System Of Nonlinear ODEs Research Articles

Analytical Models of Experimental Artefacts in an Ill-Posed Nonlinear ODE System

We discuss different approaches for the analytical description of a mechanical system used in control theory, aiming at the analytical modelling of experimental artefacts observed in the implementation of ideal searched trajectories. Starting from an established analytical solution, we develop an alternative analytical model for this solution with minimal deviations and then extend this starting point to a more flexible toolbox that incorporates a variety of phenomena that typically occur in real implementations of this mechanical system, thus providing an important step towards bridging the gap between theoretical models and experimental reality.

Hunter Self-Similar Implosion Profiles for the Gravitational Euler–Poisson System

Our result is a construction of infinitely many radial self-similar implosion profiles for the gravitational Euler–Poisson system. The problem can be expressed as solving a system of non-autonomous non-linear ODEs. The first rigorous existence result for a non-trivial solution to these ODEs is due to Guo et al. (Commun Math Phys 386(3):1551–1601, 2021), in which they construct a solution found numerically by Larson (Mon Not R Astron Soc 145(3):271–295, 1969) and Penston (Mon Not R Astron Soc 144(4):425–448, 1969) independently. The solutions we construct belong to a different regime and correspond to a strict subset of the family of profiles discovered numerically by Hunter (Astrophys J 218:834, 1977). Our proof adapts a technique developed by Collot et al. (Mem Am Math Soc 260(1255):v+97, 2019), in which they study blowup for a family of energy-supercritical focusing semilinear heat equations. In our case, the quasilinearity presents complications, most severely near the sonic point where the system degenerates.

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

New insights of heat transfer in pistons and nozzles flow of graphene-transformer oil nanofluid: A differential transform method

This work aims to study heat transfer analysis in a squeezing flow problem of nanofluid. The flow between pistons and nozzles in an automobile engine is one industrial application where the knowledge of the squeezing flow plays a crucial role. The applied magnetic field has a longitudinal inclined angle that ranges from zero to ninety degrees. In addition, viscous dissipation, chemical reaction, and suction injection are taken into consideration. Graphene nanoparticles with transformer oil are considered for Tiwari-Das Model. The transformed system of nonlinear ODEs is solved using the differential transform method. The numerical findings of the skin friction coefficient, the Nusselt and the Sherwood numbers are assessed with previously available works for accuracy. The graphical findings for relevant parameters of all profiles are analyzed. It becomes apparent that the velocity and heat transfer in compressing flows are significantly influenced by the angle of the inclination of the magnetic field. The temperature of the nanofluid and the rate of the heat transfer are both improved in the existence of the viscous-dissipation and thermophoresis. The findings show that the temperature rises with increased Prandtl and Eckert numbers. Furthermore, when Brownian movements of nanoparticles take place, the rate of mass transfer decreases.

Comparative numerical study between MHD Forchheimer nano and hybrid nanofluid flows over stretching sheet under aligned magnetic field in the presence of radiation absorption

ABSTRACT This study investigates the mass and heat transfer properties of Ag-H2O nanofluid MHD Forchheimer flows and Ag-MoS2/H2O hybrid nanofluid over two-dimensional linear stretching surfaces in aligned magnetic fields with absorption of radiation. The fluid experiences rotational motion around the vertical axis at a consistent angular speed. The system of nonlinear ODEs is derived from the set of nonlinear PDEs by use of similarity transformations. The BVP-5C shooting approach in MATLAB is then employed to solve the associated system of ODEs. As the Forchheimer number, porosity parameter, rotation parameter and aligned magnetic field exhibit an ascent, the velocity profile along the x-axis and y-axis experiences a decrement. Simultaneously, an augmentation in the magnetic parameter, thermal radiation and rotation parameter induces a heightened temperature profile. Additionally, greater values of the Forchheimer number, rotation parameter and magnetic parameters correspond to an increase in concentration profile. Furthermore, the results indicated an increased temperature profile in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid for magnetic, rotation and radiation parameters. Also, the concentration profile of magnetic and rotation parameters along with the Forchheimer number were noted to be higher in the Ag-MoS2/H2O hybrid nanofluid compared to that in the Ag-H2O nanofluid.

