4th Order Runge-Kutta Method Research Articles

Investigation of the highly complex nonlinear problems via modified energy balance method

In this paper, the modified energy balance method is proposed as an effective analytical approach to achieve more precise approximate results for a non-linear oscillator in a nano-electromechanical system (NEMS). The proposed solution is compared with previous approximate methods, including the Hamiltonian approach, the homotopy perturbation technique, and the numerical solution using Runge–Kutta 4th-order method. The results obtained from the modified energy balance method show excellent agreement with the numerical solutions. This technique is both efficient and precise in solving such nonlinear problems, offering significant advantages. The novelty of this paper lies in demonstrating the utility of the modified energy balance method for studying nano-electromechanical systems, enabling the construction of higher-order analytical solutions. This study allows for a thorough examination of the results obtained through these analytical solutions.

Calculation of Four-Dimensional Unsteady Gas Flow via Different Quadrature Schemes and Runge-Kutta 4th Order Method

Calculation of Four-Dimensional Unsteady Gas Flow via Different Quadrature Schemes and Runge-Kutta 4th Order Method

Double Diffusive Convection with Cross-Diffusion Effects in Hybrid Nanoliquids in an Inclined Slot

This novel study examines the effects of cross-diffusion, and nanoparticles on a hybrid nanoliquid undergoing double-diffusive convection in an inclined slot. The hybrid nanoliquid considered is unique and it contains a mixture of base liquid to which nanoparticles are suspended. The thermophysical properties of the hybrid nanoliquid are calculated using phenomenological laws and mixture theory. The results are compared with the case of mono nanoliquid and hybrid nanoliquid with one base liquid and two nanoparticles. In order to study the convective system, the shooting method along with Runge-Kutta 4th order method is employed, and the results are illustrated graphically. It is noticed that the solutal Rayleigh number, Soret parameter and inclination of the plates delay the onset of convection, while the Dufour parameter, diffusivity ratio, and the addition of nanoparticles advance the onset of the convection.

The Power Factor Improvement Modellingwith the Runge-Kutta 4thOrder, Analytical, and Experimental Method

The Power Factor Improvement Modellingwith the Runge-Kutta 4thOrder, Analytical, and Experimental Method

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

Reduction in forest resources due to increasing global warming and population growth is a critical situation the World faces today. As these reserves decrease, it alarms new challenges that require urgent attention. In this paper, we provide a semi-analytical solution to a nonlinear mathematical model that studies the depletion of forest resources due to population growth and its pressure. With the help of the homotopy perturbation method (HPM), we determine an approximate series solution with few perturbation terms, which is one of the essential power of the HPM method. We compare our semi-analytical results with numerical solutions obtained using the Runge-Kutta 4th-order (RK-4) method. Furthermore, we analyze the model’s behaviour and dynamics by changing the parametric coefficients that represent the depletion rate of forest resources and the growth rate of population pressure and present these findings using various graphs.

MHD rotating flow over a stretching surface: The role of viscosity and aggregation of nanoparticles

The magnetohydrodynamic (MHD) rotating flow that occurs across a stretching surface has numerous practical applications in a variety of domains. These fields include astronomy, engineering, the material sciences, and space exploration. The combined examination of magnetohydrodynamics rotating flow across a stretching surface, taking into consideration fluctuating viscosity and nanoparticle aggregation, has significant ramifications across several different domains. It is essential for both the growth of technology and the attainment of deeper insights into the complicated fluid dynamics to maintain research in this field. Given the aforementioned motivation, the principal aim of this study is to examine the effects of variable viscosity on the bidirectional rotating magnetohydrodynamic flow over a stretching surface. Aggregation effects on nanoparticles are used in the analysis. Titania (TiO2) is taken nanoparticle and ethylene glycol as base fluid. The nonlinear ordinary differential equations and the boundary conditions that correspond to them can be transformed into a dimensionless form by using a technique called similarity transformation. To get a numerical solution to the transformed equation, the Runge-Kutta 4th order (RK-4) method is utilized, and this is done in conjunction with the shooting method. The impact of various leading variables on dimensionless velocity, the coefficients of temperature, skin friction and local Nusselt number are graphically represented. Velocity profiles in both direction increases with increasing values of φ. The Nusselt number increases with increasing values of the radiation and temperature ratio parameters. When a 1 % volume fraction of nanoparticles is introduced, the Nusselt number exhibits a 0.174 % increase for the aggregation model compared to the regular fluid in the absence of radiation effects. When the aggregation model is used with a 1 % volume fraction of nanoparticles, the skin friction increases by 0.1153 % in the x direction and by 0.1165 % in the y direction compared to the regular fluid. Tables show the variation in Nusselt numbers, as well as a comparison of the effects of nanoparticle's aggregation model without and with radiation. Moreover, the numerical results obtained were compared with previously published data, demonstrating a satisfactory agreement. We firmly believe that this finding will have extensive implications for engineering and various industries.

