Fourth-order Runge-Kutta Method Research Articles

Chemical reaction effects on MHD heat and mass convection of rotating fluid past a moving vertical plate in a porous medium with hall current

This study presents the problem of steady heat mass convective Magnetohydrodynamic (MHD) flow of a moving and rotating fluid over a vertical uniformly moving plate through a porous medium in the presence of a first-order chemical reaction and strong magnetic field where the Hall effect is present. The numerical solutions of similarity equations are obtained by Runge-Kutta fourth order method. The graphical results show the effects of different parameters such as chemical reaction, permeability, wall velocity ratio and rotation parameters on the velocity, temperature and concentration profiles and which can also show a clear difference with the literature results.

SUSCEPTIBLE VACCINATED INFECTED RECOVERED SUSCEPTIBLE MODEL: EQUILIBRIA POINTS AND APPLICATION ON COVID-19 CASE DATA IN INDONESIA

Severe Respiratory Syndrome Coronavirus-2 is the infectious agent that causes COVID-19. A vaccine program is an effort to stop the spread of COVID-19 infections in Indonesia. The susceptible vaccinated infected recovered susceptible (SVIRS) model can be used to represent the spread of infectious diseases. This study aims to construct the SVIRS model, identify the equilibria points thus apply it to COVID-19 case data in Indonesia, and determine transmission patterns, model accuracy, and interpretation. Literature and applications are the research methodologies employed. First-order nonlinear differential equations form the obtained SVIRS model. The model has two equilibrium points: a disease-free equilibrium point, and the other is endemic equilibrium point. The SVIRS model on the spread of COVID-19 in Indonesia was obtained using daily secondary data from January 11 to November 30, 2022. The model is solved by the fourth-order Runge-Kutta method. The model’s accuracy is accurate enough to explain the spread of COVID-19 in Indonesia with a mean average percentage error (MAPE) value of 43%. According to the transmission pattern, the number of COVID-19 cases in Indonesia peaked on July 27, 2022, then decreased to zero, obtaining an equilibrium point when no more cases of the disease were present.

NUMERICAL SOLUTION OF THE SEIR MODEL USING THE FOURTH-ORDER RUNGE-KUTTA METHOD TO PREDICT THE SPREAD OF HEPATITIS B DISEASE IN AMBON CITY

Hepatitis B is a dangerous type of hepatitis and has a high risk of death. This research aims to predict the spread of Hepatitis B in Ambon using the fourth-order Runge-Kutta method. The mathematical model for the spread of Hepatitis B takes the form of a system of differential equations that includes the variables Susceptible (S) namely the subpopulation that is susceptible to infection with the hepatitis B virus, Exposed (E), namely the subpopulation that is exposed to the hepatitis B virus when it comes into contact with the Infected (I) subpopulation, I, namely the subpopulation infected with hepatitis B and Recovered (R), namely the recovered subpopulation. The values ​​ , , , , , , , and are the parameter values ​​used to be solved numerically using the fourth order Runge Kutta method which was carried out in 20 iterations with step size h=1 using data from the Maluku Provincial Health Service and the Central Bureau of Statistics from 2013 to 2022. Hepatitis B is classified as a type of hepatitis disease that is dangerous and has a high risk of death. This study aimed to construct a model of the spread of Hepatitis B disease in Ambon City and solve the model using the fourth-order Runge-Kutta method. In the research results, it was obtained that subpopulation decreased significantly in the 20th year with a total of 299,239 people, for subpopulation increased in 18th year with a total of 4,309 people, and decreased in 20th year with a total of 4,298 people, for subpopulation subpopulation increased until 20th year with a total of 254 people, and for subpopulation subpopulation increased significantly in 20th year with a total of 10,776 people.

Nonlinear thermal flutter of variable angle tow composite laminates in supersonic airflow

Compared with traditional composite materials, variable stiffness composite materials with curvilinear fiber paths have higher design flexibility and the potential to improve structural efficiency. In view of this, the nonlinear flutter characteristics of Variable Angle Tow (VAT) composite laminates with thermal effect in supersonic airflow are investigated in this paper. The proposed panel flutter model is developed based on the von Kármán large deformation plate theory. The thermal stress is approximated according to the quasi-steady thermal stress theory, whilst the aerodynamic pressure is calculated based on the first-order piston theory. The aerothermoelastic equations of motion are first established using Hamilton’s variational principle, and the Rayleigh-Ritz method is then applied to solve the governing equations in the space domain. The time-domain nonlinear equations are solved through the fourth-order Runge-Kutta method. The accuracy and convergence of the proposed method are verified through the comparisons with the results available in the literature. The effects of several parameters such as fiber orientation angle, thermal load, and aerodynamic pressure on the nonlinear flutter responses of VAT laminates are comprehensively discussed. Several dynamic behaviors of VAT laminates, including stable flat, buckled, Limit Cycle Oscillation (LCO), multi-periodic motion, quasi-periodic motion, and chaotic motion, are revealed by using the time history, phase portrait, Poincaré map, and bifurcation diagram. The results presented herein may be beneficial for the design of VAT laminates in supersonic airflow.