Contribution of hall and ion slip effects with generalized mass and heat fluxes with entropy analysis on three-dimensional Prandtl model

A major challenging proffered study rely on the inquisitive optimization of entropy together with thermal as well as mass transportation towards the chemically reactive 3-D Prandtl liquid under consideration of applied magnetic field, variable thermal conductivity and also diffusivity, hall current together with ion slip features, viscous dissipation and heat generation. Moreover, in present research study, model such as Cattaneo Christov heat flux was implemented in order to explore the thermal relaxation characteristics. By employing correspondence transformations, the system of modeled equations modified into non-linear ODEs system. The Optimal Homotopy Analysis methodology (OHAM) was adopted to solve the proffered problem. The influential arising constraints demeanor within both velocities (horizontal as well as vertical), temperature and concentration profiles were discussed and also shown graphically. Moreover, outcomes of diverse parameters were also examined and presented graphically within the entropy formation rate and also Bejan number. The effects of implanted factors across the drag force, the rate of thermal as well as mass transportation are accessible in current study via tables and validate the obtained results. Both velocity profiles (horizontal as well as vertical) decreased for increment in Hartman number whilst opposite demeanor seen for other considered constraints. Temperature profile drops for Prandtl number and elastic constraints whereas enhanced for other influential considered parameters whilst concentration profile augmented with Prandtl number. Present problem novelty relies on the modeling of comprehensive system of equations which modified into nonlinear ODEs. In order to obtain the solution numerically, technique such as Optimal Homotopy was opted.

Impact of thermal energy efficiency based on kerosene oil movement through hybrid nano-particles across contracting/stretching needle

This manuscript reveals the mathematical analysis of dual simulations in tangent hyperbolic rheology with heat transfer and mass diffusion mechanisms on the needle (expanding and shrinking). Further, Darcy's Forchhiermer law is employed with the occurrence of the magnetic field. The models of hybrid nanofluid utilized are Xue- and YO-hybrid nano-fluid models. Such models apply to optical fibers, solar systems, surgical implants, electronics and biological applications. Transfer of thermal energy and diffusion related to mass species occurs employing a non-Fourier approach. The transformed system of non-linear Odes is numerically tackled with the finite element method. Furthermore, because fluid and needle movement have opposite directions, the current model has dual solutions. It visualizes that such a complicated model is not solved numerically by FEM. The applications of Xue- and YO-hybrid nano-fluid model with non-Fourier's law on a needle are implemented for the first time. It concludes that adding YO-hybrid nanofluid with base fluid increases the entropy profile, temperature profile and Bejan profile rather than the Xue hybrid nanofluid model. Fields of concentration, velocity and heat energy for the case of upper solutions are higher than concentration, velocity and heat energy profiles for lower solutions.

Hopf-bifurcation analysis of a stage-structured population model of cell differentiation

A cell differentiation model with maturity structure in progenitor cells is described by a system of nonlinear ODEs coupled to a PDE. The work mainly discusses Hopf-bifurcation for the model when the maturation rate of progenitor cells only depends on the number of mature cells, and is a continuation of the previous work (Doumic et al., 2011). Using the methods of characteristics and a change in variable, the system can be transformed into one of state-dependent delay differential equations and then into constant delay differential equations. A qualitative analysis of the solutions is performed, which includes well-posedness, non-negativity, uniform boundedness and linearized stability. Furthermore, we conduct Hopf-bifurcation analysis by taking the maturation level of progenitor as the bifurcation parameter, which means that the levels of stem and mature cells may vary with persistent oscillations. A numerical analysis stresses the role of some important parameters on the stability and Hopf-bifurcation.

Radial basis kernel harmony in neural networks for the analysis of MHD Williamson nanofluid flow with thermal radiation and chemical reaction: An evolutionary approach

The current investigative exploration exemplifies the conceptualization of a novel design intelligent computing paradigm based on artificial neural networks (ANNs) by utilizing radial basis function (RBF) to analyze magnetohydrodynamic (MHD) Williamson nanofluid two-dimensional flow along a stretchable sheet under the effect of chemical reaction as well as thermal radiation in a porous medium. This newly designed technique is an amalgam of a well-known reliable global solver named genetic algorithms (GAs) and a swift convergence generated local solver named sequential quadratic programming (SQP) used in ANNs by taking RBF as a kernel function i.e. ANNs-RBF-GASQP solver. The PDEs demonstrating the current nanofluid problem flow are transformed into the system of non-linear ODEs through a relevant similarity transformation and subsequently solved using ANNs-RBF-GASQP solver to investigate thermohydraulic properties by manipulating the values of various system parameters present in the ODEs. Moreover, the simulation results show that increasing the heat source parameter leads to a significant decrease in temperature. Additionally, an increase in the porosity parameter causes a decrease in the velocity of nanofluid, as a higher value of porosity increases fluid permeability and greater resistance to flow. The efficacy of the suggested solver is scrutinized through various statistical and convergence analyses.

