Fourth-order Runge-Kutta Method Research Articles

Predicting Multiple Numerical Solutions to the Duffing Equation Using Machine Learning

This study addresses the problem of predicting convergence outcomes in the Duffing equation, a nonlinear second-order differential equation. The Duffing equation exhibits intriguing behavior in both undamped free vibration and forced vibration with damping, making it a subject of significant interest. In undamped free vibration, the convergence result oscillates randomly between 1 and −1, contingent upon initial conditions. For forced vibration with damping, multiple variables, including initial conditions and external forces, influence the vibration patterns, leading to diverse outcomes. To tackle this complex problem, we employ the fourth-order Runge–Kutta method to gather convergence results for both scenarios. Our approach leverages machine learning techniques, specifically the Long Short-Term Memory (LSTM) model and the LSTM-Neural Network (LSTM-NN) hybrid model. The LSTM-NN model, featuring additional hidden layers of neurons, offers enhanced predictive capabilities, achieving an impressive 98% accuracy on binary datasets. However, when predicting multiple solutions, the traditional LSTM method excels. The research encompasses three critical stages: data preprocessing, model training, and verification. Our findings demonstrate that while the LSTM-NN model performs exceptionally well in predicting binary outcomes, the LSTM model surpasses it in predicting multiple solutions.

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

In this article, a very simple modified form of the harmonic balance method is used to solve a strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force. Taylor series expansion up to third term is considered for the harmonic restoring force. The first approximate solutions of the present method pleasantly agree with the numerical solution obtained by Runge–Kutta fourth order method. Accuracy and simplicity of the present method solution is established when compared with the other method solutions. The present method can be utilized to other nonlinear oscillators.

High-order geometric integrators for the variational Gaussian approximation.

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it is symplectic, conserves energy exactly, and may partially account for tunneling. However, the variational method is also much more expensive. To improve its efficiency, we symmetrically compose the second-order symplectic integrator of Faou and Lubich and obtain geometric integrators that can achieve an arbitrary even order of convergence in the time step. We demonstrate that the high-order integrators can speed up convergence drastically compared to the second-order algorithm and, in contrast to the popular fourth-order Runge-Kutta method, are time-reversible and conserve the normand the symplectic structure exactly, regardless of the time step. To show that the method is not restricted to low-dimensional systems, we perform most of the analysis on a non-separable twenty-dimensional model of coupled Morse oscillators. We also show that the variational method may capture tunneling and, in general, improves accuracy over the non-variational thawed Gaussian approximation.

Performance of a pilot-scale BDD reactor by numerical analysis of reaction rate parameters and additional numbers for mass transfers

The performance of a pilot-scale boron-doped diamond (BDD) reactor through a numerical analysis of reaction rate parameters and enhanced mass transfer has been investigated. The main objective of this research is to evaluate the efficiency of the reactor in mineralizing and degrading caffeine as an emerging contaminant. Based on the kinetic mechanisms and mass transport correlations reported in the literature, two reaction rate kinetic models for caffeine degradation are proposed and analyzed. The models consider different electrolytes (NaCl and Na2SO4) and applied current densities. The kinetic fitting process utilizes the gradient-maximal electrochemical approach, together with orthogonal placement methods, fourth-order Runge-Kutta (RK4) methods, and Nelder & Mead methods for optimization of kinetic parameters and spatial discretization of the material balance. Experimental data obtained from a factorial design with four factors and two levels (24) validate the proposed kinetic models. Caffeine degradation is achieved with NaCl and Na2SO4 electrolytes at concentrations of 60 ppm and 100 ppm, respectively. The corresponding applied loads are 1.5 AhL−1 and 3 AhL−1. Na2SO4 exhibits superior performance with a total organic carbon (TOC) removal efficiency of 99.13%, while NaCl achieves 31.47% mineralization. The behavior of caffeine degradation under the operational and scale conditions demonstrates that NaCl, as a support electrolyte, enables controlled charge transfer (current density) during the degradation process. In contrast, Na2SO4 as a support electrolyte introduces a mixed control of charge and mass transfer. The pilot-scale kinetic parameters obtained in this study provide valuable insights into the support electrolyte dynamics and current density dynamics in BDD-based Electrooxidation (EO) systems, particularly in complex matrix applications. Furthermore, the observed electrical consumption supports the potential application of EO as a viable technology for industrial-scale tertiary wastewater treatment, specifically for caffeine removal.

