System Of Nonlinear Ordinary Differential Equations 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.

A Bayesian Collocation Integral Method for Parameter Estimation in Ordinary Differential Equations

Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through simulation studies, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided. Supplementary materials for this article are available online.

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.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

Computational analysis to examine the role of nanoparticle shape on operative usage of solar energy

In thermal power and manufacturing processes, solar energy utilization has grown extensively. This work investigates the heat diffusion characteristics of an unsteady Oldroyd-B hybrid nano liquid flow across an expanding cylinder. A constant magnetic field is applied in the radial path in the presence of radiant heating, generation, or sinks, and Cattaneo-Christov (CC) heat flow. Hybrid nano liquid composed of two distinct nanoparticles (CoFe2O4 and Fe3O4) embedded in ethylene glycol as the primary fluid. A mathematical model was established and reconstructed into a nonlinear ordinary differential equations (ODEs) system, utilizing the proper likeness conversion. The Matlab inbuilt bvp5c scheme determines the problem's numerical solution. We observed the instabilities in the flow and thermal area and also the local Nusselt number impacted by the pertinent dimensionless characteristics. The findings are addressed graphically and tabularly for several nanoparticle shapes (spherical, brick, and platelet). It has been discovered that spherical shapes are found to have a faster heat conduction rate than brick and platelet nanoparticle shapes for the considered flow case. While unsteadiness parameters tend to increase the temperature field, the augmentation of the temperature relaxation factor reduces the energy profile.

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.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.

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.

Comparison of damping techniques in non-linear spring-mass systems for power transformers under short-circuit condition

PurposeThe simplified modeling of many physical processes results in a second-order ordinary differential equation (ODE) system. Often the damping of these resonating systems cannot be defined in the same simplified way as the other parameters due to the complexity of the physical effects. The purpose of this paper is to develop a mathematically stable approach for damping resonances in nonlinear ODE systems.Design/methodology/approachModifying the original ODE using the eigenvalues and eigenvectors of a linearized state leads to satisfying results.FindingsAn iterative approach is presented, how to modify the original ODE, to achieve a well-damped solution.Practical implicationsThe method can be applied for every physical resonating system, where the model complexity prevents the determination of the damping.Originality/valueThe iterative algorithm to modify the original ODE is novel. It can be used on different fields of the physics, where a second-order ODE is describing the problem, which has only measured or empirical damping.

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.

Numerical investigation of magnetized nanofluid flow with thermal radiation and homogeneous/heterogeneous reactions over a vertical cylinder

Nanofluid flow has gained attention due to its promising applications in numerous industries. Nanofluids exhibit a significant advantage over conventional fluids due to their superior heat transfer capability, attributed to the presence of nanoparticles that enhance thermal conductivity, resulting in improved heat dissipation and efficiency. Current research aims to numerically investigate the impact of homogeneity and heterogeneity on the mixed convection flow of incompressible viscous nanofluids through a vertically permeable cylinder under suction/injection circumstances. The study also considers thermal radiation and heat source/sink phenomena. Ethylene glycol is utilized as base fluid, while nanoparticles include silicon carbide SiC and Titanium dioxide TiO2. The mathematical flow model, based on nonlinear partial differential equations (PDEs), is converted into ordinary differential equations (ODEs) by using suitable similarity transformations that pronounced nonlinear system of ODEs. To deal with this nonlinear system, the bvp4c and shooting methods are used in the commercial software MATLAB to solve these ODEs numerically. The impacts of flow parameters on various quantities of interest are elaborated graphically. From obtained results it is analyzed that velocity field boosts up versus higher values of nanoaprticle volume fraction. The velocity field is decreases with increasing the amount of magnetic field. An increase in the thermal field is observe with a rise in the thermal radiation parameter. The good agreement between current results and published work is noted.

Stability analysis of stratified radiative non-Newtonian Casson nanofluid flow past on stretching/shrinking sheet using two-phase model

An incompressible steady-state 2D (two-dimensional) stratified boundary-driven Casson nanofluid flowing over the stretched/shrinked surface under buoyancy, magnetic and radiative effects is considered. The two-phase Buongiorno model is adopted to express the behavior of nanomaterial with the stratified boundary conditions. A system of nonlinear ODEs (ordinary differential equations) is extracted from the modeled PDEs (partial differential equations) by using the similarity transformations. The shooting scheme is adapted to execute the obtained equations numerically. The achieved results indicate the duality in the solutions at various ranges of the specified constraints like as suction, shrinking or stretching. The stability analysis is done to check the feasibility of solutions by BVP4c in Matlab. It is achieved that an increment in suction constraint enhanced the skin friction for shrinking case. The temperature curves are decreased with an improvement in Casson, suction, buoyancy and stratification parameters. The nanoparticles concentration reduced against the higher Casson, Brownian motion, suction, thermal and solutal stratification parameters. On other hand, an increment in the thermophoresis increases nanoparticles concentration profile.

Thermal analysis of Non-Newtonian visco-inelastic fluid MHD flow between rotating disks

Analytical techniques are used in this article to examine the forced flow and heat transfer in a non-Newtonian Reiner-Rivlin fluid between two infinitely revolving disks in a magnetic field. The partial differential equations (PDEs) emphasizing flow and heat transfer were transformed into a nonlinear ordinary differential equations (ODEs) system using the Von Karman similarity transformation. The system of nonlinear ODEs with boundary conditions was solved using the Homotopy Perturbation Method (HPM) and Akbari-Ganji's Method (AGM). The study assessed the effects of physical factors on temperature profiles and the radial, transverse, and axial velocity components, including the visco-inelastic parameter, Reynolds number, Prandtl number, forced parameter, and magnetic field. The Python programming language was used for symbolic and numeric calculations of velocity components and temperature profiles. The accuracy of the results obtained through the analytical approach validates the applicability and efficiency of the proposed approach for studying the forced flow and heat transfer in non-Newtonian fluids between rotating disks.

