Order Runge-Kutta Method Research Articles

Numerical Solution of Hybrid Fuzzy Differential Equation by using Third Order Runge-Kutta Method Based on Linear Combination of Arithmetic Mean, Root Mean Square and Centroidal Mean

Objectives: This article develops the third order Runge-Kutta method, which uses a linear combination of the arithmetic mean, root mean square, and centroidal mean, to solve hybrid fuzzy differential equations. Methods: Seikkala's derivative is taken into account, and a numerical example is provided to show the efficacy of the proposed method. The outcomes demonstrate that the suggested approach is an effective tool for approximating the solution of hybrid fuzzy differential equations. Findings: The comparative analysis was carried out using the third order Runge-Kutta method that is currently in use and is based on arithmetic mean, root mean square, and centroidal mean. Compared to other methods, the suggested method offers a more accurate approximation. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Root Mean Square, and Centroidal Mean using Khattri's formula. And the developed formula is used to solve the third order Runge-Kutta method for the first order hybrid fuzzy differential equation. All real life problems which can be modeled in to an initial value problem can be solved using this formula. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala's derivative, third order Runge-Kutta method, Arithmetic mean, Root mean square, Centroidal mean, Initial value problem

Heat and Mass Transfer in Unsteady Radiating MHD Flow of a Maxwell Fluid with a Porous Vertically Stretching Sheet in the Presence of Activation Energy and Thermal Diffusion Effects

The aim of this study is to analyze the effects of activation energy and thermal diffusion an unsteady MHD reactive Maxwell fluid flow past a porous stretching sheet in the presence of Brownian motion, Thermophoresis and nonlinear thermal. The non-linear partial differential equations that govern the fluid flow have been transformed into a two-point boundary value problem using similarity variables and then solved numerically by fourth order Runge–Kutta method with shooting technique. Graphical results are discussed for non-dimensional velocity, temperature and concentration profiles while numerical values of the skin friction, Nusselt number and Sherwood number are presented in tabular form for various values of parameters controlling the flow system. The present study is compared with the previous literature and found to be in good agreement.

Mathematical modelling of the emission characteristics of a field electron cathode in a scanning electron microscope under biosampling conditions

Currently, the use of electron microscopes in medicine is developing intensively, including scanning electron microscopes (SEM), which are designed to solve a huge number of problems in various fields with a wide range of electron accelerating voltages and electron beam energies. The development of an SEM with certain emission characteristics, with a range of lower beam energies for the study of biological samples, is an urgent task because modifying the SEM to solve problems in medicine, for example, would make it possible to obtain higher-quality images of biospecimens for diagnostics and monitoring the effectiveness of therapy. To develop new SEMs with certain characteristics, it is proposed to conduct less expensive research using numerical methods based on mathematical models of processes in electron-optical SEM systems. In this regard, this work sets the task of determining the size and shape of the beam, the main emission characteristics of the field electron cathode (FEC) of the SEM, which is under the influence of the electric field that excites electron emission and the external longitudinal magnetic field by studying the movement of the outermost electron of the beam, taking into account the influence of space charge beam electrons, external magnetic field. In the model, the FEC is approximated by a paraboloid of rotation, and the concept of a boundary “outermost” electron is introduced, the trajectory of which determines the shape and size of the beam. The problem of calculating the emission characteristics along the trajectory of the outermost electron of a FEC is solved using a mathematical model that includes the following equations: motion of the “outermost” electron, Maxwell outside and inside the beam, continuity of the current density, Fowler-Nordheim equation. As a result, a system of 18 first-order ordinary differential equations was obtained, the numerical calculation of which using the 4th order Runge-Kutta method allows us to obtain the emission characteristics of the FEC. As a result, it is suggested that it would be feasible to modify SEMs for more effective use in the medical field, taking into account their increasing use in disease diagnosis and the possible improvement of image quality through the development of FEC SEMs with more suitable characteristics.

