Nonlinear Differential Systems Research Articles

Comparative analysis to study the Darcy–Forchheimer Tangent hyperbolic flow towards cylindrical surface using artificial neural network: An application to Parabolic Trough Solar Collector

Solar thermal collectors convert sunlight into useful thermal energy by absorbing its incoming radiation. Concentrated solar power technologies use the parabolic trough solar collector to collect solar energy with temperatures ranging from 325–700 K. The tangent hyperbolic fluid model is one of the most important non-Newtonian fluid models. Laboratory studies demonstrate that this model accurately predicts the shear thinning phenomenon. In addition, tangent hyperbolic fluid has a better heat transfer performance due to its rheological bearing at various shear rates. The current study investigates the heat transmission performance of Darcy–Forchheimer tangent hyperbolic radiative inclined cylindrical film movement in parabolic trough solar collector with an irregular heat sink/source utilizing the Levenberg–Marquardt technique and backpropagated neural networks. Through the implementation of required transformations, this system is turned into an equivalent nonlinear ordinary differential system. The findings are investigated for Newtonian and tangent hyperbolic fluid cases to understand the rheological characteristics. The outcomes are considered using graphical and mathematical evaluations. Fluids featuring tangent hyperbolic rheological conductivity are obligatory for active rate of heat diffusion. As a consequence, these fluids may be employed in Parabolic Trough Solar Collector for increased heat transmission rate and operational usage of solar energy. Furthermore, We create a dataset using the Runge–Kutta fourth-order shooting technique to create the proposed multilayer perceptron artificial neural network. The data points representing the MoD values are observed to be closely clustered around the zero deviation line. Additionally, it is important to highlight that these data points have relatively small numerical values. Moreover, when calculating the average MoD values for each output, it becomes evident that they are consistently very low.

The Formulation of Scaling Expansion in an Euler-Poisson Dark-Fluid Model

We present a dark fluid model described as a non-viscous, non-relativistic, rotating, and self-gravitating fluid. We assume that the system has spherical symmetry and that the matter can be described by the polytropic equation of state. The induced coupled nonlinear partial differential system of equations was solved using a self-similar time-dependent ansatz introduced by L. Sedov and G.I. Taylor. These kinds of solutions were successfully used to describe blast waves induced by an explosion following the Guderley–Landau–Stanyukovich problem. We show that the result of our quasi-analytic solutions are fully consistent with the Newtonian cosmological framework. We analyzed relevant quantities from the model, namely, the evolution of the Hubble parameter and the density parameter ratio, finding that our solutions can be applied to describe normal-to-dark energy on the cosmological scale.

Modeling and Initialization of Nonlinear and Chaotic Fractional Order Systems Based on the Infinite State Representation

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional order system are those of its distributed integer order differential system, as proven by the numerical simulations presented in the paper. Moreover, this technique applies directly to fractional-order chaotic systems, like the Chen system. The true interest of the fractional order approach is to multiply the number of equations to increase the complexity of the chaotic original system, which is essential for the confidentiality of coded communications. Moreover, the sensitivity to initial conditions of this augmented system generalizes the Lorenz approach. Determining the Lyapunov exponents by an experimental technique and with the G.S. spectrum algorithm provides proof of the validity of the infinite state representation approach.

A computational stochastic procedure for solving the epidemic breathing transmission system

This work provides numerical simulations of the nonlinear breathing transmission epidemic system using the proposed stochastic scale conjugate gradient neural networks (SCGGNNs) procedure. The mathematical model categorizes the breathing transmission epidemic model into four dynamics based on a nonlinear stiff ordinary differential system: susceptible, exposed, infected, and recovered. Three different cases of the model are taken and numerically presented by applying the stochastic SCGGNNs. An activation function ‘log-sigmoid’ uses twenty neurons in the hidden layers. The precision of SCGGNNs is obtained by comparing the proposed and database solutions. While the negligible absolute error is performed around 10–06 to 10–07, it enhances the accuracy of the scheme. The obtained results of the breathing transmission epidemic system have been provided using the training, verification, and testing procedures to reduce the mean square error. Moreover, the exactness and capability of the stochastic SCGGNNs are approved through error histograms, regression values, correlation tests, and state transitions.

Exponential stability of impulsive conformable fractional-order nonlinear differential system with time-varying delay and its applications

This paper investigates the exponential stability of conformable fractional-order nonlinear differential systems with time-varying delay and impulses. The authors utilize the principle of comparison and the Lyapunov function method to establish sufficient conditions that guarantee the exponential stability of a specific class of conformable fractional-order nonlinear differential systems. These findings extend the existing results on systems with integer-order to a certain extent. Furthermore, the paper considers the effect of impulses in the delayed system, which was not addressed in previous literature The authors also apply the criterion for exponential stability to conformable fractional-order neural networks. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed results.

