Nonlinear Ordinary Differential System Research Articles

Mass and heat transfer in the boundary layer region of modified second-grade fluid flow over a linearly stretched sheet embedded in porous media

Here, we have improved a model to describe the flow variables, velocity, mass and heat transfer in the boundary layer region of modified second-grade fluid flow over a linearly stretched sheet in a porous media. The modified model is used to study the qualitative impact of buoyancy parameter, second-grade fluid parameter, magnetic parameters, porous parameter, power-law index, and chemical reaction parameter on the flow profiles, radial and axial velocities, temperature, and concentration. Starting with the steady-state governing equations of mass, momentum, heat, and concentration of the fluid flow, we obtained the boundary layer approximations of the flow near the linearly stretched sheet with the no-slip boundary condition. Similarity transformation has been used to convert the partial differential equations system into a nonlinear ordinary differential equations system. The radial velocities, temperature, and concentration profiles have been solved numerically, and the qualitative influence of the flow parameters on the flow variables has been simulated and graphically presented for comparison. We have observed that radial and axial velocities were increasing for the shear-thinning and shearthickening fluids with solutal Grashof number, thermal Grashof number and second-grade fluid parameters. In contrast, the porous and chemical reaction parameters slow down both fluids’ radial and axial velocities. The temperature and concentration increase with the porous and magnetic parameters. However, thermal and solutal Grashof and second-grade fluid parameters suppress temperature and concentration. In shear-thickening fluids, chemical reaction parameters enhance concentration but suppress in shear-thinning fluids.

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

We take into account the nonlinear complex generalized Zakharov dynamical system which models the spread of the Langmuir waves in ionized plasma, in the conformal sense in this manuscript. Fractional wave transformation is enforced to convert the nonlinear fractional system to a nonlinear ordinary differential equation system. The new Kudryashov method which was recently introduced and is an efficient method, is implemented to the presented equation to acquire analytical solutions. The required constraint conditions are offered to ensure the validity of the obtained solutions. To analyze the physical interpretations for some of the produced solutions, we illustrate some graphical representations. We derive the bright and singular solitons. Furthermore, 2D views of the behavior of the solitons are represented to investigate the effect of the values of the parameters in the proposed model and fractional parameters. Also, the modulation instability of the model is investigated to ensure the obtained results are stable.

Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey

This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.

New Qualitative Outcomes for Ordinary Differential Systems of Second Order

This paper deals with a nonlinear ordinary differential system of second order. In the paper, qualitative properties of solutions of the system called asymptotic stability (AS), uniform stability (US), boundedness, ultimately boundedness (UB) and integrability of solutions, are investigated by using the second method of Lyapunov. We give four new qualitative results and an example as a numerical application of the results. The results of this article extend and improve some earlier ones in the literature.

Peristaltic transport of radiative and dissipative MHD third order nanofluid through the vertical asymmetric channel with heat and mass convection

: The fundamental purpose of the present study is to provide a novel framework for investigating the peristaltic flow through a vertical asymmetric channel filled with a magnetic third order nanofluid model saturated in a porous medium under the influence of varying electrical conductivity. The essential characteristics of Brownian and thermophoresis phenomena are additionally included in the modelling of the heat equation, in besides joule heating, viscous dissipation, heat generation/absorption, and nonlinear thermal radiation. Furthermore, the effects of double diffusion convection are examined, with a small Reynolds number along with extended wavelength examples that reflect biological scientific assumptions are used. The software Mathematica's built-in command ND Solve is implemented to solve the derived nonlinear ordinary differential equations system numerically. One of the most significant phenomena concerning peristaltic motion, known as the trapping phenomenon, has also been spotlighted using contour plots and circulating bolus. The key discoveries showed that the size of the trapping bolus tends to decrease while the density of trapping bolus increases with an enhancement in the temperature- and concentration-dependent electrical conductivity coefficient. Further, for all areas under consideration, the rate of heat and mass transmission is enhanced due to an augmentation in the coefficient of temperature-dependent electrical conductivity.

Hybrid model for the optimal numerical solution of nonlinear ordinary differential systems

Hybrid model for the optimal numerical solution of nonlinear ordinary differential systems

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.

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.

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.

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.

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.

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.

