Runge-Kutta Research Articles

Determination of the characteristics of the perturbed transition orbit to transfer from LEO to GEO

This paper aims to study, the process of satellite transition between different orbits and evaluate the characteristics of the turbulent transitional orbit of transferring a satellite from Low Earth Orbit (LEO) to geostationary orbit (GEO) (hp=35790km). The measurements in this research involve considering different LEO characteristics of perigee heights (h_p=200, 400, 600,800) and eccentricities (e=0.01, 0.05 and 0.1), whereas the inclination is fixed to be (i= 23.445 degree).Also, the transfer of the satellite was studied in two stages. The first stage was from the perigee of the first orbit at height 200 km, to the apogee of the second orbit at height 945 km with eccentricity (0.01). the second stage, was the transition the satellite from the perigee of the second orbit at height 800 km to the final orbit (GEO) with the same eccentricity. Two types of turbulence were considered which are atmospheric drag and perturbation (Geopotential acceleration The perturbing acceleration effects on satellite due to the Earth’s gravity potential represent as function of J2 parameter). A program in MATLAB was designed to calculate the speed that for the transition, time and the ratio mass change, which represents the percentage of fuel as well as the extent of the effect of turbulence on the orbital elements of the transitional orbit.The perturbed equation of motion was solved using Runge–Kutta method. The results showed that the increase altitude, the lower ΔV(velocity required to transition), where the ΔV at altitude of 800 km was the best results, reaching (2.244 km/sec). As for the transition in two stage, it may need a higher energy for the transition, as the difference between it and the direct phase in (ΔV= 0.011 km/sec). As the height of the initial orbit increases, the effect of the disturbance on the orbital elements of the transitional orbit decreases, and since the transition time is less than half a day (0.44 day), the effect of the disturbance is clear, so it cannot be neglected.

A Non-Standard Finite Difference Discretization Scheme Applied to a Malaria Model

In this research work, a dynamically consistent non-standard finite difference (NSFD) scheme is developed to solve a continuous-time model of malaria transmission with herbal medicine as control strategy. We compared results from NSFD scheme with the standard finite difference methods (4th order Runge-kutta and forward Euler methods). The numerical investigation showed that the proposed NSFD method remains consistent, preserves the positivity of solutions and converges to true equilibrium points of the continuous model independent of the step size h.

Computational role of homogeneous–heterogeneous chemical reactions and a mixed convective ternary hybrid nanofluid in a vertical porous microchannel

Abstract This article mainly scrutinizes the heat transfer and flow characteristics of a mixed convection ternary hybrid nanofluid in a porous microchannel considering the catalytic chemical reaction and nonuniform heat absorption/generation. Using appropriate similarity transformations, the modeled equations are converted into reduced ones and then solved via the Runge–Kutta–Fehlberg 4th/5th order method. To strengthen this analysis, the convection mechanism has been deployed. The effect of pertinent physical parameters on the fluid motion and thermal field is displayed, including some important engineering variables like the Nusselt number, Sherwood number, and drag force. The novel outcomes display that the flow reduces with porous permeability and nanoparticle volume fraction. The temperature of the nanofluid improves with nonuniform heat absorption/generation. The concentration decreases in the presence of both homogeneous and heterogeneous reaction intensities. The heat transfer rate enhances for the Eckert number, and a similar influence on the mass transfer rate is noticed for homogeneous reaction parameters. Further, the drag force declines for the Grashof number. The outcomes show that, in all cases, the ternary hybrid nanofluid shows a greater impact than the nanofluid. The attained findings represent applications in the era of cooling and heating systems, thermal engineering, and energy production.

Iterated Crank–Nicolson Runge–Kutta Methods and Their Application to Wilson–Cowan Equations and Electroencephalography Simulations

The Wilson–Cowan model has been widely applied for the simulation of electroencephalography (EEG) waves associated with neural activities in the brain. The Runge–Kutta (RK) method is commonly used to numerically solve the Wilson–Cowan equations. In this paper, we focus on enhancing the accuracy of the numerical method by proposing a strategy to construct a class of fourth-order RK methods using a generalized iterated Crank–Nicolson procedure, where the RK coefficients depend on a free parameter c2. When c2 is set to 0.5, our method becomes a special case of the classical fourth-order RK method. We apply the proposed methods to solve the Wilson–Cowan equations with two and three neuron populations, modeling EEG epileptic dynamics. Our simulations demonstrate that when c2 is set to 0.4, the proposed RK4-04 method yields smaller errors compared to those obtained using the classical fourth-order RK method. This is particularly visible when the spectral radius of the connection matrix or the excitation-inhibition coupling coefficient is relatively large.

A Higher Order Nonconforming Virtual Element Method for the Cahn–Hilliard Equation

AbstractIn this paper we develop a fully nonconforming virtual element method of arbitrary approximation order for the two dimensional Cahn–Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge–Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

Explicit Runge–Kutta Numerical Manifold Method for Solving the Burgers’ Equation via the Hopf–Cole Transformation

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

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.

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.

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.