A fractional approach to solar heating model using extended ODE system

In contrast to fossil fuels, the sun has more than enough energy to supply the entire world's energy needs. The sole constraint on solar energy as a renewable resource is our capacity to efficiently and economically convert it to electricity. In this work, we take advantage of the fractional derivative, we introduce the dynamics of the solar heating model. We incorporated the Atangana-Baleanu derivative (ABC) in our analysis. Then the solutions of our fractional system are investigated for existence and uniqueness. In order to visualize the fractional order model solution, we used a novel and trendy numerical method to represent the dynamics of different parameters of the nonlinear ordinary differential equation system. It is shown that the proposed ODE models are valid and efficient for fractional order data for a functioning solar heating system.

A design of novel Gudermannian neural networks for the nonlinear multi-pantograph delay differential singular model

In this paper, a new stochastic numerical platform through the Gudermannian neural network (GNN) based intelligent computing solver (GNNICS) is accessible for solving the nonlinear singular multi-pantograph delay differential (MP-DD) systems. In GNNICS, Gudermannian kernel is exploited to construct the neural network models of differential operators with different neurons for the nonlinear system along with the hybrid computing via global genetic algorithm (GA) and local refinements based active set (AS), i.e., GNN-GAAS method. A fitness function with GNN models is formulated for solving the MP-DD equation along with the optimization of design variables of the network using GAAS. To investigate the performance of the designed GNNICS based GNN-GAAS algorithm, three different variants of the MP-DD systems are used to assess the correctness, effectiveness, and robustness. The statistical investigations based on different performance are presented to authenticate the consistent accuracy, convergence, and stability of the designed GNN-GAAS algorithm. Furthermore, the negligible absolute error that are performed as 10−06 to 10−08 for solving the GNNICS based on GNN-GAAS algorithm.

Nonsimilar modeling analysis of Carreau–Yasuda mixed convective flow in a porous medium subjected to Soret and Dufour influences

AbstractThis nonsimilar convection study is about the flow of Carreau–Yasuda (CY) nanofluid model above a vertically extendible surface. Convection in a fluid‐filled permeable medium has given due consideration because of its relevance in a variety of applications, including insulation, relocation of water from geothermal reservoirs, storage of nuclear waste, renewable energy, mechanical engineering, and enhanced oil reservoir recovery. By virtue of linear stretching and buoyancy effects, flow in a stationary fluid is induced along a vertical porous surface. In x‐momentum equation, linear buoyancy in the context of temperature and concentration is taken into consideration. Modeling of energy expression is done in the presence of Dufour and Soret influences. Governing differential system describing convection equations is changed into nonlinear partial differential system (PDE) by implementing applicable nonsimilar transformations. By making use of analytical local nonsimilarity (LNS) technique and bvp4c (numerical finite difference‐based algorithm), the transformed dimensionless nonsimilar structure is simulated numerically. At the end, the alteration of important nondimensional numbers is studied on transport quantities such as temperature, concentration and velocity field. The repercussions of relevant parameters on drag coefficient, Nusselt number and Sherwood number have been tabulated. Numerical simulations of nonsimilar model suggests that the velocity profile reduces due to rise in the values of Weissenberg number, porosity and suction parameter. The temperature profile is increased in comparison with the higher estimates, Eckert, and Dufour numbers. Because of larger values of Soret and Prandtl number, an increase in concentration profile is seen. Friction coefficient and Nusselt number increases with respect to higher estimations of porosity parameter, Weissenberg number and Prandtl number respectively, whereas they decrease against Dufour and Eckert variations.

ATSS model based upon applications of Cattaneo-Christov thermal analysis for entropy optimized ternary nanomaterial flow with homogeneous-heterogeneous chemical reactions

ATSS model for magnetohydrodynamic Darcy-Forchheimer radiative ternary nanoliquid flow by curved stretched sheet is constructed. Ternary nanoliquid consists of three different nanoparticles (gold (Au), zinc oxide (ZnO) and multi-walled carbon nanotube (MWCNT)) in ordinary liquid (Carboxymethylcellulose water (CMC-H2O)). Cattaneo-Christov heat flux and convective condition are discussed. Thermal equation has Ohmic heating, heat generation/absorption and radiation. Entropy production along with cubic autocatalysis chemical reaction is discussed. The obtained differential nonlinear systems are numerically computed by using ND-solve method. Graphical discussion for liquid flow, entropy rate and temperature via influential parameters for nanomaterial (Au/CMC-H2O), hybrid nanomaterial (Au+ZnO/CMC-H2O) and ternary nanomaterial (Au+ZnO+MWCNT/CMC-H2O) is organized. Computational outcomes for coefficient of skin friction and Nusselt number are studied. Opposite trend of velocity and skin friction for curvature is noticed. Velocity reduces against higher magnetic field. Augmentation for temperature against Eckert and thermal Biot numbers is noticed. Entropy production enhancement for radiation and diffusion parameter is observed. Thermal transport rate for radiation and thermal relaxation time has an increasing effect.