Modeling of Darcy‐Forchheimer magnetohydrodynamic Williamson nanofluid flow towards nonlinear radiative stretching surface using artificial neural network

AbstractModern industries face a new challenge in cooling processes. Traditional cooling lubricants have limited heatconducting capacity. The development of nanofluids possessing superior properties such as high thermal conductivity, homogeneity, and long‐term stability has revolutionized the cooling lubrication industry. The literature reports a wide range of applications of nanofluid, such as cooling devices, peristaltic pumps for diabetic treatments, accelerators, reactors, petroleum industry applications, solar collectors and so forth. Nanofluids like Williamson nanofluid are very important non‐Newtonian fluids that have pseudoplastic properties. Williamson nanofluid has a number of applications in the medical and engineering sciences. It is used in food processing, inkjet printing, adhesives and emulsions, coated photographic films, and many other applications. In the current study, the nanomaterial flow of the Darcy‐Forchheimer Williamson nanofluid model is evaluated using the Levenberg–Marquardt approach with backpropagated neural networks. Thermalphoresis and Brownian motion are incorporated into the nanofluid model. This system is converted into an analogous nonlinear ordinary differential system through the application of necessary transformations. A dataset for the proposed multilayer perceptron artificial neural network is generated by altering the necessary variables through a Runge–Kutta fourth‐order shooting procedure. It has been created an artificial neural network called a multiple‐layer perceptron in order to forecast the values of the multiple‐layer perceptron. It is discovered that the parameter had the highest mean deviation of and the parameter had the lowest mean deviation of . Furthermore, MSE value of ANN model developed to estimate the skin friction coefficient value as and R value as 0.99954 whereas MSE and R values of the ANN model developed for the estimation of the LNN value were obtained as and 0.99999, respectively.

Comparative appraisal of mono and hybrid nanofluid flows comprising carbon nanotubes over a three-dimensional surface impacted by Cattaneo–Christov heat flux

Carbon nanotubes (CNTs) are nanoscale tubes made of carbon atoms with unique mechanical, electrical, and thermal properties. They have a variety of promising applications in electronics, energy storage, and composite materials and are found as single-wall carbon nanotubes (SWCNTs) and double-wall carbon nanotubes (DWCNTs). Considering such alluring attributes of nanotubes, the motive of the presented flow model is to compare the thermal performance of magnetohydrodynamic (MHD) mono (SWCNTs)/Ethylene glycol) and hybrid (DWCNTs- SWCNTs/Ethylene glycol) nanofluids over a bidirectional stretching surface. The thermal efficiency of the proposed model is gauged while considering the effects of Cattaneo-Christov heat flux with prescribed heat flux (PHF) and prescribed surface temperature (PST). The flow is assisted by the anisotropic slip at the boundary of the surface. The system of partial differential equations (PDEs) is converted into a nonlinear ordinary differential system by the use of similarity transformations and handled using the bvp4c numerical technique. To depict the relationship between the profiles and the parameters, graphs, and tables are illustrated. The significant outcome revealed that the fluid temperature rises in the scenario of both PST and PHF cases. In addition, the heat transfer efficiency of the hybrid nanoliquid is far ahead of the nanofluid flow. The truthfulness of the envisioned model in the limiting scenario is also given.

Thermodynamically-Consistent Modeling of Ferromagnetic Hysteresis.

Models of ferromagnetic hysteresis are established by following a thermodynamic approach. The class of constitutive properties is required to obey the second law, expressed by the Clausius-Duhem inequality, and the Euclidean invariance. While the second law states that the entropy production is non-negative for every admissible thermodynamic process, here the entropy production is viewed as a non-negative constitutive function. In a three-dimensional setting, the magnetic field and the magnetization are represented by invariant vectors. Next, hysteretic properties are shown to require that the entropy production is needed in an appropriate form merely to account for different behavior in the loading and the unloading portions of the loops. In the special case of a one-dimensional setting, a detailed model is determined for the magnetization function, in terms of a given susceptibility function. Starting from different initial magnetized states, hysteresis cycles are obtained by solving a nonlinear ordinary differential system. Cyclic processes with large and small amplitudes are established for materials such as soft iron.

Dual solutions on three-dimensional nanofluid flow and heat transfer over a permeable non-linearly shrinking surface with second-order velocity slips