Comparison of Some Numerical Simulation Techniques for COVID-19 Model in Iraq

The aim of our study is to solve a nonlinear epidemic model, which is the COVID-19 epidemic model in Iraq, through the application of initial value problems in the current study. The model has been presented as a system of ordinary differential equations that has parameters that change with time. Two numerical simulation methods are proposed to solve this model as suitable methods for solving systems whose coefficients change over time. These methods are the Mean Monte Carlo Runge-Kutta method (MMC_RK) and the Mean Latin Hypercube Runge-Kutta method (MLH_RK). The results of numerical simulation methods are compared with the results of the numerical Runge-Kutta 4th order method (RK4) from 2021 to 2025 using the absolute error, which proves that the MLH_RK method is the best and closest to the expected values. The results have been discussed after being tabulated and represented graphically. Epidemic behavior for the next two years until 2025 has been projected using the proposed methods.

Optimal system of solution using group invariance technique for shock wave in a non-ideal self-gravitating gas in rotating medium in presence of magnetic field

Abstract This work demonstrates the study of the optimal system of solutions for shock wave propagation in a non-ideal self-gravitating gas in rotating medium with magnetic field (axial or azimuthal) for the adiabatic flow in cylindrical geometry by applying the group invariance technique. Using the group invariance technique, we have obtained the one-dimensional (1-D) optimal system of sub-algebra for the basic governing equations. The infinitesimal group optimal classes are obtained and the similarity solution in four possible cases (two cases for perfect gas and two cases for non-ideal gas) with exponential law shock path are discussed. The numerical solution by using the Runge Kutta 4th order method is obtained and the distribution of physical variables are shown via graph. The impact of the rotational parameter, non-idealness parameter, shock Cowling number, similarity exponent and gravitational parameter on the strength of the shock and flow variables are investigated. With an increase in the shock Cowling number, non-idealness and rotational parameters, the shock strength decreases, i.e., they have decaying impact on shock wave; whereas the shock strength increases with gravitational parameter and similarity exponent. Also, the strength of the shock is reduced by considering the magnetic field to be axial instead of azimuthal.

Significance of heat generation and impact of suction/injection on Maxwell fluid over a horizontal plate by the influence of radiation

The current research paper aims to investigate the Maxwell fluid flow over a flat plate under slip conditions and suction/injection parameters. Radiation, velocity slip conditions, and heat generation are also considered for this study. The governing equations are changed into ordinary differential equations with the appropriate similarity transformations. Then these equations are answered using the shooting technique and the Runge-Kutta 4th-order method. The impact of Deborah number (De), suction/ injection (fw), thermal radiation (Rd), viscous dissipation (Ec), velocity slip parameter (λ) and heat generation (Q) are displayed through graphs and tables. The main target of this investigation is to discuss the impact of the Nusselt number profile for the suction parameter (fw) against momentum slip (λ), thermal slip conditions (β) and radiation parameter. Skin friction profile for suction parameter also discussed against Maxwell fluid parameter (De). Numerical findings are compared to previously published research using bvp4c in MATLAB programme. The critical solutions of this study are that the Nusselt number outline increased for increasing the value of momentum slip, radiation parameter and thermal slip conditions. Still, it got decreases for viscous dissipation (Ec) against radiation. Skin friction profile decreased for increasing value of suction parameter against Deborah number.