Comparative study of CUDA-based parallel programming in C and Python for GPU acceleration of the 4th order Runge-Kutta method

In this paper, a comparative study is presented on the application of General-purpose Computing on Graphics Processing Units for solving the point reactor kinetics equations through the utilization of the 4th Order Runge-Kutta (RK4) method using the programming languages C and Python. Sequential and parallel algorithms of the RK4 method were developed in C/C++ and Python, with parallel algorithms specifically designed to operate on Graphics Processing Units (GPUs) utilizing the NVIDIA Compute Unified Device Architecture (CUDA) as the programming platform. As an experiment, the execution time for the sequential and parallel algorithms were compared for a reactivity value of ρ = 0.003 and a simulation time of t = 100 s. The parallel C and Python algorithms achieved, respectively, speedups of 9.33 and 409.7 when comparing the execution time on the best GPU utilized (RTX 3070Ti) with the best CPU (3600XT), while still maintaining numerical precision.

Dissipative MHD flow of ternary‐hybrid nanofluid over an inclined stretching sheet with heat source and mass flux due to temperature gradient

AbstractAn analysis has been made to illuminate the unsteady MHD slip flow of ternary‐hybrid nanofluid of water conveying spherical Cu, cylindrical Al2O3, and platelet Ag nanoparticles over an inclined stretching sheet. The above complex scenario explores how the presence of a magnetic field affects the behavior of ternary‐hybrid nanofluid over an inclined stretching surface. Understanding this interaction is crucial in advanced thermal systems like cooling mechanisms and energy generation technologies. The study can provide insights into optimizing heat transfer rates, enhancing thermal conductivity and improving efficiency in practical applications by manipulating the flow characteristics and temperature distribution in the system. The heat transfer has been analyzed incorporating Darcy dissipation and heat source in the presence of 1st order velocity slip. The presence of Soret effect makes the study more congenial. Using similarity transformation technique, the governing partial differential equations are converted into a system of ordinary differential equations. The reduced system is dealt with 4th order Runge–Kutta method with shooting technique. The comparison with earlier results gives a reliability of present results. The important findings are: velocity of the flow decelerates for higher values of 1st order slip parameter, velocity of the flow increases with increase in Soret number, increasing values of chemical reaction parameter lead to lower concentration in both the cases, that is, with or without velocity slip.

Novel Approximate Solutions for Nonlinear Blasius Equations

The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

Tribological perceptions into shear thinning fluid flow: A quantitative analysis using rotating disk phenomenon

In the realm of tribology, the growing landscape of engineering applications has brought about significant developments but, despite such advancements in industrial applications, there remains a gap in understanding the numerical investigation of the flow, surface frictional forces as well as thermal characteristics in a rotating disk with nanoparticles, incorporating chemical reactions and velocity slip conditions. So, in the context of rising concerns, this article is of paramount importance for the rheology and friction caused by a rotating disk with the above physical phenomena. Numerical solutions are presented with the help of standard shooting scheme in conjunction with the 5th order Runge-Kutta method. The governing partial differential equations (PDEs) for the fluid flow are transformed into ordinary differential equations (ODEs) using new variables to facilitate the utilization of the Runge-Kutta 5th order method. The primary focus is to obtain the significance of fluid parameters, partial slip, chemical reactions, Brownian motion, and thermophoresis parameters. The implications of physical parameters are elucidated through graphical representation and discussed in detail.

PENERAPAN METODE RUNGE-KUTTA ORDE 4 MODEL PENYEBARAN DEMAM BERDARAH DENGUE DI KOTA MAKASSAR

This research was conducted based on the increasing rate of spread of dengue hemorrhagic fever in the city of Makassar. The aim of this research is to obtain a model for the spread of dengue hemorrhagic fever and find out the numerical solution of the mathematical model. The method used in this research is a quantitative method. The data is processed in the form of quantitative data in the form of the number of people infected, dead and recovered from dengue fever in the city of Makassar in 2021. This research is to predict future dengue hemorrhagic fever cases using the Runge-Kutta Order 4 method. The mathematical model for dengue hemorrhagic fever is in the form of a system of differential equations that includes the variables Sh (Susceptible Host), Ih(Infected Host), Rh (Recovered Host), Sv (Susceptible Vector), dan Iv (Infected Vector). Then look for the parameters that will be used. The research results obtained in january 2022 with Dt = 0,01 using the Fourth Order Runge-Kutta Method with initial values of Sh0= 1426454, Ih0 = 583, Rh0 = 582, Sv0 = 13165, Iv0 = 1215 is Sh100 = 1420142, Ih100 = 2949, Rh100 = 4529, Sv100 = 13706, Iv100 = 751.

