Fourth-order Runge-Kutta Research Articles

Multi-peak soliton dynamics and decoherence via the attenuation effects and trapping potential based on a fractional nonlinear Schrödinger cubic quintic equation in an optical fiber

Fiber optic research continues to develop due to the need for fast information technology. The input signal in an optical fiber is in the form of a soliton wave that propagates in the fiber with a balanced index of refraction, dispersion, and diffraction. Fiber optics may lose some of the signal over time if they solely use a nonlinear refractive index. This study aims to examine into the dynamics and decoherence of multi-peak solitons caused by electromagnetic signal attenuation and potential trap in the propagation via optical fibers. The methods used start with the stationary solution of the nonlinear Schrödinger Cubic Quintic equation, the Newton-Raphson method, and the fourth-order Runge-Kutta. In stationary solutions, the p (state energy) and ρ (Lèvy index) values affect the number, pulse width, and height of soliton peaks. Increased p and ρ values lead to increased instability, causing the data transmitted into the optical fiber to collapse and become unreadable by the receiver. In the excited state (p > 0), the decoherence phenomenon occurs and causes wave instability and destruction. In term of energy intensity, a stable form of transmission in the ground state has a pattern of continuously weakening, decreasing, and smoothing, while in the excited state there is a spike in the intensity of the electromagnetic wave. Based on this study, multi-peak solitons can only be applied for propagation distances, which tend to be short due to their dynamics and decoherence.

A Nonstandard Finite Difference Scheme for a Mathematical Model Presenting the Climate Change on the Oxygen-plankton System

This paper presents a mathematical model describing climate change in the oxygen-plankton system. The model consists of a system of non-linear ordinary differential equations. The Nonstandard Finite Difference (NSFD) method is applied to discretize the non-linear system. The stability of the continuous and discrete model is presented for the given parameters in the literature. The compatibility of the results has been seen. Moreover, the model is solved by both the NSFD method and the Runge–Kutta–Fehlberg (RKF45) method. The numerical results are compared. Furthermore, the efficiency of the NSFD method compared to classical methods such as the Euler method and the fourth order Runge-Kutta (RK4) method for the bigger step size is shown in tabular form.

A novel comprehensive method for weak signal blind detection based on adaptive quasi-periodic potential stochastic resonance and its application in bearing fault detection

In order to solve the common output saturation of stochastic resonance systems and the limitation of classical index SNR for blind detection, a novel adaptive quasi-periodic potential stochastic resonance blind detection method is proposed. First, a model of quasi-periodic potential stochastic resonance (QPPSR) possessing infinite steady state is constructed and analyzed for its structure change pattern. The superior performance of the model is verified by using the fourth-order Runge–Kutta algorithm. Secondly, the mechanism of QPPSR is analyzed using the probability flow method, which reveals the relationship between system parameters and performance. Again, a novel comprehensive blind detection index (CBDI) is exquisitely constructed to make up for the shortcomings of each indicator. Finally, CBDI and QPPSR are constructed into an adaptive blind detection system and applied to bearing fault detection. The results analyzed by experiments verify the good engineering application prospect of CBDI-QPPSR.

Numerical study of MHD flow over stretching cylinder with variable Prandtl number and viscous dissipation in ternary hybrid nanofluids with velocity and thermal slip conditions

Industrial applications in domains such as warm rolling, crystal development, thermal extrusion and optical fiber illustration are seeing a significant increase. These applications specifically focus on addressing the challenge of a cylinder in motion inside a fluid environment. Elevated temperatures may affect the viscosity and thermal conductivity of fluids. Understanding the relationship between temperature and the properties of fluids is crucial. In light of these presumptions, the primary goal of this study is to examine, under transverse magnetic field, shape factor, velocity, thermal slip conditions and viscous dissipation, how temperature-dependent fluid properties could enhance the heat transfer efficiency and performance evolution of ternary hybrid nanofluid. In order to study flow fluctuations, the impact of nanoparticle addition and improvements in heat transfer, a variable Prandtl number is also included. The use of similarity variables converts the controlling flow model from partial differential equations (PDEs) to ordinary differential equations (ODEs). Mathematica’s shooting strategy solves ODEs using the fourth-order Runge–Kutta (RK-IV) method. Numerical calculations were done after setting parameters to acquire the desired results. Analytical data are provided in tables and graphs for convenient usage. The results showed that the velocity profile increases as the values of [Formula: see text], Pr, M, Re and S grow, and decreases when the values of [Formula: see text] decrease. Re, Pr and S lower the temperature profile, whereas [Formula: see text], [Formula: see text] and Ec raise it. The skin friction profile steepens as [Formula: see text], S, Re and M increase relative to the stretched cylinder, and flattens as [Formula: see text] and [Formula: see text] decrease. The Nusselt number profile rises as [Formula: see text], Pr, S and Re decrease with [Formula: see text], Ec and [Formula: see text]. When the Prandtl number goes from 3.0 to 6.2 in a ternary hybrid nanofluid with brick-shaped nanoparticles, the Nusselt number goes up by around 55.7%.

