RK4 Method Research Articles

Impact of High-Power Cosh-Gaussian Beam on Second Harmonic Generation in Collisionless Magnetoplasma

The impact of high power Cosh-Gaussian (ChG) beam on Second harmonic generation (SHG) in Collisionless magnetoplasma is explored in present work. Whenever the input beam propagates along external magnetic magnetic field direction, then there are two propagation modes viz. extraordinary mode and ordinary mode. The modification in magnetic field strength causes redistribution of carriers. The density gradients get established in plasma in a normal direction to input wave due to ponderomotive force. Further, there is production of electron plasma wave (EPW) at input wave frequency due to density gradients. EPW nonlinearly interacts with pump wave causing generation of 2nd harmonics. The 2nd order differential equation (ODE) for beam width of and efficiency of 2nd harmonics are derived through well-known paraxial theory approach. RK4 method is employed for carrying out numerical calculation of nonlinear ODE along with efficiency of 2nd harmonics. Impact of change in selective laserplasma parameters and externally applied magnetic field on beam waist of input wave and efficiency of SHG are also explored.

Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Existing literature established stability in a delayed logistic Lotka–Volterra predator- prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size h=1m,m∈Z+. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay τ−sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.

Harmonic response analysis of a circular plate subjected to moving point loads using Green’s function approach

Dynamics of circular plates under moving loads are an important class of problems in manufacturing industry, for example, cutting thin annular plates using point cutting tools and wood/bar/pipe cutting saws. These tools are subjected to excessive vibration during operation. Its dynamic behavior needs to be analyzed rigorously. In the present work, dynamics of a circular plate subjected to moving harmonic point load is analyzed using Kirchhoff’s plate hypothesis. In order to capture the geometric nonlinearity, the von Kármán strain displacement relation is used. Constitutive relations are developed using a complex modulus. Further, extended Hamilton’s principle is used to obtain the nonlinear coupled governing equations in radial, tangential, and transverse directions. Governing equations are discretized using Galerkin’s method. The novelty in the present work is to develop Green’s function incorporating viscous damping to obtain the plate response subjected to arbitrary moving load for general boundary conditions. The proposed method is capable to obtain the plate response due to single/multi point moving loads. In order to verify the accuracy of the proposed method, Green’s function solution is compared with the numerical solution obtained from the RK4 method and shows good agreement. Further, resonance instabilities are studied, which occur when a single point load rotates with critical angular velocity. Subsequently, the response of the circular plate under three-point loads, each having a magnitude of 1/3rd of the single moving point load moving circularly in a constant phase has been discussed. In the end, a parametric study of hysteresis damping on the dimensionless deflection of the plate is carried out.

Analyzing the effectiveness of Nields constraint and stratification on dynamics of non-Newtonian fluid by executing numerical and machine learning approaches

Stratified flows are commonly observed in numerous industrial processes. For example, a gas-condensate pipeline typically uses a stratified flow regime. However, this flow arrangement is stable only under a specific set of operating conditions that allows the formation of stratification. In this study, the authors analyzed the flow attributes of Prandtl Eyring liquid past an inclined sheet immersed in a stratified medium. The flow also characterizes the features of the magnetic field along with a first-order chemical reaction. Convective boundary constraints associated with the thermosolutal exchange at the extremity of the domain are also prescribed. The fundamental equations of the study are formulated in dimensional PDEs and converted into dimensionless ODEs via similar variables. The numerical solution of the modelled setup is acquired by executing computations using shooting and RK-4 methods. The intelligent computing paradigm working on the mechanism of the back-propagated Levenberg-Marquardt strategy is also capitalized to forecast the behavior of related physical quantities. Graphs and tables are drawn to elaborate the impression of pertinent factors on flow distributions. It is perceived that the momentum profile diminishes with the magnetic field effect, whereas the opposite behavior is observed for the skin friction coefficient. The thermal and concentration distributions were found to dominate in the absence of stratification. Consideration of convective heating and concentration tends to elevate thermal and mass distributions.

