Fifth-order Runge Kutta Method Research Articles

An approach to the effects of stellar rotation on the theoretical apsidal motion constants. Calculations from 0.40 M_sol to 25.0 M_sol

The most reliable sources for determining absolute stellar parameters are the double-lined eclipsing binary systems. Some of these systems also show apsidal motion, characterized by the variable log k$_2$. This point grants us the possibility of investigating the stellar interior, specifically the degree of stellar mass concentration. The first studies carried out about four decades ago on this topic showed appreciable discrepancies not only with respect to the comparison between the observed absolute dimensions and their theoretical counterparts, but mainly concerning the degree of mass concentration through the analysis of their apsidal motions. Fortunately, this scenario has been gradually improving with the advances in the quality of observational techniques and advances in the input physics of the evolutionary stellar models (e.g. opacities, thermonuclear reactions, equations of state, numerical techniques). These new developments in the input physics has improved the comparison between observations and the values predicted by theory, including the apsidal motion rates. This progress has lead us to investigate second-order effects such as rotation and dynamic tides. In this paper we deal with the effects of rotation on the degree of mass concentration. The stellar models were calculated using the MESA package. The mass range studied here was 0.40 to 25.0 M$_ odot $ for a generic chemical composition characterized by X = 0.70 and Z = 0.02. The present models were computed without taking into account core overshooting in order to highlight the effects of rotation. Each model was followed from the pre-main sequence until the central hydrogen content is of the order of or less than 0.03, covering the range of masses and evolutionary status of the double-lined eclipsing binaries showing apsidal motion. Regarding the calculation of the internal structure coefficients bf we integrated the Radau equation using the fifth-order Runge-Kutta method and a tolerance level of 10$^ $. As an auxiliary tool, homology transformations have been used to explain, qualitatively, how the equation of state, thermonuclear reactions, and convection impact the degree of mass concentration of the models. In our past studies on the effects of rotation on log k$_2$ an average correction was used for all models together. For this a small range of masses (2.0, 7.0, 15.0 M$_ odot $) has been used. In the present paper the corrections due to the effects of rotation in log k$_2$ are presented for each mass individually and for three evolutionary phases. This point is particularly important bf given that these corrections show a clear dependence on the mass and on the evolutionary status. Such corrections can be easily introduced into the theoretical calculation of apsidal motion rates through a linear equation. A typical correction due to the rotation effects is of the order of -0.03 in log k$_2$.

Effects of Hall, ion slip, viscous dissipation and nonlinear thermal radiation on MHD Williamson nanofluid flow past a stretching sheet

This work aims to explore the effects of ion slip and Hall current on the flow of MHD Williamson nanofluid over a stretching plate in the presence of chemical reaction, viscous dissipation, ohmic (Joule) heating, nonlinear thermal radiation, and heat generation/absorption. After using appropriate similarity transformations, the highly nonlinear governing partial differential equations are transformed to a system of ordinary differential equations. The subsequent non-linear problems are treated to numerical results by reducing the converted nonlinear ODE in to first order. The fifth-order Runge–Kutta method along with the shooting approach is implemented using the python programming tool to analyze the problem. Graphical representations have been used to investigate the impacts of the derived physical parameters on the temperature, velocity, and concentration distributions of nanoparticles with the goal of giving each parameter a physical interpretation. The study of the effects of various physical parameters on the skin friction coefficient, the local Nusselt number and the Sherwood number was finally explained with the help of graphs and tables. It is discovered that when the Williamson parameter increases, both the principal and secondary velocity profiles decline but the temperature and concentration profiles rise. Both the Hall parameter and ion slip effect produce similar results with increasing principal velocity profile and decreasing secondary velocity, temperature and concentration profiles. Furthermore, it is seen that the temperature and concentration profiles increase in response to an increase in the nonlinear thermal radiation parameter. The increment in Eckert number (viscous dissipation term) also showed enhancements in temperature and concentration profiles. Additionally, the primary skin friction coefficient and rate of mass transfer are accelerated by raising the magnetic field parameter, Williamson parameter, or Darcy number, but the heat transfer rates are slowed down in the boundary layer. Lastly, the obtained results were cross-referenced with earlier findings, considering specific assumptions, to establish the credibility of the results. The comparison revealed that the new results provided valuable and profound insights into the studied phenomenon.