Entropy generation on inclined magnetize double diffusive convective transportation of radiative Casson nanofluid in porous medium with source/sink

This research intends to investigate the entropy generation on the magnetized double diffusive heat and mass transfer flow of the Casson nanofluid under the influence of an inclined magnetic field in a porous medium. Additionally, the combined impact of heat absorption, chemical reaction, Brownian diffusion, source/sink, and thermophoresis phenomena is also taken care of. The fluid flow involves convective boundary conditions for both temperature and concentration instead of a constant value at the surface. The flow-regulating system involved nonlinear PDEs that are turned into nonlinear systems of ODEs by using scaling variables and then solved this system numerically in Matlab using the bvp4c strategy, which is a collocation technique based on the Lobatto 3-stage FDM algorithm. Graphical representations illustrate the behavior of fluid velocity, entropy generation, concentration, and temperature in response to changes in flow parameters. Physical quantities like skin friction coefficient, Nusselt number, and Sherwood number have been investigated using 2D and 3D plots. Here, we concluded that the inclined magnetic field decimates the flow velocity gradually and greater values of the magnetic field lead to an increased rate of entropy generation. Furthermore, it has been noted that the temperature profile improves as the Brownian motion of particles increases, and the distribution of energy also enhances with larger values of the thermophoresis. The obtained key findings are discussed in a physical manner using graphic representation.

FRACTAL-FRACTIONAL SIRS MODEL FOR THE DISEASE DYNAMICS IN BOTH PREY AND PREDATOR WITH SINGULAR AND NONSINGULAR KERNELS

In this study, we consider a mathematical model for the disease dynamics in both prey and predator by considering the Susceptible–Infected–Recovered–Susceptible (SIRS) model with the prey–predator Lotka–Volterra differential equations. Carrying capacity, predation, migration, and immunity loss are also taken into account for both species. Using the law of mass action, the physical model is transformed into a nonlinear coupled system of ODEs. The classical/integer order ODE system is then generalized through the fractal-fractional differential operators of power law and Mittag-Leffler kernels. For the model under consideration, we additionally check for positivity, boundedness, the basic reproduction number, and equilibrium points. Graphical results are obtained by use of a numerical approach, and the existence and uniqueness of the solution are also established theoretically. The graphical solutions has been displayed via 2D and 3D phase plots. It has been shown that when the fractional order reduces, the amplitude of the chaotic attractor dynamics shrinks, along with the range of limit cycles and periodic trajectories while a drop in the fractal dimension parameter causes an increase in the time period of the chaotic attractor dynamics. This study not only improves the understanding of epidemic breakouts in predator–prey systems, but also highlights the efficiency of fractal-fractional calculus in ecological modeling.

Numerical simulation of time-dependent viscous fluid flow with upward and downward fluctuation of spinning disk

The parametric approach towards time-dependent viscous fluid flow across a gyrating disk with upward and downward fluctuation. The major goal of this research is to assess fluid flow under the influence of magnetic fields and heat propagation processes. Because they provide a thorough description of electromagnetic interactions. Maxwell's equations are at the heart of all contemporary information and communication technologies. The governing equations comprising Navier Stokes equation, energy, concentration, and Maxwell equations have been represented appropriately for this purpose. The governing equations are turned down to the system of non-linear ODEs through a resemblance framework. The obtained system of differential equations has been resolved via numerical procedure Parametric Continuation Method (PCM). For the scale reliability purpose, the outcomes are compared to another numerical Matlab scheme boundary value solver. In the current analysis, the presence of convective boundary conditions correlated to mass and energy is of physical relevance. The numerical findings are provided in tabular and graphical forms. The consequences of suction and wall injection have been also highlighted. The upward motion of the spinning disc is thought to lead to comparable findings as in an injection scenario, whilst the downhill motion is thought to contribute to wall suction-like effects.

Entropy generation analysis of microrotating Casson’s nanofluid with Darcy–Forchheimer porous media using a neural computing based on Levenberg–Marquardt algorithm

