System Of Ordinary Differential Equations Research Articles

A hybrid CPU‐GPU paradigm to accelerate reactive CFD simulations

AbstractThe solution of reactive computational fluid dynamics (CFD) simulations is accelerated by the implementation of a hybrid central processing unit/graphics processing units (CPU/GPU) Finite Volume solver based on the operator‐splitting strategy, where the chemistry integration is treated independently of the flow solution. The integration of ordinary differential equations (ODEs) describing the finite‐rate chemical kinetics is solved by an adaptive multi‐block explicit solver on GPUs, while the load of the fluid solution is distributed on a multicore CPU algorithm. The resulting speed‐up for reactive CFD simulations is up to 10; the performance gain increases with the size of the mechanism. The proposed implementation is general and can be applied to any CFD problem where the governing equations for the fluid transport are coupled with an ODE system. Code validation is performed against reference solutions on a selection of test cases involving reacting flows.

Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

A toxicant-taxis reaction–diffusion model for the interaction between two species and a toxicant in an aquatic ecosystem

In this paper, we develop a toxicant-taxis reaction–diffusion model that describes the interaction dynamics between two species and a toxicant in a polluted aquatic ecosystem. We analyze the existence and stability of the equilibria of the corresponding space-absent ordinary differential equation system. We then perform linear stability analysis for our reaction–diffusion system to obtain sufficient conditions on model parameters under which the stable spatially homogeneous steady states become unstable. Numerical simulations are performed to illustrate that spatially inhomogeneous patterns and spatiotemporal patterns between the two species and the toxicant will typically emerge. We also numerically analyze the effects of spatially varying toxicant input and the intensity of toxicant-taxis on the population persistence and species diversity. It turns out that heterogeneous toxicant inputs and substantial toxicant taxis can boost the coexistence of two species, thereby promoting species diversity in ecosystems. By comparing the impact of a toxicant on a single species versus its impact on two species, we find that the collective sharing of the toxic burden among multiple species proves advantageous for biodiversity.

Numerical Scheme by Legendre-Galerkin for Characterization of Micropolar-Maxwell Unsteady Fluid Model

In this dissertation, Micropolar-Maxwell unsteady fluid flow prior an expanding sheet is asymptotically investigated using an efficient Legendre-Galerkin technique in the existence of magnetic field. By employing fitting similarity mapping to the controlling partial differential equations, a system of nonlinear ordinary differential equations is obtained. In comparison to the prior studies, the suggested asymptotic outcomes are guaranteed. The impacts of elasticity/material parameters on velocity/micro-rotation profiles are shown and explored with the help of tables and figures to enhance the mechanical properties of the sheet.

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

НЕЙРОННА МЕРЕЖА ТИПУ АВТОКОДУВАЛЬНИК ДЛЯ ВКЛАДЕННЯ ОДНОВИМІРНИХ ЧАСОВИХ РЯДІВ

The problem of time series embedding is a universal one. It is the main prerequisite when it comes to modeling of dynamical processes using systems of autonomous ordinary differential equations (ODEs) because they have hard requirements for the dimensionality of the problem. One-dimensional ODE can only exhibit 3 types of behavior while two-dimensional ODE can exhibit 9. This is why it is important to increase the dimensionality of the problem before starting the modeling to allow for wider range of possible behaviors in the final model. One way to increase the dimensionality is to delay-embed the time series data but this approach can be extended to allow the use of an autoencoder neural network that would associate a higher-dimensional vector to each point in the time series and will allow the modeling to be performed in higher dimension.

Uniform propagation of chaos for a dollar exchange econophysics model

Abstract We study the poor-biased model for money exchange introduced in Cao & Motsch ((2023) Kinet. Relat. Models 16(5), 764–794.): agents are being randomly picked at a rate proportional to their current wealth, and then the selected agent gives a dollar to another agent picked uniformly at random. Simulations of a stochastic system of finitely many agents as well as a rigorous analysis carried out in Cao & Motsch ((2023) Kinet. Relat. Models 16(5), 764–794.), Lanchier ((2017) J. Stat. Phys. 167(1), 160–172.) suggest that, when both the number of agents and time become large enough, the distribution of money among the agents converges to a Poisson distribution. In this manuscript, we establish a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which justifies the validity of the mean-field deterministic infinite system of ordinary differential equations as an approximation of the underlying stochastic agent-based dynamics.