Simulation based analysis to study the effect of compression ring crown height in ring liner assembly on the performance of four stroke petrol engine

Among different piston rings of an Internal Combustion Engine (IC), the top compression ring-liner assembly shared a momentous amount of the total frictional losses, particularly at the Top Dead Center (TDC) and Bottom Dead Center (BDC) where boundary lubrication subsists. In the present study, the lubrication mechanism in piston ring cylinder liner conjunction for top compression ring using one dimensional Reynolds equation assuming axisymmetric contact was studied. The hydrodynamic and mixed lubrication regimes were taken during the study. Gumbel boundary conditions were used for Reynolds equation solution. For boundary pressure calculation, first, combustion chamber pressure variation was determined, then, interring pressure was calculated assuming orifice volume method by solving mass flow rate through crevice volumes using Runge-Kutta 4th order method. The Reynolds equation was solved using finite difference method. Whole solution process was repeated for different engine speed and crown height of top compression ring to study the effect of them in wear and power loss. First, the solution was found for 3000, 5000 and 7500 RPM keeping constant crown height of 10 µm. Average power loss was found to be increased with the increasing engine speed. There was increased in oil film thickness around mid-stroke with the increased engine speed, but there was no significant change in film thickness with the engine speed at the TDC and BDC positions. Then, the solution was again repeated for different RPM varying the crown height. The compression ring with lower crown height showed the better wear performance by increasing the minimum film thickness in critical zones of engine cycle for all RPM. The crown height of 5,7 and 9 µm gave the minimum power loss (77.89, 163.47 and 297.33 W) for 3000, 5000 and 7500 rpm. Moreover, the power loss in the range of 5-9 µm had no significant differences.

Entropy generation on chemically reactive hydromagnetic oscillating flow of third grade nanofluid in a porous channel with Cattaneo-Christov heat flux

The heat and mass transfer characteristics along with Cattaneo-Christov heat flux on oscillating hydromagnetic flow of third grade nanofluid through a permeable channel under entropy generation analysis have been examined in this paper. The impacts of chemical reaction, Brownian motion, thermophoresis, Ohmic heating, and radiative heat have also been taken into consideration. The Buongiorno nanofluid model has been utilized for the present analysis.The investigation related to the present study is helpful in biomedical engineering, manufacturing industries as coolants, energy conservation, cancer treatments (like hyperthermia), dynamics of physiological fluids, biomedicines, and nano-drug suspension in pharmaceuticals. The system of nonlinear ordinary differential equations (ODEs) have been attained and solved by applying the Runge-Kutta 4th order method with the help of shooting process after the execution of the perturbation procedure on nondimensional partial differential equations (PDEs). The results of the present study have been deliberated by plotting graphs for the effects of various non-dimensional parameters on entropy, Bejan number, concentration, velocity, and temperature. The numerical values of heat and mass transfer rates for different physical parameters have been computed.

A wavelet approximation method for solving nonlinear ship roll damping equations: An operational matrix of derivative approach

The effect of roll damping on ship dynamics is more significant. Here, an efficient Lucas wavelet method is applied to estimate the parameters of ship roll motion models. For the first time, a learning model was approached for forecasting over ship roll angle with damping and restoring moments where the results are obtained from Lucas wavelet method. A barge-like vessel fitted with two spherical tank models and FPSO tank model equations with different damping and restoring coefficients are investigated to estimate the validity and applicability of the proposed LWM for different loading conditions. With the help of operational matrices of derivatives, the nonlinear differential equations are converted into a system of algebraic equations using an appropriate collocation point. The proposed wavelet approximation results are compared with experimental data, frozen cargo, Runge-Kutta method of 4th order (RKM) and Homotopy Perturbation Method (HPM) results. Moreover, numerical experiments are provided to show that the proposed wavelet method is an effective and convenient method for solving nonlinear differential equations in ship dynamics.

Waves in waterways generated by moving pressure field in Boussinesq equations using unstructured finite element model

A finite element model for depth integrated form of Boussinesq equations is presented. The equations are solved on an unstructured triangular mesh using standard Galerkin method with mixed interpolation scheme. The elemental integrals are calculated analytically and time-stepping is done using Runge–Kutta 4th order method. It is extended to simulate ship-generated waves using moving pressure fields. The unstructured formulation provides the flexibility of mesh refinement as needed, for capturing wave transformation or moving pressure field. The model is verified against experimental and numerical results for wave transformation over the Whalin shoal. The results for moving pressure field are compared against numerical results from FUNWAVE. Further, a simulation of ship navigating a curved path is presented. Finally, a real-life application and validation against field measurements is provided for waves generated by a fast ferry moving along a GPS tracked path in Tallinn Bay, Estonia.