SEIRS model for malaria transmission dynamics incorporating seasonality and awareness campaign

Malaria, a devastating disease caused by the Plasmodium parasite and transmitted through the bites of female Anopheles mosquitoes, remains a significant public health concern, claiming over 600,000 lives annually, predominantly among children. Novel tools, including the application of Wolbachia, are being developed to combat malaria-transmitting mosquitoes. This study presents a modified susceptible-exposed-infectious-recovered-susceptible (SEIRS) compartmental mathematical model to evaluate the impact of awareness-based control measures on malaria transmission dynamics, incorporating mosquito interactions and seasonality. Employing the next-generation matrix approach, we calculated a basic reproduction number (R0) of 2.4537, indicating that without robust control measures, the disease will persist in the human population. The model equations were solved numerically using fourth and fifth-order Runge-Kutta methods. The model was fitted to malaria incidence data from Kenya spanning 2000 to 2021 using least squares curve fitting. The fitting algorithm yielded a mean absolute error (MAE) of 2.6463 when comparing the actual data points to the simulated values of infectious human population (Ih). This finding indicates that the proposed mathematical model closely aligns with the recorded malaria incidence data. The optimal values of the model parameters were estimated from the fitting algorithm, and future malaria dynamics were projected for the next decade. The research findings suggest that social media-based awareness campaigns, coupled with specific optimization control measures and effective management methods, offer the most cost-effective approach to managing malaria.

Hydrothermal analysis of Ag and CuO hybrid NPs suspended in mixture of water 20%+EG 80% between two concentric cylinders

This paper investigates the mixture of hybrid nanofluid (NF) flow between two concentric cylinders numerically. Ethylene glycol (80%) and water (20%) comprise the base fluid. Additionally, copper oxide (CuO) and silver (Ag) NPs are added. The hybrid NF flows between the two coaxial cylinders is studied. The inner cylinder is fixed and the outer cylinder is rotating. A radial magnetic field also affects fluid flow, and joule heating joule heating is also considered. The transformed system of nonlinear ordinary differential equations is solved numerically using the fifth-order Runge-Kutta method. Numerical solutions are obtained for fluid velocity and velocity gradient, fluid temperature and temperature gradient, pressure, surface friction coefficient and Nusselt number under the influence of different variables. The computer code is validated by comparing the variation of temperature of hybrid nanofluid with the numerical published data, a good agreement was observed. As the magnetic field parameter is increased, the fluid's velocity reduces. The fluid temperature increases as the magnetic field parameter and Brinkman number were increased. The skin friction coefficient increases with the enhancement of volume percentage and the magnetic field parameter. The Nusselt number rises with increasing of the magnetic field and Brinkman number.

Unsteady MHD hybrid nanofluid flow over a convectively heated linear stretching cylinder with velocity slip: A comparative study

This paper provides a comparative numerical analysis of unsteady hybrid nanofluid ([Formula: see text]/H2O) and nanofluid (TiO2/H2O) flows through a permeable linear stretching cylinder placed in a porous medium has been presented in the presence of oblique Lorentz force. The flow regulating boundary layer equations has been governed due to the sway of the suction, viscous-Ohmic dissipation, heat source, thermal radiation and first-order velocity slip with imposing convective boundary conditions on the surface. The self-similar transformation has been employed to convert the governing coupled nonlinear PDEs into a set of ODEs and finally solved them numerically by fifth-order Runge–Kutta method with shooting technique. Finally, a comparative study has been made on the thickness of the momentum and thermal boundary layers due to the effect of governing parameters for simple and hybrid nanofluids. Further, a numerical observation on local skin friction and thermal-transport coefficients is presented with the help of tables. The result indicates that the progressiveness of the velocity profile reduces for the unsteadiness parameter, Hartree pressure gradient, velocity slip parameter, inclination parameter, porous medium parameter, magnetic parameter and suction parameter while improving for velocity ratio parameter and temperature profiles increases with Biot number, Eckert number, heat generation parameter and thermal radiation parameter, but decreases for Hartree pressure gradient and velocity ratio parameter. This work has found various applications in high-temperature and cooling processes, paints, conductive coatings, medicines, space technology, biosensors, conductive coatings and so on.