PurposeThe purpose of this paper is to showcase the utilization of the magnetohydrodynamics-microrotating Casson’s nanofluid flow model (MHD-MRCNFM) in examining the impact of an inclined magnetic field within a porous medium on a nonlinear stretching plate. This investigation is conducted by using neural networking techniques, specifically using neural networks-backpropagated with the Levenberg–Marquardt scheme (NN-BLMS).Design/methodology/approachThe initial nonlinear coupled PDEs system that represented the MRCNFM is transformed into an analogous nonlinear ODEs system by the adoption of similarity variables. The reference data set is created by varying important MHD-MRCNFM parameters using the renowned Lobatto IIIA solver. The numerical reference data are used in validation, testing and training sets to locate and analyze the estimated outcome of the created NN-LMA and its comparison with the corresponding reference solution. With mean squared error curves, error histogram analysis and a regression index, better performance is consistently demonstrated. Mu is a controller that controls the complete training process, and the NN-BLMS mainly concentrates on the higher precision of nonlinear systems.FindingsThe peculiar behavior of the appropriate physical parameters on nondimensional shapes is demonstrated and explored via sketches and tables. For escalating amounts of inclination angle and Brinkman number, a viable entropy profile is accomplished. The angular velocity curve grows as the rotation viscosity and surface condition factors rise. The dominance of friction-induced irreversibility is observed in the vicinity of the sheet, whereas in the farthest region, the situation is reversed with heat transfer playing a more significant role in causing irreversibilities.Originality/valueTo improve the efficiency of any thermodynamic system, it is essential to identify and track the sources of irreversible heat losses. Therefore, the authors analyze both flow phenomena and heat transport, with a particular focus on evaluating the generation of entropy within the system.

Legendre wavelet integration operational matrix method for the numerical solution of unsteady MHD nanofluid flow between moving parallel plates

This study proposes a novel method based on Legendre wavelet integration operational matrix for solving a system of nonlinear ODE’s to model the two-phase, unsteady nanofluid flow and heat transfer between parallel plates in presence of magnetic field. To validate the applicability of proposed method, system of nonlinear BVP with exact solutions is employed as a test problem. Test problem results are compared with exact solution, and other numerical methods. The governing equations were then changed using similarity transformation into a set of nonlinear ODE’s. . Following this, the LWOMM is applied for the proposed problem. Findings demonstrated that, for a constant Reynolds number, skin friction coefficient value increases with rising Hartmann and squeezing numbers. Furthermore, it is demonstrated that Eckert number is a reducing function of the squeeze number and Nusselt number is an increasing function of Hartmann number.

Dynamics of magnetized viscous dissipative material of hybrid nanofluid with irregular thermal generation/absorption

The intriguing thermophysical properties of nanoparticles have considerable potential in thermal transportation for engineering applications. The inclusion of solid nanoparticles in the base fluid raises the fluid's thermal conductivity, which improves thermal transfer characteristics. Thus, an analysis is performed by assuming the Tiwari and Das model, for the flow and energy transportation aspects of hybrid nanoparticles in a shrinkable porous medium. The current study focuses on the thermal heat transfer performance on the dynamic of ethylene glycol-based titania and alumina nanoparticles. The flow-controlling system of PDEs is transformed into a system of non-linear ODEs by using similarity transformations, and the resulting equations are then solved numerically through the solver bvp4c in Matlab function with a limit of tolerance 10−6. The multiple branches of solutions are noted for distinct values of suction and shrinking parameters. The outcomes reveal that the inclusion of titania and alumina nanocomposites improves the thermophysical properties of the base fluid significantly. The findings indicate that hybrid nanofluids are more effective for increasing the thermal transport rate. Further, in the upper branch solution, the friction drags, and energy transportation rate raises for the higher influence of suction strength while it declines for the lower branch.

Radiation-influenced magnetohydrodynamic third-grade nanofluid flow around non-linearly stretched cylinder

Abstract This analysis considers the magnetized third-grade fluid stream and microorganisms over a non-linear stretchy cylinder. The radiation impacts are taken into consideration. The effects of the governing flow at the cylinder are represented in the form of PDEs employing boundary layer approximations. The system of the PDEs is further reduced in dimensionless form after applying the similarity transformations. The dimensionless system of non-linear ODEs is solved through the numerical technique bvp4c. The effects of radiation and magnetism on the third-grade liquid over a non-linear extending cylinder are highlighted in graphs and numerically in tabular form. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade coefficients, and Reynolds number, is illustrated and explored. Suitable ranges for the parameters $( {1 \le \eta \le 10,\ 0.2 \le {{\alpha }_1} \le 0.5,\ 0 \le {{\alpha }_1} \le 1.5,\ 0.1 \le \beta \le 0.3,\ 0.1 \le \gamma \le 1.6,\ 0.05 \le M \le 0.15,\ 0.5 \le \delta \le 2.0,\ 0.7 \le Pr \le 1.3,\ 0.1 \le Rd \le 0.4,0.1 \le}$ ${e \le 0.4} )$ are chosen depending upon the convergence of the numerical method. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. The temperature curve declines when boosting third-grade parameters, thermal stratification, and Hartmann number while boosting up for curvature and radiation parameters.