Radiative squeezed flow of magnetized Prandtl-Eyring fluid over a sensor surface under Soret effect

The central aim of this numerical analysis is to describe the effects of magnetic body force and thermal radiation on the boundary layer flow of Prandtl-Eyring fluid over a sensor surface suspended between two parallel plates. The considered fluid flow is two-dimensional unsteady electrically conducting and viscous incompressible in nature. A novel Soret effect is included in the concentration equation to reveal the significant features of mass diffusion mechanism under the influence of external squeezing mechanism. Surface permeable velocity concept is also included in the boundary conditions to describe the suction/injection process. However, squeezing flows find their abundant applications in bio-medicine, medical and bio-technology areas. Particularly, the micro-cantilevers with physical or chemical sensor surfaces are effectively used in the bio-medical applications for the finding of various hazardous or bio-warfare agents’, infections, hazardous virus, nuclear reactor cooling, polymer processing, pharmaceutical, blood flow, injection molding and many more. Thus, inspired by these practical applications of squeezing flows, the present problem is modeled based on the investigated geometry. The present physical problem gives the highly nonlinear coupled unsteady two-dimensional partial differential equations and which are not amenable to any of the direct methods. Owing to this reason, the appropriate similarity transformations are used to diminish the dimensional complexity of the emerged partial differential equations (PDEs). Thus, a robust Runge-Kutta -4th order scheme (RK-4) is used as a basic tool to obtain the similar solutions of the governing equations. The graphical visualization is used to analyze the computer generated numerical data in the boundary layer regime for the different set of physical parameters. It is observed that, the enhancing magnetic number ( 0.1 ≤ M ≤ 2.7 ) in the flow region 0 ≤ η ≤ 3 increases the velocity field and decreases the temperature and concentration fields. Increasing squeezed flow index ( 0.1 ≤ b ≤ 2.5 ) in the flow domain 0 ≤ η ≤ 3 decreased the fluid velocity, temperature and concentration fields. This result may be useful in sensor surface cooling process. Rising Prandtl-Eyring fluid parameters in the range 0 ≤ η ≤ 3 decreased the velocity field and enhanced the temperature and concentration profiles. Increasing Soret number ( 0.1 ≤ Sr ≤ 0.9 ) in the flow domain 0 ≤ η ≤ 3 enhances the concentration field. Enhancing magnetic and squeezed flow numbers amplifies the skin-friction coefficient on the sensor surface. Enhancing radiation parameter increases the heat transport rate. Increasing Soret number magnifies the concentration transport rate. Finally, it is explored that, the current similar results displaying good agreement with the formerly published results in the literature and this fact confirms the accuracy and guarantee of the RK-4 method and present similar results.

Structural seismic responses prediction using the gradient-enhanced hybrid PINN

To rapidly and effectively assess the bridge seismic-resistant capability, it is essential to conduct efficient predictions of bridge seismic responses. Recently, physics informed neural network (PINN) has made great progress and utilized to solve differential equations in different fields. However, how to increase its accuracy and efficiency still remains an open challenge. In this work, a novel gradient-enhanced Fourth-Order Runge-Kutta PINN (gRK4-PINN), as a powerful hybrid PINN, is utilized to achieve this goal. As for gRK4-PINN, the physical information is not simply embedded into the loss function; instead, the RK4 method and the physical model is intricately integrated with the neural network. In addition, to improve the predictive performance, additional gradient equation is directly embedded in loss function. A large-span continuous girder high speed railway (CGHSR) bridge is adopted as numerical experiment to validate the fidelity of the proposed method. Results reveal that the Mean Absolute Error (MAE) of the predicting seismic responses is relatively small, whose value is below 0.014 in most of the time. These small MAE values indicate that the proposed gRK4-PINN performs well in predicting the seismic responses of the CGHSR bridge.

Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle (CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field (AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments, demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

Unveiling multi-wave patterns: dynamic characterization and sensitivity analysis of the Yu-Toda-Sasa-Fukuyama model in lattice and liquid

In this study, an examination of the Yu-Toda-Sasa-Fukuyama equation is undertaken, a model that characterizes elastic waves in a lattice or interfacial waves in a two layer liquid. Our emphasis lies in conducting a comprehensive analysis of this equation through various viewpoints, including the examination of soliton dynamics, exploration of bifurcation patterns, investigation of chaotic phenomena, and a thorough evaluation of the model’s sensitivity. Utilizing a simplified version of Hirota’s approach, multi-soliton pattens, including 1-wave, 2-wave, and 3-wave solitons, are successfully derived. The identified solutions are depicted visually via 3D, 2D, and contour plots using Mathematica software. The dynamic behavior of the discussed equation is explored through the theory of bifurcation and chaos, with phase diagrams of bifurcation observed at the fixed points of a planar system. Introducing a perturbed force to the dynamical system, periodic, quasi-periodic and chaotic patterns are identified using the RK4 method. The chaotic nature of perturbed system is discussed through Lyapunov exponent analysis. Sensitivity and multistability analysis are conducted, considering various initial conditions. The results acquired emphasize the efficacy of the methodologies used in evaluating solitons and phase plots across a broader spectrum of nonlinear models.

Numerical approximation of the typhoid disease model via Genocchi wavelet collocation method

AbstractIn this paper, we have considered the fractional typhoid disease model and obtained the numerical approximation of the model via the innovative wavelet scheme called the Genocchi wavelet collocation method (GWCM) with the help of Caputo fractional derivative for the fractional order. The approach under consideration is a powerful tool for obtaining numerical solutions to fractional-order nonlinear differential equations. The GWCM approach yields accurate solutions that are very close to exact solutions for highly nonlinear problems by avoiding data rounding and just computing a few terms. The Genocchi wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main benefit of this method lies in its capability to reduce the computational complexity associated with solving systems of ODEs, resulting in accurate and efficient solutions. The results of the developed technique, the RK4 method, and the ND solver have been compared. The numerical outcomes demonstrate that the implemented technique is incredibly effective and precise for solving the Typhoid model of fractional order. This paper contributes to numerical analysis by introducing the Genocchi wavelet method as a robust tool for solving biological models.

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

Impact of length and radius of carbon nanotubes on flow and heat transfer in converging and diverging channels with entropy generation

This research examines the complex interplay between the carbon nanotubes' length and radius and how that affects flow patterns, heat transfer properties, and entropy creation in both convergent and divergent channels. Our research elucidates the mechanisms at play by revealing a crucial link between carbon nanotube dimensions and the efficacy of heat transfer processes. In particular, carbon nanotubes with a longer length have better heat transmission properties, while carbon nanotubes with a smaller radius generate less entropy. From microfluidics to energy-efficient materials, these results show significant promise for optimizing thermal systems and improving performance. This research looks on the differences between the thermal transports of single- and multi-walled carbon nanotubes in water. Thermo physical characteristics of carbon nanoparticles are one of a kind and essential for this purpose. The fluid is assumed to be incompressible and to flow steadily a two-dimensional plane. The effects of thermal radiation on the heat transfer of a water-based fluid in convergent and divergent channels with decreasing and expanding walls have been studied. In order to quickly sketch the equations, the RK-4 method with shooting approach is used, and similarity transformation is employed to reduce their dimensionality. The key factors influencing hydrothermal performance have been visualized through the use of graphs. The production numbers of entropy are also shown and explained at length. The obtained data supports the reliability of the approach used. Surface heat transport was found to be negatively impacted by elongation. Colloidal single-walled carbon nanotube (SWCNT) suspensions in water had higher thermal conductivity than water itself.