Kinetic modelling of anthocyanins and vitamin C degradation in a maqui-citrus beverage during storage for different sweeteners and pasteurization treatments

Kinetic models to describe the degradation of anthocyanins (first-order model) and vitamin C (system of ordinary differential equations) in a novel beverage, has been used as a tool to identify the most relevant variables for the degradation kinetics and to support shelf-life estimation. The models were based on an empirical dataset, accounting for the effect of a prior processing treatment (HHP at 400 MPa- 180s; HHP at 600 MPa-180s; thermal treatment at 85 °C for 15s), storage temperature (4 or 20 °C), and the use of two different sweeteners (sucralose or stevia). Using model selection (based on AIC), we conclude that the degradation kinetics depends on a complex combination of process and product parameters. A strong temperature dependence on anthocyanins and vitamin C was observed during pasteurization. Products treated by HHP resulted in higher vitamin C concentrations during storage than those thermally treated. Finally, an interaction with the sweetener resulted in a slight stabilization of bioactive compounds. In conclusion, this study demonstrates the complexity of the degradation process of bioactive compounds in these beverages, advocating that nonlinear kinetic models serve as a tool to identify the most relevant parameters, ultimately supporting product formulation and shelf life estimation.

Effects of particle creation rate in two-fluid interacting cosmologies

Abstract In this work, a two-fluid interacting model in a flat FLRW universe has been studied considering particle creation mechanism with a particular form of particle creation rate $\Gamma =\Gamma _0 H+\frac{\Gamma _1}{H}$ from different aspects. Statistical analysis with a combined data set of SNe Ia (Supernovae Type Ia) and Hubble data is performed to achieve the best-fit values of the model parameters, and the model is compatible with current observational data. We also perform a dynamical analysis of this model to get an overall qualitative description of the cosmological evolution by converting the governing equations into a system of ordinary differential equations considering a proper transformation of variables. We find some non-isolated sets of critical points, among which some usually are normally hyperbolic sets of points that describe the present acceleration of the universe dominated by dark energy mimicking cosmological constant or phantom fluid. Scaling solutions are also obtained from this analysis, and they can alleviate the coincidence problem successfully. Statefinder diagnosis is also carried out for this model to compare it with the ΛCDM, and any other dark energy models byfinding various statefinder parameters. Finally, the thermodynamic analysis shows that the generalized second law of thermodynamics is valid in an irreversible thermodynamic context.

Double shooting method for FRCM reinforced systems in debonding problems

The debonding process in FRCM (Fiber Reinforced Cementitious Matrix) strengthened brittle substrates, such as concrete and masonry, is characterized by complex failure modes that include debonding at both the external and internal interfaces of the composite, as well as damage to the mortar matrix. This study addresses this problem classically as a mode-II fracture process by accounting for the one-dimensional non-linear interfacial shear stress-slip relationship combined with matrix axial non-linearity. The approach employed considers the longitudinal displacements of both the inner and outer layers of mortar, along with that of the fiber textile, as unknowns, and analytically solves the second-order ordinary differential equation (ODE) system that governs the debonding problem. Then, the boundary value problem (BVP) is transformed into an initial value problem (IVP) and a double shooting method is proposed to find displacements and stresses along the bonded length in the presence of non-linearity. The appropriate initial values of matrix layer displacements, satisfying the required boundary conditions, are efficiently determined through a two-dimensional bisection method working on rectangular and triangular patches. The softening behavior of both mortar layers and interfaces between mortar and fiber are taken into account using jagged constitutive relationships to enhance convergence. The numerical calculation speed, robustness, and key simulation parameters are thoroughly discussed. Moreover, the proposed numerical approach is compared with existing experimental data and models, on both concrete and masonry FRCM-reinforced specimens. The influence of mortar failure and interfacial strength is also explored. The model demonstrates its ability to effectively reproduce the global bond behavior observed in experimental studies, while capturing the local behavior and simulating various failure modes observed in the tests.

Does mutual interference stabilize prey–predator model with Bazykin–Crowley–Martin trophic function?

We investigated a system of ordinary differential equations that describes the dynamics of prey and predator populations, taking into account the Allee effect affecting the reproduction of the predator population, and mutual interference amongst predators, which is modeled with the Bazykin–Crowley–Martin (BCM) trophic function. Bifurcation analysis revealed a rich spectrum of bifurcations occurring in the system. In particular, analytical conditions for the saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations were derived for the model parameters, quantifying the strength of the predator interference, the Allee effect, and the predation efficiency. Numerical simulations verify and illustrate the analytical findings. The main purpose of the study was to test whether the mutual interference in the model with BCM trophic function provides a stabilizing or destabilizing effect on the system dynamics. The obtained results suggest that the model demonstrates qualitatively the same pattern concerning varying the interference strength as other predator-dependent models: both low and very high interference levels increase the risk of predator extinction, while moderate interference has a favorable effect on the stability and resilience of the prey–predator system.