Numerical Simulation Study of Thermal Performance in Hot Water Storage Tanks with External and Internal Heat Exchangers

This paper presents a numerical analysis of two hot water storage tank configurations—one equipped with an external heat exchanger (Tank-1) and the other with an internal heat exchanger (Tank-2). The objective is to evaluate and compare their thermal performance during charging and discharging processes. The numerical model is developed by solving a system of ordinary differential equations using the 4th-order Runge–Kutta method, implemented in the Python programming language. The results indicate that Tank-1 demonstrated a higher charging efficiency of 94.6%, achieving full charge in approximately 2 h and 20 min. In comparison, Tank-2 required 3 h and 47 min to reach full charge, with a charging efficiency of 85.9%. During discharge, both configurations exhibited similar behavior, with an efficiency of 13.63% over approximately 33 min. The analysis showed that the external heat exchanger configuration led to more effective thermal stratification, supported by the Richardson number analysis, which indicated a significant effect of buoyancy during charging. This design advantage makes Tank-1 particularly suitable for applications requiring rapid heating and minimal heat loss, such as in cold climates or intermittent demand systems. The numerical model demonstrated reliable predictive accuracy, achieving an RMSE of 6.1% for the charging process and 6.8% for the discharging process, thereby validating the model’s reliability. These findings highlight the superior performance of the external heat exchanger configuration for fast and efficient energy storage, particularly for applications in cold climates.

Runge–Kutta methods determined from extended phase space methods for Hamiltonian systems

Runge–Kutta methods determined from extended phase space methods for Hamiltonian systems

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.

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.

Stability and convergence analysis of stochastic Runge–Kutta and balanced stochastic Runge–Kutta methods for solving stochastic differential equations

Stability and convergence analysis of stochastic Runge–Kutta and balanced stochastic Runge–Kutta methods for solving stochastic differential equations

Entropy optimization in a radiative and chemically reactive EMHD flow of a nanofluid coexisting Ohmic dissipation and multiple slips

Purpose This research focuses on the controlling irreversibilities in a radiative, chemically reactive electromagnetohydrodynamics (EMHD) flow of a nanofluid toward a stagnation point. Key considerations include the presence of Ohmic dissipation, linear thermal radiation, second-order chemical reaction with the multiple slips. With these factors, this study aims to provide insights for practical applications where thermal management and energy efficiency are paramount. Design/methodology/approach Lie group transformation is used to revert the leading partial differential equations into nonlinear ODE form. Hence, the solutions are attained analytically through differential transformation method-Padé and numerically using the Runge–Kutta–Fehlberg method with shooting procedure, to ensure the precise and reliable determination of the solution. This dual approach highlights the robustness and versatility of the methods. Findings The system’s entropy generation is enhanced by incrementing the magnetic field parameter (M), while the electric field (E) and velocity slip parameters (ξ) control its growth. Mass transportation irreversibility and the Bejan number (Be) are significantly increased by the chemical reaction rate (Cr). In addition, there is a boost in the rate of heat transportation by 3.66% while 0.05⩽ξ⩽0.2; meanwhile for 0.2⩽ξ⩽1.1, the rate of mass transportation gets enhanced by 12.87%. Originality/value This paper presents a novel approach to analyzing the entropy optimization in a radiative, chemically reactive EMHD nanofluid flow near a stagnation point. Moreover, this research represents a significant advancement in the application of analytical techniques, complemented by numerical approaches to study boundary layer equations.

Novel exponentially fitted two-derivative Runge–Kutta methods for solving the radial Schrödinger equation

Novel exponentially fitted two-derivative Runge–Kutta methods for solving the radial Schrödinger equation

Nonlinear transient analysis of water lubricated bearing in the turbulent regime

Influence of turbulence on the stability characteristics of a water lubricated journal bearing is analyzed. Locus of the journal centre is plotted using the non – linear transient analysis method for different flow regimes of the bearing subjected to three different types of loading. The bearing groove angles of 18o and 36o is considered for the analysis. Turbulent coefficients are incorporated in the modified form of Reynolds equation which is solved using the 4th order Runge-Kutta method. The path traversed by journal centre is generated for different loading conditions and the stability of the bearing is examined. For periodic and fluctuating rotating loads, the journal centre traversed shorter path and reached stable position when the flow regime changed from laminar to turbulent.

Nonlinear system optimization of cutting tools with dynamic vibration absorbers in deep hole boring: A stability analysis

Deep hole boring vibrations pose challenges due to low stiffness of long cantilever bars, impacting accuracy and standards. Nonlinear modeling provides deeper insights into chatter. Integrating dynamic vibration absorbers enhances boring bar stability. In this study, a nonlinear model is developed to investigate the vibration stability of the boring process, with the model's dimensionless parameters analyzed and solved using the Runge-Kutta method. The impact of nonlinear parameters associated with the damping and stiffness of the absorbers on the vibration characteristics of boring bars is thoroughly examined. The findings reveal that there exists an optimal range for the damping and stiffness of the absorbers, where vibration amplitude is significantly reduced, especially under conditions of high excitation force. Moreover, it is demonstrated that the introduction of cubic stiffness elements can further diminish vibration amplitude. These results offer valuable guidance for the design and optimization of DVA in deep hole boring, contributing to improved machining performance and accuracy.