DJM solution of MHD flow of ternary hybrid nanofluid between nonparallel a porous media channels with velocity slip and radiation effects

This investigation on heat transfer and Jeffery Hamel ternary hybrid nanofluid flow across nonparallel a porous media channels while accounting for radiation, velocity slip, and mixture fluid ( H 2 O − C 2 H 6 O 2 ) effects is presented in this work. The nonlinear PDEs such as continuity, momentum and heat equations transformed to a system of nonlinear ODEs using similarity transformations then solved numerically and analytically, the analytical solution has been constructed using by Daftar Dar–Jafari method (DJM), the present results in particular cases are compared to results obtained by the HAM-based Mathematica package and by the Runge–Kutta Fehlberg fourth–fifth order (RKF-45) for validation. The effects of active parameters such as the nanofluid volume fraction, velocity slip parameter, Darcy number (Da), and thermal radiation parameter (Rd), are investigated on the velocity and temperature, skin friction, and Nusselt numbers. It is found that the velocity of ternary hybrid nanofluid upsurges for both convergent and divergent channel with the increment in the Hartman number (Ha). Also, results obtained reveal that the temperature profile experiences an upward shift with an increase in the Ha, and a downward shift with an increment of the Da. Furthermore, new modeling and the reliable analytic treatment via DJM approach for the ternary hybrid nanofluid is a sizable accomplishment of the current analysis.

ReMKiT1D - A framework for building reactive multi-fluid models of the tokamak scrape-off layer with coupled electron kinetics in 1D

In this manuscript we present the recently developed flexible framework for building both fluid and electron kinetic models of the tokamak Scrape-Off Layer in 1D - ReMKiT1D (Reactive Multi-fluid and Kinetic Transport in 1D). The framework can handle systems of non-linear ODEs, various 1D PDEs arising in fluid modelling, as well as PDEs arising from the treatment of the electron kinetic equation. As such, the framework allows for flexibility in fluid models of the Scrape-Off Layer while allowing the easy addition of kinetic electron effects. We focus on presenting both the high-level design decisions that allow for model flexibility, as well as the most important implementation aspects. A significant number of verification and performance tests are presented, as well as a step-by-step walkthrough of a simple example for setting up models using the Python interface. Program summaryProgram title: ReMKiT1DCPC Library link to program files:https://doi.org/10.17632/j47ym66xzj.1Developer's repository link:https://github.com/ukaea/ReMKiT1D and https://github.com/ukaea/ReMKiT1D-PythonLicensing provisions: GPLv3Programming language: Fortran, PythonSupplementary material:https://doi.org/10.14468/fdq7-z869Nature of problem: The flexible generation and modification of 1D models pertaining to multi-fluid simulations of the tokamak Scrape-Off Layer (SOL) with electron kinetics and reaction support. This would then allow both for rapid iteration on reduced models as well as the evaluation of kinetic electron effects in equilibria and during transients, following the formalism previously developed for SOL-KiT [1]. The framework was not only envisioned as the successor to SOL-KiT, but a tool that would allow users to construct their own models coupled with electron kinetics capabilities.Solution method: The framework is written heavily utilizing Object-Oriented design principles, in particular using an extended version of the puppeteer pattern as presented by Rouson et al. [2], as well as the heavy use of the strategy pattern/dependency injection. The Fortran code is MPI parallel and utilizes the PETSc library for implicit time-stepping. MPI parallelization is extended to distribution function Legendre harmonics, allowing for improved strong scaling. Initialization of the Fortran framework is done using JSON configuration files generated by an accompanying Python interface, and data analysis is standardized using widely used data formats such as HDF5.Additional comments including restrictions and unusual features: The present manuscript focuses on the design and high-level implementation of the framework, as well as the demonstration of the workflow and various verification and performance benchmarking tests. Some details are avoided for the sake of brevity at various points, and these are meant to be available as part of the general code documentation or tutorials offered on the main repositories.

Quasilinearization variational iteration method for system of nonlinear ODEs

In this manuscript, we discuss a new technique for solving system of nonlinear differential equations, which is a modification of the variation iteration method (VIM) implemented using the quasilinearization method and Adomian’s polynomial. The quasilinearization variational iteration method (QVIM) is the name given to this proposed method. The proposed method’s convergence analysis in Banach space is also discussed here. Three application problems, including the Genesio-Tesi system, are considered to test the applicability of our approach. We also discuss the case study of the chaotic and non-chaotic solutions of the Genesio-Tesi system (GTS). The convergence behaviour of the method is studied for various values of parameter x. To assess the viability and efficacy of QVIM, we compare it to the existing well-known Adomian decomposition method. The results show that the proposed method is highly efficient and simple to implement.

Magnetized heat transfer visualization through computational modeling of third-grade fluid via exponentially stretching cylinder

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do.