NUMERICAL SOLUTION OF NON-SIMILAR BOUNDARY-LAYER FLOW OVER A CYLINDER

This paper presents а numerical solution of a non-similar two-dimensional boundary-layer flow over a cylinder. The assumptions and related two-dimensional flow equations are presented. The fourth-order Runge-Kutta method with the backward differentiation formula (BDF) method to separation point is implemented in the numerical solution using MATLAB software. Numerical solution results are compared with well-known analytic solutions. Shear stress diagram, friction ratio based on θ and x, and output results are illustrated. The results of numerical solution demonstrate good consistency with analytic solutions.

Deep learning in bifurcations of particle trajectories

We show that deep learning algorithms can be deployed to study bifurcations of particle trajectories. We demonstrate this for two physical systems, the unperturbed Duffing equation and charged particles in magnetic reversal by using the AI Poincaré algorithm. We solve the equations of motion by using a fourth-order Runge–Kutta method to generate a dataset for each system. We use a deep neural network to train the data. A noise characterized by a noise scale L is added to data during the training. By using a principal component analysis, we compute the explained variance ratios for these systems which depend on the noise scale. By plotting explained ratios against the noise scale, we show that they change at bifurcations. For different values of the Duffing equation parameters, these changes are of the form of different patterns of growth-decline of the explained ratios. For the magnetic reversal, the changes are of the form of a change in the number of principal components. We comment on the use of this technique for other dynamical systems with bifurcations.

Influence of Dofour and Soret on Eyrig-Powell Nanofluid Flow from A Circular Cylinder with Viscous Dissipation

The impact of Viscous dissipation on the heat and mass transfer characteristics of an Eyring Powell nanofluid flow past a horizontal circular cylinder is intensively investigated in the presence of Dufour and Soret effects. The free laminar flow is subject to a uniform transverse magnetic field. The continuity, momentum, energy, and concentrations equations are transformed into a nonlinear system of partial differential equations using appropriate non-similarity variables. The transformed system was solved numerically using the fourth order Runge Kutta method. The effect of parameters including Prandtl number, Dufour effect, Soret effect and Schmidt number were studied and presented graphically. Nusselt and Sherwood numbers have also been derived and discussed numerically.

Numerical simulation along with the experimental work for an underdamped oscillator using fourth order Runge–Kutta method. An undergraduate experiment

Experiments based on oscillatory motion are an essential part of the curricula for students of physics and engineering in their undergraduate studies, such as the determination of spring constant for a well-known classical oscillator i.e., spring-mass system. Moreover, it is important for students to understand the physics of damped oscillators because, in real-world scenarios, a system involving oscillations cannot be completely analysed without understanding conditions of damped oscillations i.e., underdamped, overdamped, and critical damped. In this work, we design a computational physics lab for the simulation of the underdamped oscillator using an Excel spreadsheet by employing 4th order Runge–Kutta method. Further, we construct a simple experimental setup to observe the damped oscillation and obtaindata for the underdamped system. Final analysis was based on comparison between the experimental data and the numerically estimated data.

Thermal convection and entropy generation analysis of hybrid nanofluid slip flow over a horizontal poignant thin needle with an inclined magnetic field: A numerical study

Hybrid nanofluids have emerged as a promising field of research in recent years, finding applications in various industries and sectors. The entropy generation and impact of an inclined magnetic field on a water-Ethylene Glycol (50:50) mixture-based nanofluid over a poignant thin needle has been investigated under slip conditions. The flow is swotted using thermal and velocity slip boundary conditions. The numerical effects of an inclined magnetic field on the flow of ferrous and aluminum oxide nanoparticles in a hybrid nanoliquid with as base fluids have been examined. The non-dimensional ODEs along with boundary conditions are solved by numerical technique based on the Runge–Kutta fourth-order method. Velocity profile has been analyzed for the magnetic parameter, inclined angle and volume fractions. The Bejan number provides insights into the balance between heat transfer and entropy generation in a system. From the numerical values of skin friction coefficient and Nusselt number, it is analyzed that [Formula: see text]O3 possesses 1.01% and 1.06% of high heat transfer rate and high surface drag force than [Formula: see text]O4. Heat transfer profile has been analyzed for entropy generation and the Bejan number. The applications of hybrid nanofluids are vast, and one specific area where they can be utilized is in horizontal needle systems.