General solution of phase mismatched second harmonic conversion efficiency in periodically poled nonlinear medium

In this work we provide for the first time a novel and more accurate second harmonic generation (SHG) conversion efficiency equation for periodically poled nonlinear medium. We believe the equation derived in this work is more useful than equations reported in previous works which are limited because it is considering the case of either nondepleted pump regime or perfect phase matching. The equation derived in this work tackles the general case: depleted pump regime and phase mismatching which makes it important for theoretical – experimental study. We used MgO:PPLN crystal as an example to illustrate SHG conversion efficiency as function of crystal length, laser power density, crystal temperature and beam divergence. We included fundamental and second harmonic waves reflection at entrance and exit surfaces respectively and performed numerical solution using fifth-order Runge-Kutta method. The numerical results show excellent match with analytical results obtained from the derived formula. We compared our theoretical results with experimental results reported in the literature and the comparison shows a good match. Finally, we performed a comparison between depleted and nondepleted pump regime. The results show a difference of 20% between two efficiencies making nondepleted efficiency equation, which used in many theoretical – experimental comparisons, not accurate at high power densities.

Impact of Artificial Compressibility on the Numerical Solution of Incompressible Nanofluid Flow

The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies. Therefore, this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow. To improve the stability and convergence speed of time discretizations, the fifth-order Runge-Kutta method is applied. A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids. The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation. The optimum artificial compressibility is found to be between 1 and 5. These findings can be utilized to enhance the performance of commercial numerical simulation tools, including ANSYS and COMSOL.

Fluid Flow and Mixed Heat Transfer in a Horizontal Channel with an Open Cavity and Wavy Wall

Heat exchangers are utilized extensively in different industries and technologies. Consequently, optimizing heat exchangers has been a major concern among researchers. Although various studies have been conducted to improve the heat transfer rate, the use of a wavy wall in the presence of different types of heat transfer mechanisms has not been investigated. This study thus investigates the mixed heat transmission behavior of fluid in a horizontal channel with a cavity and a hot, wavy wall. The fluid flow in the channel is considered laminar, and the governing equations including continuity, momentum, and energy are all solved numerically. The numerical solution is stabilized by using a first-order multi-dimensional characteristic-based scheme in combination with a fifth-order Runge-Kutta method. The flow and heat transfer effects of varying Richardson numbers, Reynolds numbers, wave amplitude, wavelength, channel height, and cavity width are examined. The results indicate that the mean Nusselt number increases with an increase in Reynolds number, wave amplitude, and cavity width, while it decreases with an increase in Richardson number, wavelength, and channel height. The minimum Nusselt number is calculated to be 0.7, whereas the maximum Nusselt number is 27.09. The Nusselt number has only increased by 40% in the higher depths of the cavity, despite the Richardson number being 10,000 times larger. But this figure increases to 130% at lower depths. The mean Nusselt number is thus significantly influenced by channel height and cavity width. The influence of wave amplitude on the mean Nusselt number is twice that of wavelength.

A C++ code for simulations of RHEED intensity oscillations within the kinematical approximation