Phase Portraits of Families VII and VIII of the Quadratic Systems

The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes.

MHD thermo-radiative heat transfer characteristics of carbon nanotubes based nanofluid over a convective expanding sheet in a porous medium with variable thermal conductivity

ABSTRACT The impacts of thermal radiation (non-linear) and thermal conductivity (variable) on magnetohydrodynamic (MHD) heat transfer over a flat stretchable surface by taking single-walled carbon nanotubes (SWCNTs) and multiple-walled carbon nanotubes (MWCNTs) inside the base liquid (water) embedded in a porous media have been analysed. We have utilized an appropriate set of similarity transformations to rehabilitate the governing partial differential equations into a non-linear ordinary differential equations system. These equations are then solved using a Runge–Kutta–Fehlberg technique with a shooting algorithm. The influences of several critical physical parameters on the velocity (V) and temperature (T) fields, coefficient of skin-friction (C f ), and local Nusselt number (Nu x ) have been examined in detail by plotting several graphs and tables for both SWCNTs and MWCNTs. The acquired outcomes show that SWCNTs in water-based nanofluid has remarkable effects on skin friction and the heat transfer rate when contrasted with MWCNTs. Likewise, the variable thermal conductivity enhancement diminishes the Nu x , whereas the inverse impacts the expanding T ratio parameter.

Some stability results on non-linear singular differential systems with random impulsive moments

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well.

Nonsimilar forced convection analysis of maxwell nanofluid flow over an exponentially stretching sheet with convective boundary conditions

AbstractThis proposed work aims to describe nonsimilar forced convection analysis for the flow of Maxwell nanofluids. The forced flow is commenced due to the stretching surface (SS) at an exponential rate. Heat and mass transmission is tackled with convective and zero mass flux at the surface. The fundamental laws of conservation governing the present flow problem are expressed mathematically in the form of a nonlinear partial differential system. The differential system is then remodeled into a system of nonlinear dimensionless partial differential equations (PDEs) under the implementation of suitable nonsimilar transformations. The resulting nonsimilar system is analytically approximated by applying local nonsimilarity (LNS) and then using the finite‐difference based bvp4c algorithm, it is numerically simulated to ascertain the influences of the various parameters—including the; Deborah number, Biot number, Schmidt number, Brownian diffusion parameter, Prandtl number, and thermophoresis diffusion parameters, on velocity and temperature. The reduced Nusselt numbers and the conduct of velocity and thermal distribution for varying parameters are illustrated graphically. The reduction of the boundary layer (BL) thickness is measured because‐of the Biot number while an increase is observed with an elevation in Deborah number. The velocity configuration is reduced with expanding Deborah number. It is remarked that as the values of the thermophoresis and Brownian motion parameter rise, the thermal BL thickness expands, and at the surface, the temperature gradient demises. An excellent comparison is noticed between present works with the already published article. The novelty of this article is that most of realworld BL flow problems are nonsimilar in nature, while many research have neglected this fact and treated the problem as similar. Therefore, we talked about the Maxwell problem's nonsimilar analysis here. The author's observations indicate that the analysis of the topic under discussion has not yet been declared in published literature.

Numerical simulation for peristaltic transport of radiative and dissipative MHD Prandtl nanofluid through the vertical asymmetric channel in the presence of double diffusion convection

This work's primary goal is to give a novel thorough investigation that examines the effects of the temperature- and concentration- dependent electrical conductivity on the peristaltic flow through the vertical asymmetric rotating channel filed with magnetic Prandtl nanofluid saturated in a porous medium. In addition to joule heating, viscous dissipation, and nonlinear thermal radiation, the fundamental characteristics of Brownian motion and thermophoresis processes are also considered when modelling the heat equation. Further, the influences of the double diffusion convection are examined, and the cases of biological scientific assumptions which is known as low Reynolds number and long wavelength are applied. The software Mathematica's built-in command (ND Solve function) is used to numerically solve the consequent nonlinear coupled differential equations system. The main findings showed that as the Soret and Dufour numbers increased, consequently increased the heat resistance, whereas the dissolvent concentrations and nanoparticle volume percentage had the reverse impact.