Thermal analysis on Ferro Casson nanofluid flow over a Riga plate with thermal radiation and non-uniform heat source/sink

Ferro hybrid nanofluids can be used in electronics and microelectronics cooling applications to reduce heat accumulation and efficiently remove surplus heat. These nanofluids aid to maintain optimum operating temperatures and reduce device overheating by enhancing the heat transfer rate. With this motivation, the aim of the present numerical analysis is to study the three-dimensional incompressible hybrid nanofluid flow over a slippery Riga surface by combining the Casson fluid model. Mathematical modeling is constructed with nanoparticles as Fe3O4 and CoFe2O4 with base fluid as water. The non-uniform heat source/sink and thermal linear radiation effects are taken into account with the Hamilton–Crosser thermal conductivity model. A system of nonlinear PDEs is produced by the proposed problem and then the relevant similarity variables are implemented to transform the set of partial differential equations and their accompanying boundary conditions into the coupled nonlinear differential equations with one independent variable. These modified ordinary differential equations (ODEs) were then successfully solved with the Runge–Kutta fourth-order method by combining via the shooting technique. With the aid of graphical representations, the effects of various influencing parameters were presented and analyzed comprehensively. Furthermore, the impacts of relevant parameters on heat transfer rates and shear stress are concisely discussed and illustrated in tabular forms. The significant findings include the enhancement of the radiation parameter increases the thermal boundary layer thickness and the thickness increases whenever surface experiences slippery conditions. The axial and transverse momentum of the fluid are controlled with the Casson parameter. An effective connection is noticed once the current numerical solutions are verified under particular conditions that were previously described.

Effect of slip velocity on Newtonian fluid flow induced by a stretching surface within a porous medium

This article aims to examine an unsteady 2-D laminar flow of magnetohydrodynamic fluid caused by an elastic surface immersed in a permeable medium under the influence of thermal radiation and extended heat flux. Thermal conductivity and viscosity both are supposed to vary with temperature. This flow model also includes velocity slip, heat source, and joule heating. The governing equations of the fluid, including momentum and energy equations, of the proposed problem are transfigured into a system of interconnected non-linear ordinary differential equations through similarity transformations. The resultant equations are solved efficiently by employing the shooting technique in combination with the fourth-order Runge-Kutta method. Numerical values and the effect of numerous governing factors on the flow field, temperature distribution, local skin friction coefficient, and Nusselt number are showcased via graphs and tables. The investigation reveals that velocity slip, heat source, and porosity parameters enhance the temperature field while diminishing the velocity field. Furthermore, the velocity slip parameter notably reduces both the coefficient of skin friction and the Nusselt number.

Effect of slip velocity on Newtonian fluid flow induced by a stretching surface within a porous medium

This article aims to examine an unsteady 2-D laminar flow of magnetohydrodynamic fluid caused by an elastic surface immersed in a permeable medium under the influence of thermal radiation and extended heat flux. Thermal conductivity and viscosity both are supposed to vary with temperature. This flow model also includes velocity slip, heat source, and joule heating. The governing equations of the fluid, including momentum and energy equations, of the proposed problem are transfigured into a system of interconnected non-linear ordinary differential equations through similarity transformations. The resultant equations are solved efficiently by employing the shooting technique in combination with the fourth-order Runge-Kutta method. Numerical values and the effect of numerous governing factors on the flow field, temperature distribution, local skin friction coefficient, and Nusselt number are showcased via graphs and tables. The investigation reveals that velocity slip, heat source, and porosity parameters enhance the temperature field while diminishing the velocity field. Furthermore, the velocity slip parameter notably reduces both the coefficient of skin friction and the Nusselt number.

Large sampling intervals for learning and predicting chaotic systems with reservoir computing

Reservoir computing (RC) is an efficient artificial neural network for model-free prediction and analysis of dynamical systems time series. As a data-based method, the capacity of RC is strongly affected by the time sampling interval of training data. In this paper, taking Lorenz system as an example, we explore the influence of this sampling interval on the performance of RC in predicting chaotic sequences. When the sampling interval increases, the prediction capacity of RC is first enhanced then weakened, presenting a bell-shaped curve. By slightly revising the calculation method of the output matrix, the prediction performance of RC with small sampling interval can be improved. Furthermore, RC can learn and reproduce the state of chaotic system with a large time interval, which is almost five times larger than that of the classic fourth-order Runge–Kutta method. Our results show the capacity of RC in the applications where the time sampling intervals are constrained and laid the foundation for building a fast algorithm with larger time iteration steps.