Non-Darcy Newtonian Liquid Flow with Internal Heat Generation using Boundary Conditions of the Third Kind of Fully Developed Mixed Convection in a Vertical Channel

An analysis of the impact of internal heat generation and other heat parameters on the temperature, velocity and rate of heat transfer of the liquid moving upward in a channel is studied. A fully developed non-Darcy flow using boundary conditions of the third kind with internal heat generation in a vertical channel is selected as the mathematical model. By utilising the similarity transformation, the governing equations are reduced to non-linear ordinary differential equations (ODEs). The converted ODEs are numerically solved and analysed using the fourth-order Runge-Kutta (RK4) method, incorporating a shooting technique with Newton’s method. The generated numerical algorithms are programmed in MATLAB for velocity, temperature and the local Nusselt number analysis. The numerical results of the flow and temperature variables are presented graphically. The impact of the parameters on the Nusselt number is also graphed to determine which of the three types of boundary conditions is the best for allowing heat transmission. The severity of flow reversal is increased under the Robin and Dirichlet conditions by enhancing the Darcy and Forchheimer numbers while decreasing the Brinkman and internal heat generation values. The temperature profiles improved with the increase in Brinkman numbers. Both Nusselt numbers remained constant for the Neumann boundary condition for all parameters except internal heat generation and local heat exponent. The Robin boundary condition is found to be the best facilitates heat transmission, since it delivers more pleasing and realistic results than the Dirichlet and Neumann conditions

Nonlinear vibration of pinned FGP-GPLRC arches under a transverse harmonic excitation: A theoretical study

The non-linear in-plane primary resonance of functionally graded porous (FGP) sinusoidal arches made of graphene platelet reinforced composites (GPLRC) subjected a transverse harmonic excitation is investigated in this paper. To eliminate the coupling of inner forces and facilitate the derivation, the neutral plane-based equations of motion are established considering the force and bending moment equilibrium conditions. By utilizing an incremental harmonic balance (IHB) technique, the frequency response features, dynamic time history, and limit cycle under primary resonance and 1/2 super-harmonic resonance are determined. FE analysis is conducted to verify the accuracy of the presented results. To further examine the computational efficiency of IHB procedure, a fourth order Runge–Kutta method is used to determine the structural responses as a counterpart. Comprehensive parametric studies are then carried out, particular focus is paid to the cases of material compositions, geometric properties, and dynamic parameters on the frequency response characteristics. It is found that the increasing porosity coefficient significantly exacerbates the left-inclined softening behaviors of the frequency response curves. On the contrary, the left-inclined effect continues to weaken as the GPL weight fraction gradually grows.

Transient properties of grazing ecosystem driven by Lévy noise and Gaussian noise

This paper explores how Lévy noise and Gaussian noise affect the ecological grazing system by using the fourth-order Runge–Kutta method to simulate dynamic system and the Janicki-Weron algorithm to produce Lévy noise. Two deterministic quantities, the mean first passage time (MFPT) and the probability density function (PDF) of the first passage time (FPT), are utilized to explore the transient properties of grazing ecosystem. Our research results show that: (i) The transitions between two alternative stable states can be induced by the Gaussian noise intensity, Lévy noise intensity, Lévy stability index and Lévy skewness parameter. (ii) A higher Lévy noise intensity and a larger Lévy stability index or Lévy skewness parameter make the MFPT from desert state to sustainable vegetated state shorter; a higher Gaussian noise intensity makes it longer, which indicates that increased Lévy noise intensity can mitigate ecosystem degradation; increased Gaussian noise intensity will lead to desertification of vegetation. (iii) For larger Lévy noise intensity, the MFPT from sustainable vegetated state to desert state as a function of Gaussian noise intensity exhibits one maximum value when it transformation from the sustainable vegetated state to the desert state, the noise enhanced stability (NES) phenomena of grazing ecosystems are observed.

