4th Order Runge-Kutta Research Articles

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.

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.

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.

Spectro-spatial analysis of van der Pol-type phononic crystals

Abstract The application of phononic chains as metamaterials demonstrates their remarkable capability to manipulate the propagation of waves. These periodic structures yield frequency-dependent behavior of material comprising characteristics with many possible engineering applications. In this paper, we investigate the weak and general nonlinear behaviors of the van der Pol-type damped phononic chains. The analysis of wave propagation is initially conducted for a one-dimensional structure, and subsequently, is extended to consider the wave motion through two-dimensional and three-dimensional lattices. Results are obtained using the method of multiple scales and a Spectro-spatial analysis by employing the numerical method of the 4th-order Runge–Kutta. A new phase-diagram relation within the chain’s unit cell is also introduced aiming to enhance the numerical findings. Our results indicate that in the weakly nonlinear regime, the van der Pol-type damping closely follows the linear dispersion curve, regardless of the initial amplitude. This suggests a symmetry between energy pumping and dissipation modes, where hardening and softening behaviors align with linear characteristics of common damping mechanisms, such as viscous damping. Additionally, the formulation demonstrates the existence of limit-cycle stability in the motion of each mass. For the general damped system, it is observed that a special frequency exists where the system converges, for all wave numbers similar to the synchronization effect. Hence, the motion and the frequency of all masses are synced. Additionally, non-reciprocal wave propagation is observed, resulting in a bandgap structure with a symmetry breaking occurring near the limit cycle. These results are promising in the fields of wave emitters, wave filters, and signal encryption.

Study of MHD Casson Nanofluid Flow Over an Inclined Stretching Sheet in Porous Media with Thermal Slip, Chemical Reaction and Radiation Effects

In the current study, the behaviour of Casson nanofluid subjected to magnetohydrodynamic (MHD) flow across an inclined stretched sheet within a porous medium has been investigated numerically. The governing equations are transformed into ordinary differential equations with the corresponding boundary conditions by employing similarity transformations. The solutions of essential equations are achieved by using the 4th-order Runge-Kutta method combined with the shooting technique. The novelty and innovative contribution are showcased through illustrative graphs that scrutinize the effect of factors that impact the velocity, temperature, and concentration profiles within assorted flow scenarios. The primary focal points of the study encompass examining variations in magnetic field strength, angle of inclination, and suction intensity that affect the fluid's velocity moderation, while improved porosity and radiation parameters lead to a rise in fluid temperature. Higher Biot numbers correlate with an increase in fluid temperature. The implications of positive coefficients of heat transfer are crucial across various fields to ensure efficient thermal management ensuring optimal performance and longevity. The numerical data presented is aligned with earlier published results for comparison. Furthermore, the variations in skin friction, Nusselt, and Sherwood numbers driven by different parameters are displayed in tables to highlight significant modifications.

Analysis of boundary layer flow of a Jeffrey fluid over a stretching or shrinking sheet immersed in a porous medium

Heat transfer optimization is critical in many applications, such as heat exchangers, electric coolers, solar collectors, and nuclear reactors. The current work looks at the thermohydraulic behavior of Jeffery fluid flow along a plane containing a magnetic field, a non-uniform heat source/sink, and a porous media. Numerical solutions are derived using the Runge-Kutta 4th-order approach and the shooting method. Graphs show how Prandtl number (Pr), thermal stratification (e1), Jeffery parameter (λ1), porous parameter (λ2), magnetic field (M), and heat generation/absorption (γ, a, b) affect velocity and temperature profiles. The results show that thermal stratification increases fluid velocity and temperature, whereas heat source/sink parameters have the reverse effect on heat transfer, and raising the Jeffrey parameter reduces velocity and increases boundary layer thickness. There is extremely high agreement with experimental data from the literature. This work illustrates the utility of hydromagnetic properties in modelling fluid flow over stretching/shrinking sheets in porous media.

