speed and smaller eccentricity ratio; however, the impact of grooves on damping parameters is not as great as it is on stiffness—there exists an optimum groove width to achieve a best dynamic performance.

Transverse Fluctuations and Their Effects on the Stable Functioning of Semiconductor Devices

Semiconductor plasma is often found in chaotic unpredictable motion which shows some anomalous behaviors providing multiple challenges to work with the instabilities in a semiconductor device. Experimental studies have shown that these instabilities give rise to fluctuations and azimuthal non-uniformities, which are usually present in the semiconductor. The energy fluctuations have also been observed in some of the cases. In this paper, we have obtained the fluctuations in velocity field by integrating the linearized governing hydrodynamic equations with RungeKutta method of order four (RK4). Then, we have come up with a mathematical formulation, where these fluctuations can be obtained from a KdV family equation with homotopy-assisted symbolic simulation. We have also obtained the relative velocity between the solitary structures for different parameters. Finally, by giving a detailed explanation of the behavior of semiconductor devices, we can study the usefulness of formulating the plasma waves in the various regime, and predict their characteristics theoretically.

Elasto-plastic analysis of foundations during emergency shutdown due to blast loading

Behaviour of machine foundations is investigated in this study under harmonic loading with varying frequency and pulse loading. Owing to excessive vibrations from the pulse, the machine encounters an emergency shutdown where the machine’s exciting frequency reduces to zero exponentially. The analysis is performed considering an elastic plastic single degree of freedom system with hardening and softening behaviour. Two cases are analysed depending on the time of application of pulse loading to the foundation system. For both the cases, the governing equations of motion are acquired and solved numerically using the fourth order Runge-Kutta method. Using these solutions, displacement-time graphs are obtained for a variety of parameters such as time constant, hardening/softening index, mass of machine and the foundation block, stiffness, pulse load magnitude, damping ratio and decay coefficient to study their influence on the behaviour of the machine-foundation system. The study helps in predicting the dynamic response of foundations under the emergency shutdown conditions due to pulse loading. Further, these also help in taking the decision if there exists a need for vibration barriers to have the displacements well within the permissible limits.

The unified impact of thermal radiation and heat source on unsteady flow of tangent hyperbolic nanofluid through a permeable shrinking sheet

The tangent hyperbolic nanofluid behaviour is observed in multiple practical applications and extensively used in different laboratory experiments. It motivates to focus on the two-dimensional boundary layer flow of tangent hyperbolic nanofluid past a shrinking sheet. The goal of this article is to scrutinize the mass and heat transport of an unsteady flow of non-Newtonian tangent hyperbolic nanofluid past a shrinking surface under the influences of a source of heat, thermal radiation and chemical reaction. The flow model is mathematically framed through a coupled nonlinear partial differential equations. To transform the non-linear PDEs to ODEs, we use the similarity transformations and reduced ODEs are solved by RK-IV (Runge-Kutta method of order four) with shooting technique using the computational software MATHEMATICA. The novelty which emerges from the flow pattern lies in the fact that there exist dual solutions for certain parametric domain and the range of the existing solution may be expanded by increasing the power-law index (n) and shortened by increasing the Weissenberg number (We). Another remarkable conclusion which can be drawn from the temporal stability analysis that among these two solutions, first solution is stable and realistic whereas the second solution is not physically realistic. A diversity of the parameter estimates were used to provide solution for velocity f′(η), temperature θ(η) and concentration profiles ϕ(η) of this fluid model. In addition to the above findings, there are graphs and tables with explanation for the skin friction coefficient, the Nusselt number and the Sherwood number. These characteristics of a tangent hyperbolic nanofluid are not reported yet and may be useful in the development of existing technology.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

This paper deals with the uniformly stability (US), globally uniformly asymptotically stability (GUAS) and instability of zero solution as well as integrability and boundedness of nonzero solutions of certain nonlinear integro‐differential equations (IDEs). We prove five new theorems, which include sufficient conditions related to these fundamental qualitative properties of solutions to the IDEs considered. The main tools used in the proof are two new and suitable Lyapunov–Krasovskiiˇ functionals (LKFs). In particular cases, two numerical examples are given and solved via the fourth order Runge–Kutta method (RKM) in MATLAB to illustrate the theoretical results of this paper. Compared with the existing results on the fundamental qualitative behaviors of scalar IDEs, our results are new, original, more effective and convenient for tests and applications.

The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.