Nonlinear analysis of memcapacitor-based hyperchaotic oscillator by using adaptive multi-step differential transform method

This paper aims to present a new analytical method for behavioral analysis of the memcapacitor-based hyperchaotic oscillator by adaptive multi-step differential transform method (adaptive MsDTM). Comparisons between the MsDTM, adaptive MsDTM and Runge-Kutta 4th order method (RK4M) are made. In this paper, a new technique to calculate Lyapunov exponent (LE) based on adaptive MsDTM is proposed. The effect of different system parameters on system performance is studied analytically. For this purpose, the LEs of the system are analytically calculated according to various system parameters. Finally, the effect of additive noise on system performance is investigated.

Thermal radiation effect on unsteady three-dimensional MHD flow of micropolar fluid over a horizontal surface of a parabola of revolution

This paper explores the time-dependent magnetohydrodynamics (MHD) micropolar fluid flow over a three-dimensional variable stretching surface in the occurrence of radiation effect. The model time-dependent partial differential equations (PDE's) in three independent variables are transformed into ordinary differential equations (ODE's) by the suitable self-similarity variables. Homotopy perturbation method (HPM) and Runge-Kutta (RK) 4th order method along with shooting technique are used in the present model. And also, HPM results are compared with Runge-Kutta (RK) 4th order method along with the shooting technique. The velocity, micro rotation in x and y directions, temperature, skin friction factor and heat transfer rates are examined for the emerging parameters. The velocity profiles and momentum boundary layer thickness intensification with increasing values of the vortex viscosity parameter. The higher value of a magnetic parameter declines the skin friction coefficient. This type of investigation may be profitable to the polymer fluids, exotic lubricants, electronic chips, artificial fibers, drawing of copper wires, etc.

Nonlinear Convective Flow of Williamson Fluid Over a Slendering Stretching Sheet with Melting Heat Transfer

The main concern of this paper is to analyze the nonlinearity of the Grashof numbers (thermal and diffusion) on the Williamson fluid flow across a stretching sheet with non-uniform thickness with the consideration of melting heat transfer. The modelled equations are non-dimensionalized with the utilization of similarity transformations. We used shooting technique based on Runge-Kutta 4th order method to unriddle the ensuing equations. Influence of key parameters on the common profiles i.e., velocity, temperature and concentration are deliberated with the support of graphs. And also, we examined friction factor, local Sherwood and Nusselt numbers against the same parameters and displayed the results in tabular form. It is seen that Weissenberg number and wall thickness parameters rise the magnitude of friction factor. It is additionally qualified to specify that the temperature and concentration profiles looking high in the absence of melting parametercontrasted with the presence of melting parameter.

Analysis of Wu's slip and CNTs (single and multi-wall carbon nanotubes) in Darcy-Forchheimer mixed convective nanofluid flow with magnetic dipole: Intelligent nano-coating simulation

The analysis of viscous materials flow subject to diverse configurations with remarkable physical applications has many utilizations in the electrical, mechanical, industrial, applied physics and mathematics fields. Besides these, carbon nanotubes (CNTs) have numerous applications in energy storage, nanotechnology, chemical sensors, industry, optics, structural diverse materials and conductive plastics. Such consideration in mind, ferro-fluid flow of viscous liquid submerged in CNTs towards a stretchable surface affected by magnetic dipole interaction is addressed. Mixed convection and Darcy-Forchheimer effects are accounted. The energy relation is discussed in the presence of radiative heat flux and viscous dissipation. First and second order velocity slips are implemented at the boundary surface. The governing expressions specifying the flow are altered into ordinary ones with the assistance of appropriate similarity quantities. The obtained ordinary system is computationally tackled via Runge-Kutta 4th Order Method (RK4OM). Our obtained outcomes reveal that velocity of working fluid particles declines with an enhancement in ferromagnetic interaction parameter and Darcy-Forchheimer number. Also, behavior of temperature distribution increases more speedily for heightening of radiative parameter and Biot number. Coefficient of skin friction (surface drag force) and Nusselt number (heat transport rate) are calculated in view of important flow parameter numerically. The range of parameters are β=0.0,0.3,0.5,ε=0.1,0.5,1.0,Fr=0.1,0.5,1.0,γ1=0.0,0.5,1.0,γ2=0.0,0.1,0.2,δ=1.0,5.0,10.0,R=0.0,0.2,0.5,Bi=0.5,1.0,1.5 and.λ=0.0,1.0,3.0.