This paper proposes a new software implementation for the rapid development and deployment of computer simulations of reflection high-energy electron diffraction (RHEED) intensity oscillations during the growth of thin epitaxial films. The calculations are based on the kinematical approximation. In this model the RHEED intensity oscillations from growing layers is considered as an effect of changes of surface roughness. New version program summaryProgram Title: growth22CPC Library link to program files:https://doi.org/10.17632/zkpwhrrfng.1Licensing provisions: GNU General Public License 3Programming language: C++Journal reference of previous version: Computer Physics Communications 170 (2005) 265–286Does the new version supersede the previous version?: YesReasons for the new version:The previous version of the program was written in Object Pascal [1]. This programming language is no longer popular in the scientific community. Responding to user's feedback, the current distribution of the software has been implemented in C++. This version implements improvements for ergonomics, computational performances and code readability. In addition, the output generated by the growth22 (saved in the coverageProfiles.dat file) can be used directly by program [2].Summary of revisions:The growth22 is written in standard C++, therefore one can use the GNU g++ compiler or any ANSI C++ compliant compiler to build the executable program. The code is distributed in the form of one growth22.cpp source file. A Table 1 lists the retired Object Pascal procedures and shows replacements by C++ functions. The present version of the program enables the user to easily manage the input data. Input data are loaded from a manually prepared inputData.dat text file where the user provides information on the parameters of the simulation, i.e. the growth model, the number of growing layers, the upper limit of growth time interval, the values of the phenomenological parameters that measure the net rate of transfer from one layer to the next, and the values of the growth rate for each layer. The user can edit this file as per the requirement of the problem domain. Details of the parameters in the sample inputData.dat file are described in the readme.pdf file provided with the software.The nature of problem:Generally, in order to explain the experimental data two theories—kinematical and dynamical—associated with high-energy electron diffraction in epitaxial layers are used. The dynamical theories take into account multiple scattering within the crystal and are used whenever diffraction from a perfect epitaxial layers is considered [3]. On the other hand, the kinematical theory treats the scattering from each plane in the crystal sample as being independent of others [1,4].In the kinematical theory of RHEED it is assumed that electron wave scatters from atom in the surface layer only once and then interferes with waves scattered on other atoms. This model can be used to describe diffraction effects for small angles of incidence, when the electron beam does not penetrate the deeper layers of the crystal. In the kinematical model the intensity oscillations of specularly reflected electron beam from growing layer is considered as an effect of changes of surface roughness.In this paper we present a time-efficient computer model for kinematical calculations of RHEED intensity oscillations during the growth of thin epitaxial films.The method of solution:According to kinematical approximation the diffracted-beam intensity is given by [4]:(1)I(S‖→=0,Sz)=|∑n=0∞(θn−θn+1)exp(−iSznd)|2 where S→ is the scattering vector, d is the interlayer spacing and n is an integer. θn−θn+1 is the exposed layer coverage which is treated as the atomic distribution function gives the position of surface atoms. For Sz=π/d Eq. (1) simplifies to:(2)I=|∑n=0∞(θn−θn+1)cos⁡(nπ)|2.Variation of the coverage ratio θn of the nth surface layer is modelled by a set of non-linear differential equations [1,4]. The Cash-Karp fifth-order Runge-Kutta method [5] with error estimation and adaptive stepsize control was used for solving initial value problem for non-linear differential equations [6]. The absolute and relative error tolerances and number of integration intervals are fully user-configurable.The proposed code approaches the modelling of RHEED from two different points of view, i.e. the data and functional perspectives. The data perspective (i.e. calculating the layer coverages as a function of their growth time) is used to characterize the data sources of the program whereas the functional perspective includes services provided by the program. It should be noted that the perspective of the data can be expressed by a completely different model introduced by the user.The coverage evolution curves for the diffusive growth model, corresponding kinematical diffracted intensity and total roughness of the surface are shown in Fig. 1. During the numerical calculations showed in Fig. 1 it was assumed that the number of growing layers (numLayers) equals 15, the upper limit of growth time interval (tMax) equals 12, the growth rate equals 1 MLs−1 (for all layers), and the values of the parameter kn (that determines the amount of interlayer diffusion) changing according to the linear relation:(3)kn=k1−(k1−kN)n−1N−1, where N is the total number of growing layers.