Solution of First Order Ordinary Differential Equations Using Fourth Order Runge-Kutta Method with MATLAB.

Differential Equations are used in developing models in the physical sciences, engineering, mathematics, social science, environmental sciences, medical sciences and other numerous fields. This article examined solution of first ordinary differential equation using fourth order Runge-Kutta method with MATLAB. The fourth order Runge-Kutta method for modelling differential equations improves upon the Euler’s method to obtain a greater accuracy without the necessity for higher-order derivatives of the given function. A first order differential equation was solved using fourth order Runge-Kutta method with MATLAB and the same problem was solved analytically in order to obtain the exact solution. The MATLAB commands match up quickly with the steps of the fourth order Runge-Kutta algorithm. Slight variation of the MATLAB code was used to show the effect of the size of h on the accuracy of the solution (see figure 4.1, 4.2, 4.3).The MATLAB and exact solutions are approximately equal though the MATLAB approach is easier and faster. The obtained results are in agreement with those in existing literature and improved the results obtained by [1]

A four-stage multiderivative explicit Runge-Kutta method for the solution of first order ordinary differential equations

The development of numerical methods for the solution of initial value problems in ordinary differential equation have turned out to be a very rapid research area in recent decades due to the difficulties encountered in finding solutions to some mathematical models composed into differential equations from real life situations. Researchers have in recent times, used higher derivatives in the derivation of numerical methods to produce totally new ways of solving these equations. In this article, a new Runge-Kutta type methods with reduced number of function evaluations in the increment function is constructed, analyzed and implemented. This proposed method border on the use of higher derivatives up to the second derivative in the ki terms of Runge-Kutta method in order to achieve a higher order of accuracy. The qualitative features: local truncation error, consistency, convergence and stability of the new method were investigated and established. Numerical examples were also performed on some initial value problems to confirm the accuracy of the new method and compared with some existing methods of which the numerical results show that the new method competes favorably.

Analisis Dinamik Model Sel Kanker Prostat dengan Terapi Vaksin Kuratif

Prostate cancer is a type of cancer that occurs in men and requires an effective therapeutic approach. Treatment of prostate cancer depends on the stage at diagnosis. In advanced stages of prostate cancer can be treated with hormone therapy such as chemotherapy which is then followed by vaccine therapy which aims to help increase the body's immune system response to prostate cancer cells. This model consists of a system of ordinary differential equations with five variables used, including androgen-dependent prostate cancer cells, androgen-independent prostate cancer cells, dendritic cells, effector cells, and curative vaccines. Then two equilibrium point conditions are produced, when there is no vaccine for disease free conditions and endemic conditions , then when the vaccine there are three equilibrium conditions namely disease free , side effects and local existence between prostate cancer cells with vaccine . The results of the stability analysis for each equilibrium point show that when , the condition is global asymptotic, while the condition is stable because the eigenvalue is negative. When for the condition it is unstable because the two roots are positive, then for the condition it is global asymptotic and for the condition it is asymptotically local because all the eigenvalues are negative. The numerical simulations of equilibrium points obtained using the fourth order runge-kutta method according to different q parameter values show that the larger the dendritic cells and effector cells activated, the greater the vaccine that enters the body, resulting in immune cells that will fight prostate cancer cells.