Peridynamics contact model: Application to healing using phase field theory

This paper presents a contact algorithm in non-ordinary state-based peridynamics (NOSB-PD) framework and its application in healing phase field (HPF) theory. The idea here is to exploit the nonlocality in PD to model the contact between two deformable bodies. The proposed method does not require artificial short-range force or additional degrees of freedom, e.g., Lagrange multiplier, which are often used in the conventional classical and PD contact models. In the regions where contact is likely to take place, neighbor search is performed at every time step and extra particles entering the horizon are identified. These extra particles form artificial bonds which transfer forces only in compression. This feature is essential for rebound simulation. Envisioning surface-to-surface bonding as the opposite process of cracking, we propose an HPF theory to model high velocity impact induced bonding between two material bodies. Here an HPF based bonding criterion is postulated and integrated with the proposed contact algorithm. The solution of the governing equations is obtained using the fourth order Runge-Kutta method. The efficacy of the proposed contact algorithm is demonstrated by simulations of impact and rebound of two plates, two cylinders, and cylinder with rectangular prism and comparing the energy time histories with finite element solutions obtained using ANSYS®. The potential of the proposed HPF theory is demonstrated through simulation of impact of two plates where the plates rebound when the impact velocity is below a critical value, and above that velocity, the plates join with each other.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Weakly non-linear stability analysis of g-jitter induced Rayleigh-Bénard convection under inclined surfaces

This study has investigated the effect of g-jitter on the Rayleigh-Benard convection between inclined surfaces whose lower layer is concentrated with a solute C. The problem is formulated using the Ginzburg–Landau (GL) equation, which is time-dependent non-autonomous ordinary differential equation and derived from a weakly non-linear theory and Fredholm solvability condition. The graphical solution of the GL equation is obtained with the help of MATHEMATICA by using the Runge–Kutta fourth-order method. The impact of various flow parameters on the convective rate of heat and mass transport has been discussed. The heat and mass transport is found to be advanced by the non-dimensional parameters viz. modulations’ amplitude (δ), Lewis number (Le), and solute Rayleigh number (RC), but it is suppressed by the angle of inclination ϕ.

Peridynamics contact model: Application to healing using phase field theory

This paper presents a contact algorithm in non-ordinary state-based peridynamics (NOSB-PD) framework and its application in healing phase field (HPF) theory. The idea here is to exploit the nonlocality in PD to model the contact between two deformable bodies. The proposed method does not require artificial short-range force or additional degrees of freedom, e.g., Lagrange multiplier, which are often used in the conventional classical and PD contact models. In the regions where contact is likely to take place, neighbor search is performed at every time step and extra particles entering the horizon are identified. These extra particles form artificial bonds which transfer forces only in compression. This feature is essential for rebound simulation. Envisioning surface-to-surface bonding as the opposite process of cracking, we propose an HPF theory to model high velocity impact induced bonding between two material bodies. Here an HPF based bonding criterion is postulated and integrated with the proposed contact algorithm. The solution of the governing equations is obtained using the fourth order Runge-Kutta method. The efficacy of the proposed contact algorithm is demonstrated by simulations of impact and rebound of two plates, two cylinders, and cylinder with rectangular prism and comparing the energy time histories with finite element solutions obtained using ANSYS®. The potential of the proposed HPF theory is demonstrated through simulation of impact of two plates where the plates rebound when the impact velocity is below a critical value, and above that velocity, the plates join with each other.

Numerical study on phase change propulsion mechanism of thermal underwater glider based on spectral method

The thermal underwater glider’s propulsion power is provided by the phase change energy storage tube. To improve the propulsion performance of the glider, it is essential to study the phase change heat transfer characteristics. This work proposes a collocation spectral method to handle the spatial derivative term and introduces a fourth-order Runge-Kutta method to handle the time derivative term. The temperature distribution and liquid volume fraction of the phase change material in the energy storage tube were numerically simulated under specific boundary conditions. The simulation results were then compared to the experimental data, and a good agreement was observed.Based on these findings, the impact of incoming free stream velocity and temperature difference intensity on phase change within the vessel was examined. The results underscored the dominance of convection in solid–liquid phase change and identified key factors influencing phase change in scenarios with Stefan numbers below 0.1342. Additionally, the study highlighted the limitations of fixed-value approximations and the nonlinear nature of phase change over time. These insights provide valuable guidance for developing phase change devices that harness volumetric potential energy aboard underwater thermal gliders.