Exploratory scenarios for the differential nuclear and tidal evolution of TZ Fornacis

Aims. TZ Fornacis is a double-lined eclipsing binary system with similar masses (2.057 ± 0.001 and 1.958 ± 0.001 M⊙) but characterized by very different radii (8.28 ± 0.22 and 3.94 ± 0.17 R⊙). This similarity in terms of mass makes it possible to study the system’s differential stellar evolution as well as some aspects of its tidal evolution. With regard to its orbital elements, it was recently confirmed that its orbit is circular with an orbital period of 75.7 days. The less massive component rotates about 17 times faster than the primary one, which is synchronized with the mean orbital angular velocity. Our main objective in this work is to study both the nuclear and the tidal evolution of the system. Methods. To model the TZ For system, we used the MESA package, computing the grids using the exact observed masses, radii, and effective temperatures as input, and then varying the metallicity, the core overshooting amount, and the mixing-length parameter. A χ2 statistic was used to infer the optimal values of the core overshooting and the mixing-length parameters. The same procedure was used to generate rotating models with the GRANADA code. The respective errors in the average age of TZ For were less than 5%. On the other hand, the differential equations that govern the tidal evolution were integrated using the fifth-order Runge–Kutta method, ith a tolerance of 1 × 10−7. Results. We explored two scenarios regarding the initial eccentricities: a high one (0.30) and a case of an initial circular orbit. A good agreement has been found between the observational values of the eccentricity, synchronism levels, and orbital period with the values predicted by the integration of the tidal evolution equations. The influence of the friction timescale on the evolution of the orbital elements of TZ For is also studied here. The orbital elements most affected by the uncertainties in the friction timescale are the synchronism levels of the two components. On the other hand, we used the properties of the rotating models generated by the GRANADA code as the initial angular velocities instead of using trial values. In this case, comparisons between the theoretical values of the orbital elements and their observed counterparts also lead to a good interagreement.

Hydrothermal analysis of hybrid nanofluid flow on a vertical plate by considering slip condition

Hybrid nanofluids have attracted burgeoning attention owing to their outstanding capacity to improve heat transfer. The influence of velocity and temperature slip parameter and nanoparticls' (NPs') volume fraction on a vertical plate in the existence of suction has been explored in this work. The investigation's controlling partial differentiation equations were transformed into a conventional differential equation mechanism using resemblance modifications. Equations were then solved employing the fifth-order Runge-Kutta method. The skin coefficient of friction, temperature, and temperature gradient all rise when the volume percentage of NPs increases from 0 to 2%. Furthermore, a rise in the temperature slip variable was linked to a drop in the Nusselt number (heat transfer).The Nusselt number increased 0.15% and 5.63% respectively when the velocity slip parameter enhanced from 0 to 5 and the NPs volume percentage were increased from 0 to 1.5%. Furthermore, an increase in the temperature slip from 0 to 3 inflated the x-direction skin friction coefficient 8.2%, while inflation in the velocity slip from 0 to 5 was associated with a decline in the x-direction skin friction coefficient 95%.

Dynamics of a bubble in a power-law fluid confined within an elastic solid

The dynamics of a bubble in a non-Newtonian power-law fluid confined within an elastic solid is investigated. The model proposed by Wang for the bubble oscillation in a Newtonian liquid with solid confinement is extended to the case of a generalized Newtonian fluid. The resulting non-linear ordinary differential equation for the bubble radius with respect to time is solved using the fifth-order Runge–Kutta method. The results indicate that the behavior of the bubble in the power-law fluid can be significantly different from that in a Newtonian fluid. Furthermore, the size of the solid confinement and its elasticity have a noticeable impact on the dynamics of the bubble in the power-law fluid.

EXPLICIT FIFTH-ORDER HYBRID METHODS FOR DYNAMICAL SYSTEMS WITH DELAY

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained.
 This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