Second law and entropy generation analysis of magnetized viscous fluid flow over a permeable expandable sheet with nonlinear thermal radiation: Brownian and thermophoresis effect

The second law, thermal, magnetic field, and concentration of viscous fluid across a permeable stretching surface are the focus of this study. The transverse and longitudinal velocities, temperature, and concentration with boundary conditions are computed numerically by applying Runge-Kutta 4th-order method. For this determination the system of governing equations are first converted to the first order ordinary linear equations and then solve by RK4 built-in function in MATHLAB SOFTWARE by taking step size [Formula: see text]. The existing work is compared with the difference between existing and published work. The iteration procedure was stopped until all of the nodes in the η-direction met the convergence condition 10−5. For confirmation of the results, the BVPh2 package is also applied and excellent agreement is found. Both on longitudinal and transverse kinematics, the impact of ferromagnetic and viscoelastic factors are examined. The temperature is studied in relation to the Prandtl number, the magnetism factor, and the heat reference factor. The concentration is also shown, as well as how it varies well with Schmidt number and the magnetic factor. The entropy generation number is calculated using fluid velocity, energy, and volume fraction coefficient. Moreover, the current work is also equated with the available work for limiting cases. The longitudinal and transverse velocities are declined when the viscoelastic and magnetic parameters are increased. The temperature profile enhances as the magnetic parameters and heat source-sink parameters increase, but declines as Prandtl number increases. At the same time, concentration raises as the magnetic parameters rises. From this investigation it is also observed that the concentration falls, as the Sc enhances. The entropy generation number decreases with the increasing values of magnetic parameter. It is perceived that as Pr enhances, the Ns increases near the surface but with increasing η, the situation reverses. For higher values of [Formula: see text], the generation number [Formula: see text] declines at the surface, however, the situation reverses via η rises.

Modeling and analysis of motion with wheel slip of the mobile platform with four wheel drive

The problem of motion with wheel slip of the four wheeled mobile platform has been discussed in this work. Considering the model of dynamics, the formulation of the initial problem of the platform motion and the numerical algorithm for solving this problem has been presented. The Runge-Kutta 4th order method has been used to integrate the equations of motion. Possible cases of motion of the prototype of the mobile platform with four drive modules have been considered. Taking into account the variable values of the active forces caused by the drive torque and changes in the position of the wheels during the platform motion, as well as the resistant forces opposing the active forces at the contact points of the wheels with the ground, the platform motion simulation results have been obtained. Different wheel settings, with controlling the active forces and directions of their settings while driving have been implemented to the model. The results of motion simulations and their analysis have been included. The results presented in the paper show a significant influence of wheel slip on the parameters of the motion of the mobile platform. The computational model presented in the paper enables a good representation of the platform motion parameters, which has been confirmed by comparing the results of simulation tests with the results of experimental tests of motion in relation to a construction solution of the platform other than described in the paper. Further development of the computational model is planned, mainly in order to include the deformability of the wheel suspension systems in the dynamic model of the platform, which will allow for the implementation of wider studies of the motion of such objects, including, for example, in the case of wheels passing through obstacles.

Burn-cycle-dependent spatial distribution of nuclear fuel actinides and fission products based on the AGENT code

BRITTS (Burnup of Reactor Isotopes, Tracking in Time and Space) is a newly developed burnup module based on the Runge-Kutta 4th order method supplementing the well-known AGENT neutronics code. Unlike other burnup codes that are limited to 2D models, cumulative-only calculations, or one specific fuel type, BRITTS provides detailed spatial distributions of actinides in 2D and approximate 3D (through a 2D/1D coupled characteristic solution provided by the AGENT code) at user-specified points in a burn cycle. Using two simple MOX fuel pins, BRITTS is compared with SCALE6.2. BRITTS is also compared against a well-known OECD/NEA benchmark of UO2 fuel pin with 16 participants. These comparisons show that BRITTS provides reliable calculations for concentrations of nuclear fuel actinides and fission products in both 2D and approximate 3D and can be uniquely used to analyze the spatial distributions of the actinides during the burn-cycles. This paper presents the mathematical and numerical methods employed in BRITTS and a series of benchmark examples showing the code capabilities for reactor fuel analyses.