Transient flow and heat transfer characteristics of single-phase nanofluid past a stretching sheet under the influence of thermal radiation and heat source

This work aims to investigate the unsteady hydromagnetic free convective flow of a viscous, electrically incompressible fluid influenced by an inclined magnetic field, radiation, and heat source. Specifically, it explores the behavior of water-based nanofluids under these conditions. The Newton-Raphson shooting technique combines the 4th-order Runge-Kutta approach to obtain dimensionless and accurate solutions to the governing equations. The study assesses flow characteristics and heat transfer properties over various parameter values. The outcomes are visualized through graphical representations of velocity, temperature, skin friction coefficient, and the local Nusselt number. The present study examines the impact of copper (Cu), silver (Ag), alumina (Al2O3), and titanium dioxide (TiO2) nanoparticles in water as the base fluid on the percentage increase in skin friction and Nusselt number under various physical conditions. Suction increases skin friction by 10–15 % and the Nusselt number by a similar margin, whereas injection can reduce these metrics. Radiation enhances heat transfer, resulting in a 5–10 % increase in skin friction and a 10–15 % rise in the Nusselt number, with nanoparticles like Ag showing the most substantial effects due to their superior thermal properties.

Numerical treatment of MHD micropolar nanofluid flow with activation energy, thermal radiation, and Hall current past a porous stretching sheet

ABSTRACT This communication investigates the heat and mass transfer properties of MHD micropolar nanofluid fluid flow behaviour with activation energy, thermal radiation, and Hall current past a linear stretching sheet. The governing non-linear partial differential equations of the flow are transformed into non-linear ordinary differential equations using similarity transformation variables and solved using Runge Kutta 4th order in conjunction with the shooting methodology. The shooting approach in MAPLE is then employed to solve the associated system of ordinary differential equations. The impact of pertinent parameters is revealed and discussed on velocity, temperature, and concentration. Increasing micropolar parameters increases tangential velocity profiles while decreasing the transfer velocity profiles. The comparison table for local skin friction coefficient, Nusselt number, and Sherwood number are represented with the effects of pertinent non-dimensional variables. The model has undergone thorough validation using existing data and has shown outstanding performance. It is revealed in the study that an increase in the micropolar term increases the tangential velocity for an enhanced engineering working fluid. Also, a rise in the Hall current term boosts the heat transfer to strengthen thermal science fluids performance.

An empirical analysis of hybrid MHD ferrofluid unsteady flow of two-dimensional heat transfer across a porous channel

This research investigates the heat transfer dynamics of a two-dimensional, incompressible, laminar, magnetohydrodynamics, time-dependent flow of a hybrid ferromagnetic fluid with radiation and irregular heat source/sink conditions through two porous channels. The study investigates the unique capability of ferromagnetic solid nanoparticles to improve the thermal efficiency of the host fluid, with potential applications in medicine such as drug targeting, cell separation, and magnetic resonance imaging. A mathematical model is developed as a nonlinear partial differential equation (PDE), which is subsequently converted into a nonlinear ordinary differential equation (ODE) using similarity transformations. The numerical solution is obtained using the 4th-order Runge-Kutta method implemented in Mathematica software. We analyze the impact of various nondimensional factors on the numerical results, including skin friction coefficient and Nusselt number, as well as graphical results depicting the velocity and temperature profiles of the hybrid ferrofluid. Boosting the film thickness parameter (λ) improves the heat transfer rate at the bottom of the porous channel. Raising the irregular heat source/sink parameters ( A * and B * ) and causing opposite impacts on the heat transfer rate at the bottom porous channel. Higher the radiation (R) values, the temperature profile behaves oppositely in both porous channels. The analysis reveals that the hybrid ferrofluid exhibits a higher rate of heat transfer compared to the ferrofluid.

Irreversible and reversible chemical reaction impacts on convective Maxwell fluid flow over a porous media with activation energy

The Maxwell model of fluid flow in a rotating frame over a porous media is investigated in this paper. Binary chemical reactions and fluid movement under activation energy are both covered in this study. The impact of mass and heat transmission along the boundary layer is investigated in an equilibrium process. Using the method of similarity transformation, the controlling partial differential equations are changed into ordinary differential equations. The results are confirmed using the bvp4c Matlab built-in programme, and the altered equations are resolved utilizing a 4th order Runge Kutta based shooting method. Reversible and irreversible processes, activation energy, chemical reactions, Deborah numbers, and rotation parameters are some of the parameters for which the results are offered in tables and graphs. The prior objective of this study is to examine the impact of activation energy and chemical reactions on Maxwell fluid flow in an equilibrium setting. The concentration boundary layer for reversible flows is significantly finer than that of irreversible flows with the influence of activation energy, chemical reaction, and rotation factors. A reduced boundary layer thickness can improve the rates of heat and mass transmission in tremendous applications, such as exchangers of heat and chemical reactors. In this chemical process, sulphuric acid is utilized as a catalyst along with Maxwell fluid which affects the boundary layer reaction rate and selectivity. This is crucial for efficient catalytic process design. Controlling reaction rates using fluid elasticity and reaction kinetics is useful in operations that need accurate product production.