A mathematical model with control strategies for marijuana smoking prevention

<abstract> <p>Our goal of this study is to prevent marijuana smoking in the human population. In this manuscript, an updated mathematical model was established by incorporating two additional compartments: The hospitalized class and the prisoner's class. The updated model was validated, and it was shown to be novel compared to the non-user, experimental, recreational, and addicted (NERA) users' model. This distinction was crucial as it was challenging to prevent marijuana usage without these realistic classes. The entire population was split into six primary groups, including these new classes: non-users, experimental, recreational, addicted, hospitalized, and prisoners' class. Additionally, control techniques for marijuana prevention in the population were addressed with the aid of sensitivity analysis. The important point at which we may have determined the preliminary transmission rate of marijuana smoking was the basic reproductive number $ {\mathbb{R}}_{0} $. Utilizing MATLAB, the Runge-Kutta method of order four was employed for the numerical simulation of the updated model to investigate the impact of control measures on marijuana smoking prevention.</p> </abstract>

Реализация бикомпактной схемы для HOLO алгоритма решения задач переноса излучения в среде

Bicompact schemes for the HOLO algorithm for solving the transport equation were applied to solve model problems of radiation transport in a medium (the first and the second Fleck problems). The main idea of HOLO algorithms is to accelerate the convergence of iterative processes due to the joint solving of high order (HO) and low order (LO) kinetic equations. The schemes are constructed by the method of lines on a minimal two-point stencil and have the fourth order of approximation in space. The Runge-Kutta method of the third order of approximation was used for integration over time. Verification and validation of the schemes were carried out. Solutions of the first and the second Fleck problems by the proposed method coincided well with solutions obtained by other methods. A modification of the second Fleck problem was proposed to investigate the dependence of the velocity of the thermal wave front on the absorption coefficient in the optically thick region.

An efficient deep neural network with amplifying sine unit for nonlinear oscillatory systems

Nonlinear differential equations play a pivotal role in modeling various physical phenomena, capturing their oscillatory behaviors. Numerous analytic and numerical methods have been developed to obtain precise or approximate solutions for nonlinear dynamics. This study introduces a novel approach utilizing neural network strategies, known for their computational flexibility. This study proposed a novel architecture for deep neural networks that is intended to simulate the nonlinear oscillations of two systems. It has been modified to include an oscillatory activation function called the amplifying sine unit. This study compares the results from our customized neural networks with those from the conventional stable and reliable numerical technique of the Runge-Kutta method of order four. Remarkably, there is striking agreement between the outcomes of the two methods, highlighting the ability of deep neural networks to identify nonlinear dynamical behaviors without requiring explicit conversion of mathematical models into an equation system.

Acoustic emission of pulsating bubbles in viscous media

<sec>The classical single bubble’s acoustic emission equation has been used to describe the sound filed radiated by bubble for a long time. Because this formula does not consider the influence of the medium viscosity in the process of sound wave propagation, it is more reasonable to modify it in some special cases.</sec><sec>Based on the boundary condition of the bubbles, i.e. the vibration velocity of the bubble wall is equal to the particle vibration velocity of the external medium at the bubble boundary, the acoustic wave equation in spherical coordinate system in viscous medium is solved, and the modified acoustic emission formula of the bubble in the viscous medium is given.</sec><sec>The bubble radius <i>R</i>(<i>t</i>) is obtained numerically from the bubble dynamics equation by using the fourth-fifth order Runge-Kutta method. Then the bubble's radiation sound field is obtained by using the direct substitution method and the finite element (The pressure acoustics module; two-dimensional (2D) axisymmetric geometric model) method, respectively. The modified expression <i>p</i><sub>present</sub> given in this work is more accurate to describe the bubble’s radiation than the classical expression <i>p</i><sub>classical</sub> in the cases of high-viscosity, high-frequency and long-distance. In these cases, continuing to measure the acoustic emission of bubbles by using the classical expression may have an influence on the characteristics of cavitation, such as the inaccurate descriptions of parameters such as cavitation intensity and cavitation threshold.</sec>