Strip-based numerical analysis of CFRP-reinforced steel plates with multiple debonding defects using the Runge-Kutta method

In recent years, carbon fiber-reinforced polymers (CFRP) have earned extensive attention in the field of structural rehabilitation. Adhesive bonding, as a prevalent technique, plays a crucial role in utilizing CFRP for strengthening steel structures. Nevertheless, debonding defects, commonly arising within the bond region between CFRP and host structures, hold the potential to induce the premature debonding failure of adhesively bonded joints. As a result, this research attempts to propose a strip-based numerical method using the explicit 4th-order Runge-Kutta technique (strip-based RK4) to investigate the full-range bond behavior of CFRP-to-steel single-lap joints with multiple arbitrary debonding defects. Also, three-dimensional (3D) finite element models (FEMs) of fully bonded single-lap joints were established in conjunction with cohesive zone modeling (CZM) to simulate the single-lap shear test reported in literature, based on which joints containing debonding defects were further modeled. Predictions from the proposed methodology were subsequently compared with test results and FE analysis to confirm its accuracy. Thereafter, the parametric analysis was carried out to investigate the effects of parameters, including bond length and defect position, on the bond behavior of the single-lap joint. The results indicate that the proposed strip-based RK4 method is accurate, with errors within 2% compared to FE analysis, and computationally efficient, with runtime being less than 50% of FE analysis. The findings of this study can be used as a guide for the design or damage assessment of CFRP-repaired structures in the future.

Mathematical model with sensitivity analysis and control strategies for marijuana consumption

The current study focuses on reducing the use of marijuana in the population. Marijuana seems to represent a serious threat to public health in developing countries since it is an illegal narcotic with various harmful health effects. The novelty of this manuscript is to modify the non-user, experimental users, recreational users, and addict's (NERA) model for marijuana consumption by incorporating a new class of addicted users under treatment, called the hospitalized class. This real-world issue is expressed in mathematical terms using first-order non-liner ordinary differential equations, and as a result, a mathematical model for marijuana consumption is established. The general population is divided into two main groups: marijuana users and non-users. Subsequently, four subgroups of abusers are created, each reflecting a different stage or degree of substance misuse. The invariant region and basic reproduction number R0 are those parts that are used for the validation of the proposed modified model and to find out the initial transmission rate of marijuana smoking in the general population, respectively. The essential factors influencing cannabis growth were identified based on the results of sensitivity analysis. Additionally, strategies were constructed in the form of prevention objectives to prevent individuals from experiencing increasing effects of marijuana. The effectiveness of these control strategies was subsequently confirmed through numerical simulations, and the numerical simulations were carried out with the help of MATLAB using the RK-4 method.

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.

Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods

This paper presents two standard numerical methods for solving second order initial value problems for ordinary differential equations (ODEs). The Euler and the Runge-Kutta fourth-order methods are applied without any discretization or restrictive assumptions for solving ODEs. The numerical solutions obtained by the two methods are in good agreement with the exact solutions. The convergence and error analysis which are discussed demonstrate the effectiveness of the methods. The results obtained from the two numerical methods show that the RK4 method is appropriate, consistent, convergent, quite stable, and more accurate than the Euler's method.

Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system’s imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r2-cut of the r1-plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM’s efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU).

Some appropriate results for the existence theory and numerical solutions of fractals–fractional order malaria disease mathematical model

Investigation of a mathematical model of malaria is considered in this manuscript. The concerned problem is investigated via Caputo–Fabrizio fractal fractional derivative. We derive the necessary conditions for the existence and uniqueness of solution for the proposed model of malaria. A powerful numerical procedure called Adams–Bashforth method is utilized for the simulations of our results. The considered technique is an excellent mathematical tool which is more efficient than the Euler and RK4 method. To discuss the dynamics of considered model, we provided the graphical presentations of our results by using various fractals and fractional orders values.