Enhanced heat transfer rate on the flow of hybrid nanofluid through a rotating vertical cone: a statistical analysis

The recent study needs the optimal conditions for maximizing heat transfer rate with minimizing energy consumption. This is a basic objective of several industries in their production processes along with in improving the efficiency in electronic gadgets. Therefore, hybrid nanofluid plays an important role in enhancing heat transport phenomena. The current study investigates the improved heat transfer properties of a water-based hybrid nanofluid containing Al2O3 and Cu nanoparticles flowing through a rotating vertical cone embedded in a porous medium. The convective flow is enhanced by a magnetic field and thermal radiation, contributing to the complexity of the flow dynamics. However, the dimensional physical quantities are transformed to non-dimensional parameters by the implementation of suitable similarity rules. Furthermore, the transformed equations are numerically solved using a fourth-order Runge-Kutta shooting method. Henceforth, to get the optimal heat transfer rate, a robust statistical technique called response surface methodology (RSM) and for the validation a hypothetical test is organized by using analysis of variance. The parametric behavior is conducted via graphs and the physical description is presented briefly. Further, the major outcomes of the study are: both the particle concentrations attenuate the axial velocity whereas the fluid temperature enhances significantly. Increasing radiative heat augments the heat transport phenomena and the observation reveals that the case of hybrid nanofluids overrides the case of nanofluids.

F-PICNN: A physics-informed convolutional neural network for partial differential equations with space-time domain

The physics and interdisciplinary problems in science and engineering are mainly described as partial differential equations (PDEs). Recently, a novel method using physics-informed neural networks (PINNs) to solve PDEs by employing deep neural networks with physical constraints as data-driven models has been pioneered for surrogate modelling and inverse problems. However, the original PINNs based on fully connected neural networks pose intrinsic limitations and poor performance for the PDEs with nonlinearity, drastic gradients, multiscale characteristics or high dimensionality in which the complex features are hard to capture. This leads to difficulties in convergence to correct solutions and high computational costs. To address the above problems, in this paper, a novel physics-informed convolutional neural network framework based on finite discretization schemes with a stack of a series of nonlinear convolutional units (NCUs) for solving PDEs in the space-time domain without any labelled data (f-PICNN) is proposed, in which the memory mechanism can considerably speed up the convergence. Specifically, the initial conditions (ICs) are hard-encoded into the network as the first time-step solution and used to extrapolate the next time-step solution. The Dirichlet boundary conditions (BCs) are constrained by soft BC enforcement while the Neumann BCs are hard enforced. Furthermore, the loss function is designed as a set of discretized PDE residuals and optimized to conform to physics laws. Finally, the proposed auto-regressive model has been proven to be effective in a wide range of 1D and 2D nonlinear PDEs in both space and time under different finite discretization schemes (e.g., Euler, Crank Nicolson and fourth-order Runge-Kutta). The numerical results demonstrate that the proposed framework not only shows the ability to learn the PDEs efficiently but also provides an opportunity for greater conceptual simplicity, and potential for extrapolation from learning the PDEs using a limited dataset.

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.

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.

Stability and numerical solutions for second-order ordinary differential equations with application in mechanical systems

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

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.

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Transient Response Analysis in Spacecraft Thermal Protection Structures Under Periodic Thermal Disturbances

This study introduces frequency-domain analysis and transfer functions into spacecraft thermal analysis, presenting a fast approach for analyzing the temperature response of spacecraft thermal protection structures under periodic external heat flow. By employing Fourier transform, the complex external radiation boundary was simplified into the input signal of harmonic superposition. A transfer function was established, correlating the conduction and radiation properties of the structure with its damping effects. The analytical solution was validated against fourth-order Runge–Kutta numerical solutions, with less than 1% relative errors in mean temperature values and less than 2% in fluctuation amplitudes and high phase consistency. This method enables the rapid computation of temperature responses, providing valuable guidance for the engineering inverse design of thermal protection structures. The practical applicability of the approach is demonstrated through simulations of a specific satellite model, ensuring that the working temperature and fluctuations of the electronic devices meet the preset design requirements.