Study of the Stability of the Mathematical Model of the Bound Pendulums Motion

The article presents a study of the dynamics of an oscillatory dissipative system of two elastically coupled pendulums in a magnetic field. Nonlinear normal modes of oscillation of a pendulum system have been studied, taking into account the resistance to the medium and the damping moment created by the elastic element. A system with two degrees of freedom is considered, in which the masses of the pendulums differ significantly, which leads to the possibility of localization of oscillations. In the following study, the mass ratio is chosen as a small parameter. For approximate calculations of magnetic forces, the Padé approximation is used, which best satisfies the experimental data. This approximation provides a very accurate description of the magnetic excitation. The presence of external influences in the form of magnetic forces and various types of loads that exist in many engineering systems significantly complicates the analysis of vibration modes of nonlinear systems. Studies have been carried out of nonlinear normal modes of oscillations in this system, one of the modes being a coupled mode, and the second being a localized mode. The oscillation modes are constructed using the multiscale method. Both regular and complex behavior when changing system parameters have been studied. The influence of these parameters was studied for small and large initial angles of inclination of the pendulum. Analytical solution based on the fourth order Runge-Kutta method compared with numerical simulation results. The initial conditions for calculating the vibration modes were determined by the analytical solution. Numerical modeling, consisting of constructing phase diagrams, trajectories in configuration space, and amplitude-frequency characteristics, allows one to evaluate the dynamics of a system, which can be either regular or complex. The stability of oscillation modes was studied using numerical analysis tests, which are implementations of the Lyapunov stability criterion. In this case, the stability of the oscillation modes is determined by assessing the orthogonal deviations of the corresponding trajectories of the oscillation modes in the configuration space.

Automation of the Formation of a Mathematical Formulation of Kinetics for Multistage Chemical Reactions and Numerical Solution to a Direct Problem

Introduction. The basis for research, analysis and mathematical optimization of any chemical process is an adequate mathematical model that takes into account the kinetics of the object. Kinetic analysis is a challenge in chemical technology, since it allows for optimizing synthesis processes and predicting their efficiency. Numerous chemical processes involve several stage reactions. For successful design and optimization, a mathematical model that describes each stage is needed. Creating such a model manually can be time-consuming and costly, since it requires processing a large amount of information. The modern level of automation makes it possible to accelerate the obtaining of a mathematical formulation of the kinetics of multistage reactions. In this case, working with data is greatly simplified, and the probability of making mistakes is reduced. The resulting mathematical model can be applied for further analysis and optimization of the process. The paper considers the industrial reaction of catalytic reforming of gasoline, which occupies an important place in the modern scheme of oil refining, since it is a source of high-octane components of commercial gasolines and individual aromatic hydrocarbons. This process is characterized by the participation of a large number (up to 300) of various hydrocarbons, a change in the number of moles, and non-isothermality in it. Mathematical modeling of such processes involves detailing the stages to the required level. The detailing of up to 173 stages is considered. In this setting, automation of the formation of a mathematical formulation of kinetics for catalytic reforming of gasoline has not been carried out before. Therefore, the presented work aimed at implementing effective numerical methods and algorithms for automating the building of a mathematical model taking into account kinetics, thermodynamics, and changes in the number of moles.Materials and Methods. The mathematical formulation of the kinetics of multistage reactions was developed on the basis of the mass action law. The kinetic parameters values were taken from literary sources. The direct kinetics problem was solved using algorithms: the Gear method, the Runge-Kutta method of the 4th order, and the scipy.odeint() method of the Python language. The automation concept was implemented using the IDEF0 methodology. The software was written in the Python programming language.Results. A new software was created to automate the process of forming a mathematical model, taking into account the kinetics, thermodynamics, and the volume of the reaction mixture. The program results were presented by the example of catalytic reforming of gasoline. The model implemented the possibility of taking into account the intermediate heating of the mixture in the reactor cascade. Numerical values of temperature changes corresponding to industrial data were obtained.Discussion and Conclusion. The results obtained through modeling chemical transformations in the cascade of gasoline catalytic reforming reactors confirmed the exothermic nature of the reaction. The developed software product provides displaying changes in the concentrations of reactants, as well as temperature variations in the reactor, and it can be used in scientific research organizations for the analysis of multistage catalytic processes. The results of the reaction kinetics modeling will be used in the subsequent optimization of the process conditions in production.