A Novel Numerical Scheme for Fractional Bernoulli Equations and the Rössler Model: A Comparative Analysis using Atangana-Baleanu Caputo Fractional Derivative

This study aims to use a novel scheme for the Atangana-Baleanu Caputo fractional derivative (ABC-FD) to solve the fractional Bernoulli equation and the fractional R ̈ossler model. Furthermore, the suggested technique is compared to Runge-Kutta Fourth Order (RK4). The proposed method is efficacious and generates solutions that are indistinguishable from the approx-imate solutions generated by the RK4 method. Therefore, we can adapt the approach to various systems and develop results that are more accurate. On top of that, the new technique (ABC-FD) can identify chaotic situations. Consequently, this approach can be used to enhance the performance of other systems. In the future, this technique can be employed to determine the numerical solution for a multitude of models applicable in the fields of science and engineering.

Significance of nanoparticles shape for enhanced heat transfer over on elastic sheet

This study analyzes the impact that the shape of nanoparticles has on the rate at which heat is transferred over an elastic sheet, which is one of the most important aspects of thermal management. It addresses the pressing demand for effective heat dissipation and insulation in a variety of industries, including energy systems and electronic devices, among others. The importance of this work rests in the fact that it has the potential to make conventional methods of heat transmission obsolete. By gaining an understanding of how the morphologies of nanoparticles affect their thermal properties, we may pave the way for the development of novel materials and applications, which will ultimately result in increased energy efficiency and high-performance technology. The originality by arguing that the study fills a knowledge gap on how different nanoparticle shapes influence heat transport in elastic sheets. The importance of this property for applications such as electronics and materials research should be emphasized. Within the scope of this study, a nanofluid composed of copper and water is utilized to investigate the movement of heat between the stretching sheets. This objective is supported by the utilization of the Hamilton Crosser Model as a tool. Platelets, cylinders, and blocks are some of the shapes and sizes of the nanoparticles that are utilized in this process. A magnetic field is applied, which changes the thermal properties of the nanofluids that are being used in the process of exchanging heat between two objects. This makes the process go more quickly. A similarity transformation is used to convert the governing equations into a collection of ordinary differential equations (ODEs). By using the shooting method, the boundary value problem can be converted into an initial value problem. This is achievable since the shooting technique is a shooting method. After that, a numerical solution is found for this issue using the RK-4 method. We give visual data that demonstrates how the flow pattern and temperature profile change as a result of a variety of different causes. Plots are provided for both the Nusselt number and the skin friction coefficient. As the inquiry goes on and the values of the key parameters are changed, one thing that happens consistently across all forms of nanoparticles is an increase in the velocity profile. This is a pattern. In addition, the nanofluid that is formed of platelet-shaped nanoparticles (which has a bigger value of shape factor) is shown to have the greatest temperature. This finding demonstrates a clear association between temperature and shape factor.

Optimization and control in rubella transmission dynamics: A boundedness-preserving numerical model with vaccination

Rubella is an infectious disease that can spread globally. It spreads in the subtropics as well as the tropics. Although it is commonly thought to be a non-fatal condition, there are several circumstances in which it can be fatal. Pregnant women infected with the Rubella virus face a high risk of fetal development. The main goal of this study is to look into a model for the spread of Rubella while considering a vaccination campaign as a control measure. The positivity and boundedness of the re-infection rubella transmission and vaccination model are investigated. The local and global stability of the equilibrium points is examined. The sensitivity of parameters is also investigated. Three different techniques forward Euler, RK-4, and non-standard finite difference (NSFD) method are developed for the numerical solution, and their simulation results are examined. Among these three, the NSFD method is superior due to its convergence and positive behavior for all step size values. For some values of the step size, the other two techniques failed to produce positivity and convergent solutions. Analytically, the proposed model's convergence, positivity, boundedness, and consistency are investigated. Finally, the impact of the Rubella vaccine on infected populations has been examined, revealing that vaccination is one of the most effective ways to prevent rubella transmission.