Nonlinear vibration characteristics of thermo-viscoelastic coupling of moving PET films with temperature-dependent elastic modulus

In order to improve the transport stability of roll-to-roll produced polyethylene terephthalate films (PET films) in high temperature environments, this paper investigates the nonlinear vibrational properties of moving PET films under a uniform temperature field by considering the temperature dependence of the elastic modulus. Firstly, constant tensile tests were conducted to test the elastic modulus values of PET films at different hot air temperatures, and the Wachtman model was improved to establish a temperature-dependent elastic modulus model for PET films. Secondly, based on the Kelvin-Voigt model, the nonlinear vibrational equilibrium differential equations of the system were established by applying the Bubnov-Galerkin method of discretization according to D’Alembert’s principle. Finally, numerical simulations based on the 4th-order Runge-Kutta method were carried out to analyze the effects of films width, external excitation frequency, external excitation amplitude, transfer velocity, ambient temperature, damping coefficient and viscoelasticity coefficient on the nonlinear vibration of the system. This provides a theoretical basis for the commissioning of stabilized transfer parameters for roll-to-roll production of PET films.

Similarity solution for the magnetogasdynamic shock wave in a self-gravitating and rotating ideal gas under the influence of radiation heat flux

The Lie invariance method is used to analyze the one-dimensional, unsteady flow of a cylindrical shock wave in a rotating, self-gravitating, radiating ideal gas under the influence of an axial or azimuthal magnetic field, with an emphasis on adiabatic conditions. The analysis assumes a stationary environment just ahead of the shock wave and considers variations in fluid velocity, magnetic field, and density within the perturbed medium just behind the shock front. In the governing equations, the impact of thermal radiation under an optically thin limit is integrated into the energy equation. Utilizing the Lie invariance method, the set of partial differential equations governing the flow in this medium is transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity variables. Two distinct cases of similarity solutions are obtained by selecting different values for the arbitrary constants associated with the generators. Among these cases, one yields similarity solutions assuming a power-law shock path and the other an exponential-law shock path. For both cases, the resulting set of nonlinear ODEs are numerically solved using the 4th-order Runge–Kutta method in MATLAB software. The article thoroughly explores the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), σ (ambient density exponent), l1 (rotational parameter), and G0 (gravitational parameter) on the flow properties. The findings are visually presented to offer a comprehensive insight into the effects of these parameters.

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.

SEVERE CYCLONIC STORM SITRANG: A METEOROLOGICAL ANALYSIS OF ITS FORMATION, INTENSITY, AND IMPACT ON BANGLADESH

Employing a non-central difference method of lines in tandem with the 4th order Runge-Kutta technique, this study adopts a sophisticated computational approach, ensuring precision and efficiency to resolve the shallow water equations and predict water levels caused by a cyclone in the coastal area of Bangladesh. To discretize spatial derivatives, a 4-point backward finite difference method utilized while keeping time derivatives continuous. The authors transformed the shallow water equations with boundary conditions into an initial value problem and used the Runge-Kutta(4,4) method for solving this transformed initial value problem. To effectively include the land-sea boundary and bottom structure at a reasonable charge, the authors used a high-resolution approach. To create a regular and stable tidal oscillation in the area, the authors applied the tidal component to the southern boundary of the primary model. Then ran the surge model to estimate water elevations caused by the nonlinear interaction between tide and surge. Using numerical experiments, the authors simulated water heights generated by tide, pure surge, and tide surge interaction caused by the intense cyclonic storm SITRANG. The study's simulated results demonstrate a commendable alignment with the reported data.

Analysis of Transient Instability on a 132KV Transmission grid Power System Using 4th Order Runge kutta Technique

This study examines the transient response of synchronous machines in a 132kV transmission grid power system utilizing 4th order Runge Kutta algorithms. In order to determine the critical clearance angle (CCA) and critical clearing time (CCT) of the transmission grid, a three-phase fault was intentionally introduced. Data were gathered from the Transmission Company Nigeria (TCN) for the specific purpose of analysis and simulation. The Electrical Transient and Analysis Program (ETAP 19.1) is utilized with circuit breaker and relay time settings of (0.00, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.2) to examine the differential variations in generator rotor angles, alterations in generator bus voltage, and changes in system frequency. The obtained results demonstrate that the 4th order Runge Kutta numerical technique is stable and accurate. It exhibits a fast response critical clearing time (CCT) of 0.02s and a critical clearing angle of 76.9 degrees, which is very close to the expected value of 81.5 degrees. The average critical angle was determined from the simulated results. Additionally, a lower percentage error of -51% was observed at the moment of disturbance. The 4th order Runge-Kutta numerical technique is superior due to its sustainability, faster computation, and stability in the presence of transient disturbances, as demonstrated in the study case. It is crucial to promptly and accurately coordinate the circuit breakers and protection relays in order to rapidly clear a symmetrical 3 phase fault at any bus. This will improve the stability margin of the network.