Contribution of rational design and fractional rheology on magnetic nanoparticle-based drug targeting in a microvessel with Saffman force

Magnetic nanoparticles based drug targeting is one of the prominent drug targeting methods due to non-invasive treatments, and recovery efficiency. The aim of the present problem is to analyze the trajectories of drug carrier particles and the total volume fraction of magnetic nanoparticles required for capturing purposes close to the tumor region. Due to the complex nature of blood, the viscoelastic Maxwell fluid relation is used to describe blood, and the flow of nanoparticles is investigated using momentum equations with the time-variant Caputo-Fabrizio fractional order expression to capture memory effects. The rational design of the nanoparticles such as size and shape with different aspect ratio along particle-particle interaction, Saffman force and external magnetic force are incorporated in the governing equation. The velocity profile of the fluid is represented in the form of the inverse of Laplace and Hankel transform. The velocity profile is solved by numerical integration and trajectories of the drug carrier particles analyzed by using the fourth order Runge-Kutta method by developing an algorithm. The conclusion drawn from the study shows that drug carriers lead towards the tumor region with long term memory effects, and it is easily captured at the tumor region. Viscoelastic rheology capture possibly opposes the drugs carrying magnetic nanoparticles to capture at tumor regions following a similar phenomenon by Saffman force. Spherical shaped drugs carrying magnetic nanoparticles are more prominent to be targeted to the tumor region than other shaped drugs carrying magnetic nanoparticles. The present study will help biomedical engineers and nanomedicine researchers develop magnetic devices and the next generation of drug carrier particles to treat cancerous tumors.

Komparasi Skema Numerik Euler, Runge-Kutta dan Adam-Basforth-Moulton untuk Menyelesaikan Solusi Persamaan Osilator Harmonik

This article discusses the comparison of different numerical schemes to visualize the solution of 2nd-order differential equations. One-step methods such as the Euler method and the 4th-order Runge-Kutta method are combined with the 3rd-order Adam-Bashforth-Moulton method to solve the solution of 2nd-order differential equations. This combination of methods solves the Harmonic Oscillator equation, an 2nd-order differential equation widely applied in various oscillation contexts. The order of accuracy and order of approximation error are determined analytically. Finally, simulations are given with different steps for the three methods to confirm the behavior of the solution to the Harmonic Oscillator equation. The results show that the Euler method with the lowest order of accuracy has good accuracy at the beginning of the oscillation but not when time t is increased. The Runge-Kutta method, with the highest order of accuracy, shows excellent and consistent accuracy and solution stability, while the Adam-Bashforth-Moulton method, although it has a lower accuracy than the Runge-Kutta method of order 4, can be improved by choosing a one-step method with a high order of accuracy to approximate some of the required initial solutions. All three methods can provide approximation values with excellent accuracy and stability if a small step, h, is chosen, but this step can increase the time duration to display the solution. Thus, it is necessary to choose the right h according to the context of the equation and the method used to obtain accurate solutions with optimal time duration.

Synchronization and chaos control in coupled Non-identical systems: application in Wind Turbine-Grid coupled power systems

The increasing number of renewable energy systems coupled to the grid can lead to electrical energy losses when the currents or voltages of the two systems are not synchronized. Many mathematical models have investigated the phenomenon of synchronization in coupled systems. Here, we mathematically model the dynamics of a wind turbine-grid coupled system as a periodically driven Duffing resonator coupled to a Van der Pol oscillator with both position and velocity coupling. We consider the fluctuating nature of the wind as the only external driving force. We integrate the coupled system of equations under different coupling strengths and driven frequencies using the Runge-Kutta method of order 4(RK4). The result suggests that synchronization can be achieved at higher coupling strengths even with small values of the driven frequencies than at lower coupling strengths. At higher values of the driven frequency, the system exhibits chaotic behavior for both strong and weak couplings but with synchronization maintained only for the strong coupling case. Our results suggest chaos and synchronization can be controlled in this system by turning appropriate parameters.