Sub-harmonic and simultaneous resonance instability of a thin-walled arch under a vertical base excitation at two frequencies

This paper presents the first attempt to investigate the in-plane sub-harmonic and simultaneous resonance instability of a thin-walled arch under a vertical base-excitation at a combination of two frequencies. When the base excitation reaches a critical value, the arch may experience abrupt sub-harmonic and simultaneous resonance instability in an in-plane antisymmetric mode, leading to unpredictable and catastrophic consequences. To understand the sub-harmonic and simultaneous resonance instability behavior of the arch, its in-plane kinematic equation under a vertical base-excitation with two frequencies is derived by employing Hamilton principle and then solved by utilizing Galerkin’s method in conjunction with the fourth-order Runge–Kutta method and finite element method to determine the instability regions. The in-plane sub-harmonic and simultaneous resonance instability mechanisms that have not yet been reported in the existing literature are discussed in detail. Some conclusions on the dynamic instability region are come to by comparison between the single- and two-frequency base-excitation. The in-plane sub-harmonic and simultaneous resonance instability analysis can help optimize the design of a thin-walled arch.

Effect of linear and nonlinear stiffness support on the dynamic response of high-speed railway bridges

In this paper, the effect of linear and nonlinear support vertical stiffness of elastic support on the dynamic response of the railway bridge is investigated. The bridge is modeled by a Euler-Bernoulli beam supported by linear and nonlinear supports stiffness. A constant moving load model HSLM-A is utilized to simulate the moving real train, and based on Hamilton’s principle, the governing partial differential equation of motion of the studied system is derived. A method to deal with the nonlinear boundary conditions is proposed, the fourth-order Runge-Kutta method in combination with the Galerkin method is used to obtain the dynamic response of the bridge. The finite difference method (FDM) is used to validate the proposed solution. The obtained results have shown that the dynamics of the system are governed essentially by a Duffing-like oscillator and proved that the elastic bearings and the cubic nonlinearities of supports have a significant effect on the global behavior of the railway bridge particularly at resonance.

A fractional-order model for computer viruses and some solution associated with residual power series method

Awareness of virus spreading is an important issue for building various defence strategies and protecting personal computers, smart devices, network devices, etc. In this research work, we develop epidemiological models to address this problem and introduce certain modified epidemiological fractional SAIR model, where we consider the fractional derivative in Caputo sense. We utilize the residual power series method to construct approximate solutions for the governing system. To show the efficiency and suitability of the proposed technique, we introduce a comparative between the obtained solutions and those solutions that are constructed using the fourth-order Runge–Kutta method. We derive some numerical results by considering specific values for the parameters in the governing model, and then we depict some of these results into two and three dimensions.

MHD slip effects on (50:50) hybrid nanofluid flow over a moving thin inclined needle with consequences of non-linear thermal radiation, viscous dissipation, and inclined Lorentz force

Due to their enormous implementation in engineering scrupulous, hybrid nanofluids have become a prevailing substance of consideration in mathematical and physical research. The augmentation of thermal conductivity is an exclusive emphasis of hybrid nanofluids. The fluid prototype substance probing in the existing study is the Ethylene Glycol–water (50:50) as base fluid and a mixture of nanoparticles are magnetic (Fe3O4) and non-magnetic (Al2O3) hybrid nanofluid. It is found in heat transfer and cooling in electronic devices, engines, refrigerants, energy storage and the automotive industry. To obtain solutions, the Partial Differential Equations with slip boundary conditions that describe the flow are converted into Ordinary Differential Equations with appropriate transformations using a numerically based technique known as the fourth-order Runge–Kutta method with shooting techniques. The illustration diagram of the gained outcomes accentuates the consequence of numerous physical variables on the flow dynamics in relation to fluid momentum, and thermal silhouette. Also, the consequence of heat transfer rate and surface drag force in multiple variables like Eckert number, magnetic variable, and radiation constant, are tabulated. These studies explain the decline in the thermal outline of hybrid nanofluid under slip situations when upsurges the non-linear radiation parameter. By analyzing these studies of hybrid nanofluids, it is possible to improve heat transfer efficiency and control temperature gradients in systems where nonlinear thermal radiation and viscous dissipation are important factors.

A machine learning model for fast approximation of free-surface Green's function and its application

In potential flow theory, the accurate and efficient calculation of free-surface Green's functions is essential for solving hydrodynamic issues. Given the impressive performance of machine learning methods in nonlinear function fitting, the present study utilizes an effective machine learning model called StripeGF for numerical approximation. In this model, equidistant horizontal datum lines are arranged in the computational domain away from the singularity, and Green's function and its derivatives on each line are fitted by a multi-layer perceptron (MLP) with a single input. Based on the first-order ordinary differential equation (ODE) that they satisfy, the fourth-order Runge-Kutta method is used to solve the Green's function and its derivatives between adjacent lines. In the domain nearing the singularity, a double-input MLP is applied. Use the Romberg quadrature to create a double-precision data set for training and validation, the numerical results demonstrate that StripeGF outperforms all 4 comparison methods in terms of efficiency and has accuracy of at least 4 digits in more than 99.9% of all zones. The boundary element program improved by StripeGF is verified in the hydrodynamic calculation of S175, showing good accuracy and reliability.

