System Of ODEs Research Articles

Mathematical Modelling of the Thermosolutal Effect on Blood Flow through a Micro-Channel in the Presence of a Magnetic Field

Substances such as mass concentrations in the blood cause circulation challenges. Blood is a substance made up of 55% plasma and 45% hemoglobin, which flows through the blood vessels and is aided by the heart in pumping to all organs and other tissues and cells in the human body. Mathematical modelling plays an important role in understanding the blood circulation problem in the human body, and it helps transform the real-life cardiovascular problem into a system of mathematical equations. In this research, we transformed the real-life cardiovascular problem of the heat effect on mass concentration and blood flow through blood vessels into a system of partial differential equations representing blood momentum, energy, and mass concentration equations after modifying the Navier-Stoke equation. The models are scaled from dimensional to dimensionless using specific scaling parameters. The scaled dimensionless equations were further perturbed and reduced to a system of ordinary differential equations, and the system of ODEs was solved using the Laplace method, where mathematical models representing blood velocity profiles, temperature profiles, and mass concentration profiles were obtained. A numerical simulation was performed using Wolfram Mathematica, version 12, where the effect of the pertinent parameters on the flow profiles was investigated and the results were presented graphically. The variation of the pertinent parameters such as the Grashof number, solutal Grashof number, Prandtl number, Soret number, Schmidt number, Darcy number, and magnetic field affect the flow profile such that the increase in Grashof number and solutal Grashof number causes the blood velocity to increase substantially at the centre of the vessel before decreasing to zero as the wall boundary layer increases, while the increase in Prandtl number and Soret number increases the blood temperature and velocity.

Irreversibility analysis of Eyring Powell nanofluid flow in curved porous channel

AbstractThe aim of this study is to minimize entropy in the MHD Eyring‐Powell fluid through a semi‐porous curved channel. The flow phenomena are examined under the consideration of joule heating, viscous heating, thermophoresis, and Brownian motion. The motivation of this research is to minimize entropy production in curved porous channel because thermal systems become more efficient as a result of decreased energy consumption, operational expenses, and experimental costs. This approach is useful in designing industrial equipment that requires efficient thermal management, such as fuel cells, chemical processing, and advanced refrigeration systems. The coupled boundary layer equations of the problem are highly nonlinear PDEs, which are transformed into systems of coupled ODEs by using similarity variables. The solution of a coupled system of ODEs is obtained numerically via Bvp4c. The effect of several physical parameters for entropy analysis, Bejan number, concentration, and velocity/temperature are illustrated and analyzed using graphs. Furthermore, the computational outcomes of physical quantities, for example, heat and mass transfer rate are also presented. Results indicated that the increasing value of the Reynolds number increases the entropy generation rate, while the reverse tendency is noticed for the Eyring‐Powell parameter. The rising values of the Brownian parameter increase the Bejan number after its decrease, while reverse behavior is observed for the thermophoresis parameter.

Exploring the influence of morphology on magnetized Ree–Eyring tri‐hybrid nanofluid flow between orthogonally moving coaxial disks using artificial neural networks with Levenberg–Marquardt scheme

AbstractThe present study presents an analysis of Ree–Eyring tri‐hybrid nanofluid flow between two expanding/contracting disks with permeable walls by applying the computing power of Levenberg–Marquardt supervised neural networks (LM‐SNNs). The effects of thermal radiation, Brownian motion, and thermophoresis were also thoroughly examined. The results are presented for tri‐hybrid nanofluid with SWCNT and MWCNT and Fe2O3 and H2O base fluid. The coupled non‐linear PDE system is transformed into a system of ODE associated with convective boundary conditions by applying the appropriate transformations. This is then accomplished numerically by using the finite difference‐based BVP‐4c MATLAB code that implements the three‐stage Lobatto IIIA formula. The results are novel and have been validated with LM‐SNNs outcomes. It has been observed that both numerical outcomes and LM‐SNNs produce equivalent results, and both approaches exhibit a drop in the velocity profile for the magnetic field near the lower plate and a rise near the upper plate. The skin friction against the Prandtl number increases, whereas the Nusselt number decreases at the upper disc. Compared to BVP‐4c numerical approaches, the given LM‐SNNs model is more dependable, efficient, and time‐saving because it requires less work and produces results quickly.

Singular Perturbation Analysis for a Coupled KdV–ODE System

Singular Perturbation Analysis for a Coupled KdV–ODE System

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.