Unsteady Hydromagnetic Flow over an Inclined Rotating Disk through Neural Networking Approach

The goal of this research is to examine how a magnetic field affects the unsteady flow of a hybrid nanofluid over a spinning disk that is inclined and stretched while the flow is surrounded by a non-Darcy porous medium. Furthermore, for heat transmission mechanisms, Joule heating and viscous dissipation are considered. The current article is made more realistic by imposing thermal radiation to enhance the heat transmission system under the effects of convection. Moreover, thermal and velocity slip conditions have also been incorporated into the current study. The equations that administer the flow problem along with constraints at the boundaries are converted to dimension-free form by employing a set of appropriate similarity transformations, which are then solved by the numerical technique Runge-Kutta method of order four (RK-4). The new and advanced trend for the convergence of the obtained results is validated through a neural networking approach. The temperature of hybrid nanofluid is augmented by an increase in the porosity parameter, the unsteadiness factor, the Eckert number, the magnetic field, and the Forchheimmer number, while for the values of the radiation factor, the thermal heat is decreasing near the disk and increasing away from the disk. The precision of the obtained results has been ensured by comparing them with established results, with good agreement among these results.

Impact of needle size and distinct flow conditions on thermal performance of TiO2–MWCNTs hybrid nanofluid flow past thin needle using Casson fluid model

AbstractHybrid nanofluids comprising carbon nanotubes (CNTs) are finding applications in advanced heat exchange technologies due to the unusually high thermal conductivity and chemically inert nature of CNTs. The thermal performance of a hybrid nanofluid, comprising MWCNTs and TiO2 nanoparticles and flowing past a thin hot needle, is investigated in the present article. The Sakiadis and Blasius hybrid nanofluid flow scenarios around a parabolically shaped thin hot needle are modeled by using a non‐Newtonian Casson fluid model coupled with the energy equation. The governing equations are coupled non‐linear partial differential equations which are solved by the shooting method along with the 4th order Runge–Kutta method. The main aim of the study is to perform a comparative analysis among the base fluid (which is a 50:50 mixture of ethylene glycol [EG] and water), TiO2/EG–water nanofluid, and TiO2–MWCNTs/EG–water hybrid nanofluid and to analyze which fluid is the best transporter of heat. Apart from this, the impact of needle size and different flow conditions on heat transport is investigated. The numerical results reveal that the rate of heat transfer is maximum for TiO2–MWCNTs/EG–water hybrid nanofluid among the base fluid, nanofluid, and hybrid nanofluid. The magnitude of the Nusselt number is high for the case of the Blasius flow than for the Sakiadis flow.

A Runge-Kutta numerical scheme applied in solving predator-prey fuzzy model with Holling type II functional response

The predator-prey model has been extensively studied, but only studies models in a certain environment, where all parameters and initial values involved in the model are assumed to be certain. In real practice, some parameters and initial values are often uncertain. To overcome this uncertainty problem, a model can be made by using a fuzzy theoretical approach. In this paper, we develop a numerical scheme for solving two predator-prey models with a Holling type II functional response by considering fuzzy parameters and initial populations. The behavior of the model was studied qualitatively using the 5th order Runge-Kutta method of which was modified for the fuzzy system using the Zadeh extension principle. The numerical simulation results show that, when the initial populations of prey and predators are fuzzy, the behavior of the fuzzy model would be qualitatively the same as the crisp model. Finally, we conclude that the resulting fuzzy behavior represents a generalization of crisp behavior. This gives more realistic results since the solution is obtained by explicitly considering the problem of uncertainty.

A high-order finite difference method with immersed-boundary treatment for fully-nonlinear wave–structure interaction

In order to predict nonlinear wave loading on marine structures, a fully nonlinear higher-order finite difference based potential flow solver with all boundary conditions treated by an immersed boundary method has been developed in this paper. The solver adopts high-order finite difference schemes for the spatial derivatives and a 4th order Runge–Kutta method for time stepping. Test cases of forced oscillation of a cylinder in an infinite fluid domain are first studied, which reveal the advantage of the acceleration potential method in terms of wave load computation. Then a wave generation problem using a piston type wave maker is tested. Special attention is paid to the intersection point between the free surface and the body surface, and a scheme which best meets the accuracy and stability requirements is suggested from several proposals. A novel hyperviscosity filter, which works for both uniform and non-uniform grids, is introduced to stabilize the time-domain solution of the wave maker problem. Finally, a forced heaving cylinder on the free surface is considered, and the nonlinear wave loads on the cylinder are analyzed and compared to benchmark results.