PurposeThis paper aims to present the steady dual solutions on three-dimensional flow and heat transfer of nanofluid over a permeable non-linearly shrinking surface with two-order velocity slips conditions. Boundary layer assumption is considered in the mathematical modelling. Authors comprehend from previous studies and papers that the shrinking surfaces are extremely important in current engineering and environmental systems.Design/methodology/approachUsing appropriate similarity variables, the full partial differential equations (PDF) are modified into a specific set of ordinary (similar) differential equations (ODE). The resulting non-linear ordinary differential system is then solved both analytically for some particular cases and numerically for the general case using the function bvp4c from MATLAB for characteristic values of the parameters which govern the equations. The transformed mathematical model is analysed using the bvp4c procedure. Based on the given assumptions, this study is able to produce multiple solutions of the problem.FindingsThe ordinary (similarity) differential equations have two branches solutions, upper and lower branch solutions, given some interval of shrinking and velocity slip parameters. The authors consider here a temporal stability analysis, as they want to establish which of the solutions are stable and which are not. In a distinct paragraph, the authors discuss in detail and present in a graphical manner the effects of shrinking and second-order slip flow model on the skin friction coefficient, surface wall heat flux and dimensionless velocity and temperature profiles. The analysis reveals that the second order slip has a big influence on the flow and heat transfer characteristics.Originality/valueThe present discoveries are unique and truly new for the research of the three-dimensional stretching/shrinking forced convection flow and heat transfer nanofluids. The nanofluid is a water-based nanofluid (H2O), which contains one type of nanoparticles, namely, copper (Cu). Of course, the analysis can be further extended considering other types of nanoparticles such as alumina (Al2O3). The authors assume that the thermal equilibrium is reached for the base fluid together with the suspended nanoparticles and that the nanoparticles are uniform in dimension and form.

Numerical analysis of the high-order scheme of the damped nonlinear space fraction Schrödinger equation

Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.

CONSTRUCTION OF NEURAL NETWORK BASED INTELLIGENT COMPUTING FOR TREATMENT OF DARCY-FORCHHEIMER SISKO NANOFLUID FLOW WITH ROSSELAND'S RADIATIVE PROCESS

A generalization of Newtonian and power-law fluids is the Sisko model. It foretells dilatants and fluid pseudoplasticity. It was first suggested to use the Sisko fluid model to gauge high shear rates in lubricating greases. Three constants in this model are easily selectable for certain fluids, and it is demonstrated that the model is a good predictor of shear thickening and thinning. The study of nanofluids is gaining popularity quickly because of unique thermal, mechanical, and chemical characteristics of nanomaterials. Sisko nanofluids are also required for the production of nanoscale materials because of the superb wetting and dispersing capabilities they possess. In the present investigation, the Levenberg-Marquardt method with backpropagated neural networks is used to evaluate the nanomaterial flow of Darcy-Forchheimer Sisko fluid model. Thermophoresis and Brownian motion effects are considered when developing the nanofluid model. By applying the necessary transformations, the original nonlinear coupled partial differential system representing fluidic model are converted to an analogous nonlinear ordinary differential system. For different fluid model scenarios, a dataset for the proposed multilayer perceptron artificial neural network is produced by altering the necessary variables via the Galerkin weighted residual approach. An artificial neural network called a multilayer perceptron has been created in order to forecast the multilayer perceptron values.

Investigation of second grade viscoelastic non-Newtonian nanofluid flow on the curve stretching surface in presence of MHD

MHD extended flow of second-grade viscoelastic nanofluid is studied in two-dimensional mode on a curve stretching surface. Items like Joule heating and curvature parameter have been applied to inspect the heat transfer and mass transfer rates. To produce the nonlinear ordinary differential system, appropriate transformations are used. The quasi-linearization method is utilized to derive the solutions numerically. The impact of specific variables on the properties of fluid has been studied. Conclusions indicate that the increase in Schmidt number halted the fluid concentration, whereas temperature grew because of an increment in radiation parameter. Viscous fluid velocity and concentration are reduced faster in comparison to viscoelastic fluid. The surface drag force is an increasing function of the non-Newtonian fluid. When values of Nt and Nb are increased, the heat transfer rate of second-grade fluid increases compared to Newtonian fluid. Schmidt number and chemical reaction parameters highly affect the mass transfer of second-grade fluid compared to Newtonian fluid.

Non-linear radiative heat flux of Williamson nanofluid with gyrotactic microorganisms, activation energy and bioconvection

Gyrotactic microorganisms and nanoparticles with bioconvection play an important mantle in heat exchanger, biosensors, cooling devices, biotechnology, heating, pharmaceutical, drying food and many more. The main objective and novelty of this study is to compute the numerical solutions for the transient bioconvection flow of non-Newtonian Williamson nanofluid over a permeable contracting/expanding cylinder under the effects of Arrhenius activation energy and gyrotactic microorganisms. Additionally, binary chemical reactions and nonlinear radiation aspects are considered. Furthermore, Buongiorno nanofluid model and revised nanofluid model are utilized to investigate heat and mass transportation. Numerical solutions of highly nonlinear ordinary differential system are developed through RK45 scheme. It is interesting to investigate that motile number is an enhancing function of bioconvection Lewis number but reverse trend is noticed for higher estimation of Peclet number and microorganisms difference parameter in both cases of stretching and shrinking cylinder.