Impact of magnetic induction on the flow of self-rewetting power-law fluid over a disk surface: Onset of Marangoni convection

In the present investigation, the onset of Marangoni convection in the flow of self-rewetting fluid over a stretching disk has been discussed. The self-rewetting fluid is considered to be electrically conducting and the flow problem is discussed under the influence of an induced magnetic field. The fluid flow is mainly induced due to two different mechanisms: (i) onset of Marangoni convection due to presence of temperature gradient along the liquid-gas interface and (ii) induction of magnetic field due to presence of an external magnetic field, which modifies the original field. The governing partial differential equations are written, following Navier-stokes equations and Prandtl boundary layer equation. With the use of similarity transformations, the governing equations are turned into a system of ordinary differential equations. Nonlinear systems of ordinary differential equations are solved by the bvp4c technique. The reciprocal of the magnetic Prandtl number diminishes the induced magnetic profile.

Numerical scrutinization of heat transfer subject to physical quantities through bioconvective nanofluid flow via stretching permeable surfaces

Background: The mechanics of heat and mass transfer via nanofluid flow across many media are currently being discussed. “Nanofluids” are fluids that include highly heat-conductive nanoparticles, and they are essential for resolving engineering problems. Under the effects of activation energy, thermal radiation, and motile microorganisms, the process of heat and mass transfer through steady nanofluid flow crosses over stretched surfaces in this scenario.Methodology: For mathematical evaluation, the system of partial differential equations (PDEs) is used to describe this physical framework. By introducing suitable similarity variables with a set of boundary conditions, this mathematical system of PDEs has become a system of ordinary differential equations (ODEs). To obtain numerical results, the MATLAB built-in program “bvp4c” is used to solve the system of first-order equations.Results: In the findings and discussion section, the resulting outcomes are thoroughly examined and visually shown. The flow rate in these systems increases due to the erratic movement of microorganisms. The graphical representation shows the impacts of involving physical factors on the microorganism, thermal, concentration, and momentum profiles. Variations/changes in these profiles can be observed by adjusting the parametric values, as depicted in the graphs. Consequently, thermal transport is boosted by 25%. Additionally, the skin friction, Nusselt, Sherwood, and microbe density numbers are determined numerically. The findings demonstrate that increasing the magnetic field parameter causes the velocity profile to decrease, increasing the radiation parameter leads to an increase in temperature description, and increasing the Lewis number causes the microorganism profile’s transport rate to decrease.

Thermal inspection of hybrid nanofluid flows over a stretched cylinder at an oblique stagnation point with variable characteristics

AbstractThe enhanced thermal characteristics of hybrid nanofluids make them more versatile compared to conventional fluids. These improved thermal properties render hybrid nanomaterials highly practical for a wide range of applications, including solar systems, energy production, and cooling processes. In line with this perspective, the current study concentrates on evaluating the thermal performance of two unique hybrid nanofluid flows that impinge obliquely on a stretched cylinder. Two base fluids, FC‐77 and a binary mixture of water and ethylene glycol (50:50)%, have been considered, with the addition of nanoparticles such as single‐walled carbon nanotubes (SWCNTs) and multiwalled carbon nanotubes (MWCNTs). The said model's novelty is enhanced by the temperature dependent viscosity and thermal conductivity. Appropriate transformations are applied to derive a system of ordinary differential equations (ODEs), which are then solved numerically using the bvp4c method. A thorough examination is conducted on the physical phenomenon of pertinent parameters, accompanied by graphical representations. The results revealed that, for the FC‐77 coolant‐based hybrid nanofluid, mounting the particle volume fraction leads to a significant reduction in temperature distribution. Additionally, it is perceived that the presence of a variable viscosity parameter causes a reduction in the surface drag coefficient as well as the axial and tangential velocities. The validity of the proposed flow model is demonstrated by comparing the results with those from an earlier study.

Averaging of a normal system of ordinary differential equations of high frequency with a multipoint boundary value problem on a semi-axis

A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semi-axis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semi-axis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov-Bogolyubov averaging method on the semi-axis is justified.