CAG-NSPDE: Continuous adaptive graph neural stochastic partial differential equations for traffic flow forecasting

Traffic flow forecasting, which aims to predict future traffic flow given historical data, is crucial for the planning and construction of urban transportation infrastructure. Existing methods based on discrete dynamic graphs are unable to model the smooth evolution of spatial relationships and usually have high computational cost. In addition, they ignore the effect of stochastic noise on the modeling of temporal dependencies, which lead to poor predictive performance on complex intelligent transportation systems. To address these issues, we propose continuous adaptive graph neural stochastic partial differential equation (CAG-NSPDE). A continuous graph is constructed based on learnable continuous adaptive node embeddings for the modeling of evolution of spatial dependencies. To explicitly model the effect of noise, a novel architecture based on stochastic partial differential equation (SPDE) is proposed. It can simultaneously extract features from the temporal and frequency domains by reducing SPDEs to a system of ODEs in Fourier space. We perform extensive experiments on four real-world datasets and the results demonstrate the superiority of CAG-NSPDE compared with state-of-the-arts.

Novel machine learning investigation for Buongiorno fluidic model with Sutterby nanomaterial

The nanomaterials are frequently employed in a variety of heat transfer applications arising in energy generation, engine cooling, extrusion procedures, heat exchanger, thermos-chemical systems, manufacturing structures, powered plants etc. Experts and researchers in these fields often encounter such materials, leading to discernible impacts on velocity, temperature and concentration profiles. The objective of this study is to represent the mathematical structures for Sutterby nanomaterial fluidic model involving porous medium BFM-SNM using nonlinear autoregressive exogeneous networks backpropagated with Levenberg-Marquardt technique (NARX-LMT). The mathematical design of the model is originally presented by PDEs, which are converted into ODEs system through suitable modifications with alternative transformation incorporating numbers, i.e., Deborah, Prandtl and Reynold, as well as parameters, i.e., magnetic thermophoresis, radiation and temperature. The synthetic data is produced numerically by simulating Adams numerical method for BFM-SNM and the supervised computing paradigm of NARX-LMT is applied to the obtained datasets, and the results of NARX-LMT consistently align with numerical observations for each variant of the presented model, exhibiting negligible errors. The NARX-LMT performance on exhaustive experimentation is effectively illustrated through iterative convergence curves on mean squared error, control metrics of the optimization, distribution of error on histograms and regression outputs for Sutterby nanomaterial fluidic model.

Unsteady flow of nanofluid over a sheet of variable thickness with nonlinear kinematics

The transport of heat in an unsteady flow of nanofluid has been discussed in this paper. The flow is maintained over a non-flat, variably porous, and moving sheet in the present simulation. Moreover, the sheet has both constant temperature and nanoparticle concentration at its surface. Unsteadiness in all field variables (i.e. the flow, temperature, and nanoparticles concentration) has been produced by the time-dependent quantities, specified at the surface of the sheet (i.e. kinematics of sheet). The well-known Buongiorno model of nanofluid has been undertaken in the present studies. Furthermore, the simulation includes the particular cases of uniform and linear (polynomial & algebraic functions) stretching/shrinking and injection/suction velocities. Besides that observations of the same nature have been recorded for the non-flat sheet. The boundary layer form of the governing PDEs has been utilized to evaluate the exact status of field variables in the flow domain. Appropriate conditions are imposed by considering the geometry of the flow problem. The problem at hand is simplified by introducing a new set of functions for the field variables because of boundary inputs. After this treatment, the system of boundary value ODEs, which contains several dimensionless numbers (parameters), has appeared, and the problem is solved numerically for various values of parameters, involved in it. More precisely, effects of unsteadiness, wall roughness, surface deformation parameters, Prandtl, Schmidt, thermophoretic, and Brownian motion numbers have been seen on profiles of four field variables. Besides that, the numerical results are also obtained for the skin friction coefficient, Nusselt, and Sherwood numbers. The present modelled problem and its solution are compared with classical models of that particular nature and with their solutions.

End-to-End Statistical Model Checking for Parameterization and Stability Analysis of ODE Models

We propose a simulation-based technique for the parameterization and the stability analysis of parametric Ordinary Differential Equations. This technique is an adaptation of Statistical Model Checking, often used to verify the validity of biological models, to the setting of Ordinary Differential Equations systems. The aim of our technique is to estimate the probability of satisfying a given property under the variability of the parameter or initial condition of the ODE, with any metrics of choice. To do so, we discretize the values space and use statistical model checking to evaluate each individual value w.r.t. provided data. Contrary to other existing methods, we provide statistical guarantees regarding our results that take into account the unavoidable approximation errors introduced through the numerical integration of the ODE system performed while simulating. In order to show the potential of our technique, we present its application to two case studies taken from the literature, one relative to the growth of a jellyfish population, and the other concerning a well-known oscillator model.