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.

Numerical computation of mixed convective entropy optimized in Darcy-Forchheimer flow of Cross nanofluids through a vertical flat plate with irregular heat source/sink

This article seeks to provide multiple solutions for the buoyancy effects and heat transfer flow of non-Newtonian Cross nanofluids past a vertical flat plate in a Darcy-Forchheimer porous medium. The experimental properties of nanofluid are considered in this study. In addition, the entropy generation along with irregular heat source/sink is also incorporated. The complete partial differential equations (PDEs) are converted into a particular set of ordinary differential equations (ODE) employing the proper similarity variables. The consequent nonlinear ODE system is then resolved using the MATLAB function bvp4c for distinctive values of pertaining parameters. The profiles of entropy generation for different physical influential parameters are plotted. The results indicate that the boundary layer separation accelerates due to the superior consequence of the permeability parameter. The shear stress declines due to the Weissenberg number and permeability parameter for both solutions, while the rate of heat transfer uplifts owing to the larger influence of the Weissenberg number and permeability 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.

The flow analysis of Williamson nanofluid considering the Thompson and Troian slip conditions at the boundary

In this article, the impact of Joule heating on the thermal performance of Williamson nanofluid is analyzed under the influence of viscous dissipation along with the Joule heating. Also, the flow is subjected to Thompson and Troian slip conditions that directly influence the velocity of the flow at the boundary. Meanwhile, to achieve the even distribution of nanoparticles in the nanofluid, gyrotactic microorganisms are dispersed whose motion is due to the virtue of density gradient. The heat conduction at the surface is governed by the convective condition which allows the interpretation of the Biot number. The mathematical model is constructed employing these presumptions using partial differential equations, which are then subsequently reduced using similarity transformations to get the ordinary differential equations (ODEs). The RKF-45 numerical approach is used to solve the nonlinear ODE system thus acquired, and the findings are validated by comparing them to the previously published works. The results of this study showed that the higher values of thermophoresis and Brownian motion parameters cause more heat conduction. Also, the rise in the Eckert number that relates to the internal friction enhances the temperature conducted by the nanofluid. Meanwhile, the Lorentz force helps in controlling the flow velocity.

MHD mixed convection flow for Maxwell Hybrid nanofluid with Soret, Dufour and Morphology effects

We have investigated the two-dimensional mixed convective Maxwell hybrid nanofluid boundary layer mass and heat flows over a linearly stretching porous surface with the applied external magnetic flux. Thermal radiations along with the Dufour and Soret effects are also incorporated. The governing model of partial differential equations (PDE) is altered into ordinary differential equations (ODE) with an appropriate similarity transformation. The finite difference-based numerical method BVP4c is applied to solve the system of nonlinear ODEs. The flow features and the heat transfer characteristics have been illustrated with graphical representations and a numerical table. For varied values of the flow-related variables, organized and graphical data for the Nusselt number and skin friction coefficient are indicated. In most cases, spherical-shaped nanoparticles have a better influence on stream function, velocity and temperature distributions. This behavior is the opposite of the mass concentration profile. It has been observed that stream function decreases as increase the value of the magnetic field but opposite for mass concentration distribution and temperature profile. The temperature gradient is enhanced as a result of stronger convective flow when Soret number Sr values increase, which causes the boundary layer thickness to grow. A comparative study of hybrid nanofluid and nanofluid showed that the hybrid nanofluid has superior shear stress/skin friction and Sherwood number/surface mass flux than nanofluid flow.

Unsteady hybrid nanofluid (UO_2, MWCNTs/blood) flow between two rotating stretchable disks with chemical reaction and activation energy under the influence of convective boundaries

Hybrid nanofluids are extensively analyzed in recent studies due to their better performance in numerous areas such as heat and mass transfer enhancement, biological fluid movement, medical equipment, heat exchangers, electronic cooling and automotive industry. In current study the nanoparticle concentration utilized is much important in biomedical industry. Major applications include drug delivery, radio-pharmaceuticals, centrifuging blood to obtain red blood cells and plasma, medical implants, onco therapeutics and photo thermal cancer therapy. In this regard, the primary focus of this study is to simulate a blood based unsteady hybrid nanofluid flow between two rotating, stretching disks and convective boundaries. The two nanoparticles in this study are uranium dioxide UO_{2} and multi-walled carbon nanotubes MWCNTs. The hybrid nanofluid is under the influence of magnetohydrodynamic effects and chemical reaction with activation energy. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using suitable similarity transform. Homotopy analysis method is used to solve the non-linear system of ODEs and hbar -curves are plotted to find suitable region of hbar _{i} for convergent series solution. Velocity profile is examined for axial, radial and tangential direction against various fluid parameters. Temperature and concentration profiles are analyzed for both convective and non-convective cases. It is observed that convective boundaries result in elevated temperature when compared with non-convective case. Moreover, skin friction, heat and mass transfer rates are also examined with respect to changing volume fraction varphi _{UO_{2}}.The results revealed that skin friction and rate of heat transfer increases with increase in volume fraction of both nanoparticles UO_{2} and MWCNTs while the mass transfer rate depicts contrasting behavior.