Characteristics of MWCNT, SWCNT, Cu and water based on magnetized flow of nanofluid with Soret and Dufour effects induced by moving wedge: Consequence of Falkner–Skan power law

Soret and Dufour effects are utilized in designing efficient heat exchangers, optimizing separation processes, and understanding biological transport phenomena. A wedge-shaped structures aid in aerodynamics, optics, seismic resistance, and biological studies. Due to these application the consequences Soret and Dufour effects on flow of nanofluid due to wedge have been investigated. The characteristics of water ( H 2 O ) as a base fluid and single-walled carbon nanotubes   ( SWCNTs ) , multi-walled carbon nanotubes ( MWCNTs ) , and copper ( Cu ) nanoparticles are taking into account. The influence of thermal radiation and heat source/sink are part of this investigation. Similarity variables are used to sort out the complexity of fundamental governing equations. Maple via Runge–Kutta–Fehlberg method (RKF45) is used to solve resulting system of ordinary differential equations (ODEs). The graphical and tabulated results of velocity, temperature, and concentration profiles are plotted against different parameters involved in this study. The findings indicate that modifying the moving wedge factor reduces the temperature distribution. By increasing value of magnetic constraint the velocity distribution increased, the temperature and concentration distribution both are increased for varying the value of Dufour, and Soret and source/sink parameters.

Heat and mass transfer analysis of conducting non-Newtonian nanofluid flows over an elongating sheet with a non-uniform heat source

Nanofluids help in many fields to improve the performance of thermal systems by augmenting heat transfer rates through their thermophysical properties. The performance of the nanofluids with various base fluids may be different. Therefore, the study and comparison of behaviors of various nanofluids are useful in several applications such as fuel as a coolant in automobiles, and in medical and electronic equipment to reduce the thermal resistance. This research proposed a novel model to investigate the flow behavior of three different nanofluids over an elongating surface in the presence of a non-uniform heat source and thermal radiation effects. This investigation describes how the considered nanofluids behave in the presence of a transverse magnetic field, and other effects. The proposed governing boundary layer partial differential equations (PDEs) are reformed into a system of nonlinear ordinary differential equations (ODEs) by introducing the proper similarity transformation. The finalized equations are solved numerically with the help of the ND solve package in Mathematica software. We intended how the fluid flow and heat transfer are affected by non-dimensional controlling factors with the help of graphics. Further, the calculations and discussions are accompanied by the numerical values of the skin friction coefficient and heat and mass transfer rates. According to the current findings, the Maxwell nanofluid exhibits superior performance in velocity, and the Oldroyd-B nanofluid shows more concentration and less temperature. As a special case, the results of this investigation are compared with the existing results, and found a good agreement between the results.

Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

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.

Comparative study of Hiemenz flow in hybrid Carreau nanofluid (Fe3O4-Cu/kerosene-engine oil): nonlinear effects, Newtonian heating and viscous dissipation

ABSTRACT This study is important because it helps us understand how different factors affect the flow of a special liquid with nanoparticles, giving us useful insights for real-world applications in engineering and technology. This research endeavours to explore and contrast the distinctive characteristics of steady, 2-D magnetohydrodynamic (MHD) and stagnation point (Hiemenz) flow in Carreau hybrid nanofluid {Fe3O4-Cu/kerosene-engine Oil} over a nonlinear stretching-shrinking surface with its counterpart, Carreau nanofluid. Employing the laws of conservation encompassing mass, momentum, energy, and concentration, our methodology probes into the multifaceted effects induced by factors like the suction-injection, viscous dissipation, Newtonian heating, Hiemenz flow, thermophoresis, and Brownian motion. This exploration involves the transformation of governing partial differential equations (PDEs) into a system of ordinary differential equations (ODEs) through similarity transformations. Subsequently, the intricate non-linear ODEs are meticulously solved using the bvp4c technique, an adept MATLAB built-in function, facilitating a comprehensive understanding of the intricate dynamics within the system. In addition, the study considers the Newtonian heating effect and a numerical comparison is performed to verify the results. Using two different types of nanoparticles dispersed in kerosene-engine oil, it was demonstrated that increasing Weissenberg parameter values resulted in significant improvements in thermal conductivity and thermal velocity. A comparison of Carreau nanofluid thermal and concentration velocities with Carreau hybrid nanofluid showed that Carreau nanofluid values are lower (Weissenberg parameter). Higher Biot numbers are associated with reduced temperature velocity and elevated concentration velocity.