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

Artificial neural computing and statistical analysis of heat and mass transport of nanofluid flow with melting heat and thermal stratification

The melting heat phenomenon in viscous nanofluid MHD flow over a stretching sheet with variable porosity and permeability was utilized. The behavior of magneto-viscous nanofluid flow was scrutinized using the concepts of machine learning and statistics. Thermal stratification, heat source, and activation energy in the solution of tween-20 nanoparticles and an ethyl-acetate base-type fluid are examined. Brownian and thermophoretic phenomena are included. Non-linear partial differential equations (PDEs) are converted into non-linear ordinary differential equations (ODEs) via von Karman similarity variables. Dataset is generated by the 4th-order Runge-Kutta numerical method for the artificial neural network. The ratio parameter and melting parameter enhance the velocity outline, while the melting parameter and thermal stratification parameter reduce the temperature outline. An increase in the concentration outline is seen with the activation energy parameter and Brownian motion parameter both increase the concentration outline. The system's performance using metrics like regression analysis, mean squared error, and error histograms is evaluated. The impact of these factors on significant results, such as the drag coefficient and heat transfer rate, is statistically investigated using multiple linear regressions. The integration of statistical and machine learning methods is deemed crucial for enhancing understanding of complex fluid dynamics in magnetic nanofluid flows.

ANALYSIS AND OPTIMAL CONTROL OF TUBERCULOSIS DISEASE SPREAD MODEL WITH VACCINATION AND CASE FINDING CONTROL (CASE STUDY: SURABAYA CITY)

Tuberculosis is a contagious disease that infects humans. It is caused by the bacterium Mycobacterium tuberculosis (M.tb). It is the largest infectious disease in the world and has become a major global health problem. Therefore, efforts are being made to control the spread of tuberculosis disease through vaccination and case finding, with the aim of reducing the population of latently infected and actively infected individuals. This study discusses the mathematical model of tuberculosis disease spread, disease-free and endemic equilibrium points, and stability analysis around the equilibrium points. Then, using Pontryagin's minimum principle, the optimal control problem is solved numerically by the 4th-order Runge-Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the disease-free and endemic equilibrium points. Furthermore, optimal control in the form of vaccination of susceptible individuals is required to further suppress the rate of change of susceptible individuals into latent individuals, while control in the form of case finding on latently infected individuals is required until the 9th year to minimize the population size, while on actively infected individuals, it is required until the 8th year to minimize the population size. Providing optimal control resulted in a 100% increase in the susceptible population and a 100% reduction in the latent and infected populations.

Nonlinear dynamic analysis of a spring coupled double beam based piezoelectric energy harvester under parametric base excitation

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin’s method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester.

Dissipative heat transfer on MHD micropolar hybrid nanofluid flow through infinite parallel squeezing plates

The hybrid nanoparticles suspended in an ethylene glycol water mixture are modeled as micropolar nanofluids. Studying the properties of such a mixture through a squeezing channel is vital in heat exchangers and as a cooling agent in producing sheets out of molten solids. The study aims to regulate the heat transfer rate through a hybrid nanofluid of carbon nanotubes and ferrous oxide nanoparticles immersed in ethylene glycol water. The motion of the fluid and the heat transfer with the plates are studied through the Navier-Stokes equation and the first law of thermodynamics by assuming that the base fluid has micropolar fluid properties. Apart from the usual viscous dissipation, heat dissipation due to the fluid’s micropolar nature is vigorously studied by solving the governing equations through a shooting technique with Runge-Kutta 4th-order algorithm. The results of the current study are compared against the ones with earlier studies as exceptional cases, and are very close. It is noticed that an increase in the micropolar viscosity naturally lowers the momentum in the radial direction and, at the same time, it enhances the axial momentum due to squeezing phenomena. Further, an escalating behavior is noticed in both temperature and microrotation, with an increase in micropolar parameters due to the squeezing force.