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 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.

Comparison of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) for Estimating the Susceptible-Exposed-Infected-Recovered (SEIR) Model Parameter Values

Background: The most commonly used mathematical model for analyzing disease spread is the Susceptible-Exposed-Infected-Recovered (SEIR) model. Moreover, the dynamics of the SEIR model depend on several factors, such as the parameter values. Objective: This study aimed to compare two optimization methods, namely genetic algorithm (GA) and particle swarm optimization (PSO), in estimating the SEIR model parameter values, such as the infection, transition, recovery, and death rates. Methods: GA and PSO algorithms were compared to estimate parameter values of the SEIR model. The fitness value was calculated from the error between the actual data of cumulative positive COVID-19 cases and the numerical data of cases from the solution of the SEIR COVID-19 model. Furthermore, the numerical solution of the COVID-19 model was calculated using the fourth-order Runge-Kutta algorithm (RK-4), while the actual data were obtained from the cumulative dataset of positive COVID-19 cases in the province of Jakarta, Indonesia. Two datasets were then used to compare the success of each algorithm, namely, Dataset 1, representing the initial interval for the spread of COVID-19, and Dataset 2, representing an interval where there was a high increase in COVID-19 cases. Results: Four parameters were estimated, namely the infection rate, transition rate, recovery rate, and death rate, due to disease. In Dataset 1, the smallest error of GA method, namely 8.9%, occurred when the value of , while the numerical error of PSO was 7.5%. In Dataset 2, the smallest error of GA method, namely 31.21%, occurred when , while the numerical error of PSO was 3.46%. Conclusion: Based on the parameter estimation results for Datasets 1 and 2, PSO had better fitting results than GA. This showed PSO was more robust to the provided datasets and could better adapt to the trends of the COVID-19 epidemic. Keywords: Genetic algorithm, Particle swarm optimization, SEIR model, COVID-19, Parameter estimation.

Cost-effectiveness analysis of optimal control strategies on the transmission dynamics of HIV and Varicella-Zoster co-infection

HIV and Varicella-Zoster co-infection has been threatens the lives and livelihood of millions of individuals all over the world. In various nations of the world the co-infection disease is presently a major health problem. Although, various researchers carried out studies related with HIV and Zoster single infections, no one has carried out a study on the co-infection disease. Therefore, investigating the HIV and Varicella-Zoster co-infection transmission dynamics using mathematical modeling approach is relatively necessary. The main objective of this study is to investigate the cost-effective analysis of the proposed optimal control strategies on the transmission dynamics of HIV and Varicella-Zoster co-infection in the community using compartmental modeling approach. Qualitatively, we have computed the model equilibrium points and analyze their local stabilities using Routh-Hurwitz stability criteria, the effective reproduction numbers by using the next generator matrix operator method, we re-formulated the optimal control problem of the complete co-infection model by implementing five time dependent optimal control strategies, analyzed the optimal control problem by applying the Pontryagin's Maximum/Minimum principle. Sensitivity analysis and numerical simulations have been carried out using the forward sensitivity index formula and the fourth order Runge-Kutta numerical methods respectively and verified the analytic results. Finally, we have performed the cost effective analysis for the proposed controlling strategies and from the result we have observed that implementing strategy four (treatment of Varicella-Zoster infected individuals to increase their recovery rate and recovery period) is the most cost effective-strategy as compared with other alternatives.

Energy harvesting for a vibration isolation system with negative stiffness under harmonic excitation

ABSTRACT In this paper, an energy harvesting system with a negative stiffness structure is established, whose main purpose is to achieve good vibration isolation, and at the same time, the system can also achieve the purpose of energy harvesting. In order to analyse the system model, the amplitude–frequency response equations for the displacement, induced voltage and electric power of the system when the system is in the main resonance state are obtained by using the multiscale method, and their amplitude–frequency response curves are plotted. In addition, we obtained the existence conditions for the system to satisfy Smale's horseshoe chaos using the Melnikov function. And the phase diagram and Poincaré mapping of the system were obtained using the fourth-order Runge–Kutta method. Numerical results show that the system has some energy harvesting performance along with vibration isolation. Variations in the parameters of the system affect the resonance frequency band and co-oscillation amplitude of the system as well as the nonlinear characteristics of the system itself.

An Exploration of the Phonon Frequency Spectrum and Born–von Karman Periodic Boundary Conditions in 1D and 2D Lattice Systems Using a Computational Approach

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.