Modeling, analyzing and simulating the dynamics of Lassa fever in Nigeria

Lassa fever is an infectious and zoonotic disease with incidence ranging between a hundred to three hundred thousand cases, with approximately five thousand deaths reported yearly in West Africa. This disease has become endemic in the Lassa belt of Sub-Saharan Africa, thus increasing the health burden in these regions including Nigeria. A deterministic mathematical model is presented to study the dynamics of Lassa fever in Nigeria. The model describes the transmission between two interacting hosts, namely the human and rodent populations. Using the cumulative number of cases reported by the Nigerian Centre for Disease Control within the first week of January 2020 through the eleventh week in 2021, we performed the model fitting and parameterization using the nonlinear least square method. The reproduction number {mathcal {R}}_{0} which measures the potential spread of Lassa fever in the population is used to investigate the local and global stability of the system. The result shows that the model system is locally and globally asymptomatically stable whenever {mathcal {R}}_{0}<1, otherwise it is unstable. Furthermore, the endemic equilibrium stability is investigated and the criteria for the existence of the phenomenon of bifurcation is presented. We performed the sensitivity analysis of each reproduction number parameter and solutions of the developed model are derived through an iterative numerical technique, a six-stage fifth-order Runge–Kutta method. Numerical simulations of the total infected human population (E_{h}+I_{h}) under different numerical values (controlled parameters) are presented. The result from this study shows that combined controlled parameters made the total infected human population decline faster and thus reduces Lassa fever’s burden on the population.

Effect of heat and mass transfer in nanofluid flow along a vertical thin needle

The present study deals with the effects of heat and mass transfer with vertical thin needles in alumina water nanofluid. The fifth-order Runge-Kutta method with shooting technique is used to obtain solution of incompressible viscous over a vertical thin needles with mass transfer. The governing partial differential boundary layer equations in cylindrical form are first converted to nonlinear differential equations. Numerical results of velocity, temperature, share stress and concentration have been obtained and reported graphically for different parametric conditions to analysis. Significant changes have been observed in heat transfer rate and concentration.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Analysis and Numerical Simulation of SEIRS Model with Vaccination on COVID-19 Spread

This research aims to construct a mathematical model on the spread of COVID-19 in Indonesia, analyze the stability of the model, and perform numerical simulations. In this research, the SEIRS model was constructed for the spread of COVID-19. This model was constructed by considering the effect of vaccination parameters and with the assumption of decreased immunity. The model has two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The basic reproduction number is obtained through the Next Generation matrix method which is formed from the model at the disease-free equilibrium point. The stability analysis of the model was carried out by considering the Jacobian matrix of the model. To perform numerical simulation, the Fifth-order Runge-Kutta method is used to provide a numerical solution for the model. From the simulations that have been carried out, it can be seen that vaccination can slow down the spread of COVID-19 and accelerate the recovery of the population.

Accurate Numerical Method for Singular Initial-Value Problems

In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.

Fast Simulations of Highly-Connected Spiking Cortical Models Using GPUs.

Over the past decade there has been a growing interest in the development of parallel hardware systems for simulating large-scale networks of spiking neurons. Compared to other highly-parallel systems, GPU-accelerated solutions have the advantage of a relatively low cost and a great versatility, thanks also to the possibility of using the CUDA-C/C++ programming languages. NeuronGPU is a GPU library for large-scale simulations of spiking neural network models, written in the C++ and CUDA-C++ programming languages, based on a novel spike-delivery algorithm. This library includes simple LIF (leaky-integrate-and-fire) neuron models as well as several multisynapse AdEx (adaptive-exponential-integrate-and-fire) neuron models with current or conductance based synapses, different types of spike generators, tools for recording spikes, state variables and parameters, and it supports user-definable models. The numerical solution of the differential equations of the dynamics of the AdEx models is performed through a parallel implementation, written in CUDA-C++, of the fifth-order Runge-Kutta method with adaptive step-size control. In this work we evaluate the performance of this library on the simulation of a cortical microcircuit model, based on LIF neurons and current-based synapses, and on balanced networks of excitatory and inhibitory neurons, using AdEx or Izhikevich neuron models and conductance-based or current-based synapses. On these models, we will show that the proposed library achieves state-of-the-art performance in terms of simulation time per second of biological activity. In particular, using a single NVIDIA GeForce RTX 2080 Ti GPU board, the full-scale cortical-microcircuit model, which includes about 77,000 neurons and 3 · 108 connections, can be simulated at a speed very close to real time, while the simulation time of a balanced network of 1,000,000 AdEx neurons with 1,000 connections per neuron was about 70 s per second of biological activity.