Dynamics and steady state of squirmer motion in liquid crystal

AbstractWe analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.

On the Analysis of a Fractional Tuberculosis Model with the Effect of an Imperfect Vaccine and Exogenous Factors under the Mittag–Leffler Kernel

This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical model employed in this investigation is a fractional representation of tuberculosis, utilizing the Atangana–Baleanu fractional derivative in the Caputo sense. The validity of the proposed model is established through a rigorous qualitative analysis. The existence and uniqueness of the solution are rigorously determined by applying the fundamental theorems of the fixed point approach. The stability analysis of the model is conducted using the Ulam–Hyers approach. Additionally, the study employs the widely recognized iterative Adams–Bashforth technique to obtain an approximate solution for the suggested model. The numerical simulation of the tuberculosis model is comprehensively discussed, with a particular focus on the assumptions made regarding vaccination. The model assumes that only a limited portion of the population is vaccinated at a steady rate, and the efficacy of the vaccine is a critical factor in reducing disease burden. The findings of the study indicate that the proposed model can effectively assess the impact of vaccination on mitigating the spread of tuberculosis. Furthermore, the numerical simulation underscores the significance of vaccination as an effective control measure against tuberculosis.

Three-dimensional MHD flow of hybrid material between rotating disks with heat generation

In this study, we investigate the flow of electrically conducting hybrid nanofluid (Ag+Cu/H2O), due to rotating disks, along with thermal slip, heat generation, and viscous dissipation. The nonlinear differential system is modelled and transformed into dimensionless partial differential equations using suitable dimensionless variables. To obtain solutions for the considered model, a finite difference toolkit is implemented, and numerical solutions are achieved. Graphical results are presented to display the influences of different dimensionless variables on flow velocity and temperature. This research contributes to a better understanding of hybrid nanofluid flows and can inform the design of cooling systems and other practical applications.

Numerical study of MHD radiative Williamson fluid over an inclined moving plate covering of water-based hybrid nanomaterial

Modern developments in nanotechnology have provided a fantastic foundation for creating a better ultra-high-performing coolant known as nanofluids for many applications in manufacturing and engineering. Numerous scholars have been drawn in by the hybrid nanomaterials’ capacity to improve heat transmission more to examine the working fluid. This study uses the mathematical solution to explore the effect of viscous dissipative on the radiative-convective Williamson hybrid nanofluid flow through an angled moving plate with a magnetic impression. Focusing on certain reasonable assumptions, a nonlinear partial differential equation system is developed and then numerically solved using the bvp4c method. It is thoroughly explained how a particular collection of distinctive factors affects the motion features, shear stress, temperature field profiles and heat transfer. The motion declined with increasing Williamson fluid for both stretching and shrinking surfaces. An upsurge profile for energy is showing for radiation impression. The shear rate falls with the inclined plate and increases with the buoyancy impression.

Modelling COVID-19 pandemic control strategies in metropolitan and rural health districts in New South Wales, Australia

COVID-19 remains a significant public health problem in New South Wales, Australia. Although the NSW government is employing various control policies, more specific and compelling interventions are needed to control the spread of COVID-19. This paper presents a modified SEIR-X model based on a nonlinear ordinary differential equations system that considers the transmission routes from asymptomatic (Exposed) and symptomatic (Mild and Critical) individuals. The model is fitted to the corresponding cumulative number of cases in metropolitan and rural health districts of NSW reported by the Health Department and parameterised using the least-squares method. The basic reproduction number ({mathrm{R}}_{0}), which measures the possible spread of COVID-19 in a population, is computed using the next generation operator method. Sensitivity analysis of the model parameters reveals that the transmission rate had an enormous influence on {mathrm{R}}_{0}, which may be an option for controlling this disease. Two time-dependent control strategies, namely preventive (it refers to effort at inhibiting the virus transmission and prevention of case development from Exposed, Mild, Critical, Non-hospitalised and Hospitalised population) and management (it refers to enhance the management of Non-hospitalised and Hospitalised individuals who are infected by COVID-19) measures, are considered to mitigate this disease’s dynamics using Pontryagin’s maximum principle. The most sensible control strategy is determined through the cost-effectiveness analysis for the metropolitan and rural health districts of NSW. Our findings suggest that of the single intervention strategies, enhanced preventive strategy is more cost-effective than management control strategy, as it promptly reduces COVID-19 cases in NSW. In addition, combining preventive and management interventions simultaneously is found to be the most cost-effective. Alternative policies can be implemented to control COVID-19 depending on the policymakers’ decisions. Numerical simulations of the overall system are performed to demonstrate the theoretical outcomes.