Analysis of SEIQV Epidemic Model on the Spread of Covid-19 in Jember Regency

Covid-19 is a disease that attacks the respiratory system caused by infection with SARSCoV-2. Efforts to prevent the spread of Covid-19 by vaccination. The spread of disease can be modeled into a mathematical equation. The model used in this study is the SEIQV Model. The disease spread model is analyzed by finding the equilibrium point and the stability of the equilibrium point as well as the basic reproduction number. In the Covid-19 distribution model, a bifurcation analysis is carried out which is needed to determine changes in stability and changes in the number of equilibrium points. Then perform a numerical solution using the fourth-order Runge-Kutta method which is simulated using MATLAB.

Efficient Numerical Solution of the Sharma-Tasso-Olver Equation Using Radial Basis Function-Pseudo Spectral Method with Comparative Analysis

In this study, we will solve the Sharma-Tasso-Olver equation by utilizing the radial basis function-pseudo spectral approach. Pseudo spectral techniques are renowned for their exceptional precision but have constraints regarding their ability to handle geometric variations. The integration of Radial Basis Functions (RBF) with pseudo spectral methods offers a solution to surmount this particular limitation. To estimate the differentials with respect to the variable , we will use the radial basis functions, including commonly used ones like multiquadric, inverse multiquadric, and inverse quadric, have been employed. Furthermore, the problem has transformed into the ordinary differential equation (ODE), and the solution to this ODE system has been acquired through the application of the fourth-order Runge-Kutta method. Additionally, a comparative analysis has been conducted between the solutions obtained through the suggested technique and the exact solutions.

ANALYSIS AND OPTIMAL CONTROL OF HIV GROWTH MODEL IN THE BODY WITH ANTIRETROVIRAL THERAPY

Human Immunodeficiency Virus (HIV) is a virus that affects the human immune system. HIV infection causes a decrease in the body's immunity because the virus attacks immune-building cells, especially T-CD4 cells. Currently, there is no treatment that can cure or eliminate HIV, but antiretroviral therapy can be done. This study discusses the growth model of HIV in the body that is given control in an effort to maximize healthy T-CD4 cells. In this model, the infection-free and infected equilibrium points are also discussed and their stability is analyzed. Then the optimal control is solved using the Pontryagin Maximum Principle method and solved numerically using the fourth-order Runge Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the infection-free equilibrium point and unstable around the infected equilibrium point. Simulation results show that with the control of antiretroviral therapy, the T-CD4 cell population grows significantly which can improve the quality of life of patients. And the growth of HIV in the body can be inhibited until it cannot reproduce itself.

Numerical Investigation of Diffusion thermo and Thermal Diffusion on Mixed Convection flow of Williamson Nanofluid through a vertical cone with porous material in the presence Thermal radiation and Activation Energy

In this paper, analyze the impact of Activation energy, Diffusion thermo and thermal diffusion with heat and mass transfer inherent of over a vertical cone through Thermally Radiant Williamson nanofluid with saturated porous medium under the convective boundary condition in the presence of thermal radiation has been studied. The coefficients of Brownian and thermophoresis diffusions are also taken into consideration. The governing partial differential equations are reduced to a couple of nonlinear ordinary differential equations by using suitable transformation equations; these equations are then solved numerically with the use of the conventional fourth-order Runge Kutta method accompanied by the shooting technique. As a result, the effects of various physical parameters on the velocity, temperature, and nanoparticle concentration profiles as well as on the skin friction coefficient and rate of heat transfer are discussed with the aid of graphs and tables. This study has been directly applied in the pharmaceutical industry, microfluidic technology, microbial improved oil recovery, modelling oil and gas-bearing sedimentary basins, and many other fields. Further, to check the accuracy and validation of the present results, satisfactory concurrence is observed with the existing literature.

Ultra-low frequency vibration energy harvesting of piezoelectric vibration systems with an adjustable device

Energy harvesting from external Ultra-low frequency vibrations is currently attracting a lot of attention. However, for the same system, it is difficult to satisfy the need for more efficient energy harvesting in a variety of external environments. Accordingly, an adjustable gearing device connected to the energy harvesting system is used herein for adapting energy harvesting to a wide range of external environments. The device uses four symmetrical inclined springs. One end of the springs is connected to a gear and the other end to a mass element. The geometrical parameters of the system can be varied by rotating the gears, thus varying the dynamic stiffness of the system. The kinetic equations of the system are derived and their nonlinear terms are Taylor-expanded. The amplitude-frequency response curve of the system is obtained by solving using the harmonic balance method. Multi-color plots of RMS values of induced voltage and electric power are obtained using the fourth-order Runge-Kutta method. The results show that the device has excellent adjustable performance. Adjusting the rotation angle of the gears can change the soft and hard characteristics of the system's dynamic stiffness and the intrinsic frequency, thus improving the system's energy harvesting performance in different environments.

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.