Magnetohydrodynamic Mixed Convection Heat and Mass Transfer of Nanofluid Flow Over a Stretching Wedge-Shaped Surface with the Effect of Thermophoresis and Brownian Motion

The present study has been investigated to the consequence of the magnetic parameter, Grashof number, modified Grashof number, Prandtl number, thermal radiation parameter, Brownian motion parameter, thermophoresis parameter, heat generation parameter, Schmidt number, Biot number, stretching parameter, Lewis number, and chemical reaction parameter, respectively, over a stretching wedge of the magnetohydrodynamic (MHD) BL nanofluid flow. The main goal of this paper is to numerically investigate the nature of the MHD BL nanofluid flow along a stretching wedge-shaped surface with radiation, heat source, and chemical reaction parameters. The fundamental equations has been transformed into ordinary differential equations (ODEs) by the usual transformation. The numerical solutions are found by employing Runge-Kutta fourth-order method by exploiting symbolic software MATLAB via the shooting method. The novelty of the current study is implicated in the area of fluid dynamics to solve nonlinear differential equations numerically and is an important contribution to the field of nanofluids flow. Numerical solutions reveal that the concerned physical parameters lead to progress in the skin friction factor, rate of change of heat transfer as well as the rate of change of concentration. Brownian motion and thermophoresis parameters play a crucial role in the variation of temperature and concentration profiles and also in the development of thermal and concentration boundary layers.

Study on the seismic model of a special shelf structure

In this paper, the shelf structure is discretized, reduced in order, examined, and studied for seismic response by using the concentrated mass method, Lagrange method, transfer function method, and fourth-order Runge-Kutta method based on MATLAB numerical calculation. A simplified model of special equipment including supporting frame structure is proposed, and the calculation methods of equivalent mass, stiffness, and damping of supporting frame structure are given. In addition, the seismic response time-history curve of each position of the shelf structure is quickly obtained by using the lumped mass method in this paper. The conclusion is that the seismic response of the upper floor is greater than that of the lower floor, and the seismic response of the middle position of the same floor is greater than that of both sides. This paper provides a research foundation for the follow-up work. Research in this paper verifies the rationality of simplifying the complex dynamic model of the shelf structure continuum into a simple basic model, which provides a simplified model with a high fitting for the follow-up rub-impact analysis and research, and greatly improves the calculation efficiency.

OCRUN: An oppositional Runge Kutta optimizer with cuckoo search for global optimization and feature selection

The recently proposed swarm intelligence algorithm, Runge–Kutta Optimization (RUN), is rooted in the fourth-order Runge–Kutta method. Compared with its counterparts, RUN boasts an advantage of having a more concrete theoretical foundation embodying a more powerful optimization efficacy, free from any metaphor. However, RUN still has its shortcomings. The compelling enhanced solution function leads to insufficient exploration ability of the algorithm, and resulting in an imbalance between exploration and exploitation that cannot be mitigated. An improved version based on opposition-based learning and cuckoo search is proposed to compensate for the above deficiencies, called OCRUN. OCRUN is tested on 30 test functions of CEC2014 with 10 classical metaheuristics and 9 advanced metaheuristics, respectively. Combining the experimental results and the Wilcoxon signed-rank test, OCRUN exhibits excellent performance. At the same time, parameter sensitivity analysis experiments are also carried out on this test set. Furthermore, a binary implementation of the algorithm was constructed specifically for feature selection cases, labeled as BOCRUN. BOCRUN is compared with 5 existing binary metaheuristics on 15 public datasets. The experimental results show that the improved algorithm performs well in feature selection. Therefore, OCRUN is an effectively improved optimizer. Finally, the OCRUN method offers high-quality solutions to engineering problems and contributes significantly to engineering design under practical constraints. The method has been successfully applied to various design scenarios, such as reducer design, cantilever beam design, and tension/compression spring design. The OCRUN method outperforms other similar products in terms of performance and effectiveness.