Entropy analysis in squeezing flow of CoFe2O4+TiO2+MgO/H2O trihybrid nanofluid between parallel plates with viscous dissipation and chemical reaction: A numerical approach

Trihybrid nanofluid exhibits significantly higher thermal conductivity compared to individual nanoparticles. This enhanced thermal conductivity is especially notable at low concentrations, leading to improved heat transfer efficiency. Hence, trihybrid nanofluids have potential practical applications in various heat transfer systems, cooling, and heating processes. This study investigates the entropy-based thermal aspects of MHD trihybrid nanofluid composed of MgO, TiO2, and CoFe2O4 nanoparticles in combination with water as the base fluid. The analysis is conducted between two parallel plates that are subjected to squeezing motion. Novel aspects encompassing thermal radiation, viscous dissipation, reaction, and temperature-dependent viscosity are taken into consideration. The governing PDEs are transformed into a system of ODEs using appropriate similarity transformations. The numerical solution is achieved through the implementation of the RKF method with a shooting technique. The effects of emerging parameters on velocity, temperature, concentration profiles, and engineering quantities are investigated and discussed through graphs and tables. The outcomes indicate that suction and magnetic fields cause a 7.50% decrease in nanofluid velocity due to augmented squeezing and medium factors. The temperature increases by 8% in trihybrid nanofluids with stronger radiation. An increase in the reaction leads to a decrease in the concentration of CoFe2O4/H2O nanofluid by 5.53%. An escalation in the Brinkman number results in simultaneous increments of 6.04% in entropy and 4.83% in the Bejan number. Magnetic field strength reduces friction by 2.46% on both plates. Enhancing the radiation parameter leads to a 7.31% rise in heat transfer at the lower plate, but it declines by 4.49% at the upper plate. Higher concentration ratios lead to 5% higher Sherwood numbers on both plates.

Bioconvective reversible (an esterification process) Casson nanofluid flow over a radiative spinning disc with microorganism

This study describes the Casson (a non-Newtonian fluid) nanofluid bioconvection flow across a spinning disc in the presence of gyrotactic microorganisms, many slips and thermal radiation. Also, the flow is considered as a reversible flow. The esterification process is taken into account. Using the proper variables, a system of extremely nonlinear PDEs is converted into a system of ODEs. To arrive to the solution of such equations, a numerical approach is used. Using the bvp4c approach, nonlinear flow equations can be numerically solved. Investigated are the effects of different numbers on the thermal field, volumetric concentration of nanoparticles and microbiological field. The key characteristics of the parameters in relation to the profiles of the velocity, temperature, concentration and microorganisms are graphically assessed with appropriate physical effects. A graphical explanation is provided for the wall shear stress, local Nusselt number, local Sherwood number and local motile density number. Rate of motile density number shows a prominent difference between reversible and irreversible flows for Brownian motion and Peclet number. The results of the theoretical simulations have dynamic applications in the fields of biotechnology and thermal engineering.

Heat and mass transfer analysis of hybrid nanofluid flow over a rotating permeable cylinder: A modified Buongiorno model approach

Many external factors influence the properties of nanoparticles in the working fluid, which subsequently determines the efficacy attributes of the ensuing nanofluid. Incorporating these factors is essential for studying heat and mass transfer in hybrid nanofluids, as it provides a detailed representation of the mechanism. Such factors are porous medium, heat source, Thermophoresis, and Brownian motion. The rotation of a cylinder can alter the flow field and boundary layer around it, which could be used for flow management scenarios such as drag reduction or separation of boundary layers control, and the complex flow patterns generated through a rotating cylinder could be utilized for improved mixing and heat transfer in processes in industry. Thus, the current novel aspect of the study examines the impacts of thermophoresis and Brownian motion over a rotating cylinder in the presence of a porous media comprised of water-based TiO2 and Fe2O4. The physical circumstances are primarily represented as PDEs based on the considered flow conditions, which then get transformed into a system of ODEs using the relevant similarity variables. The numerical solution is obtained by using RKF-45th and the shooting procedure. The flow symmetry and the effects of the specific variables associated with the problem are examined and delivered graphically. The findings confirm that the drag coefficient improves with porous material, altering fluid dynamics. Heat transmission steadily drops as the thermophoretic parameter gets larger, but mass transfer displays a contrary trend for evolving Brownian motion parameters. Also, the current results are validated with existing past studies.

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.

Observer-Based Adaptive Control for a Coupled PDE–ODE System of a Flexible Wing

Observer-Based Adaptive Control for a Coupled PDE–ODE System of a Flexible Wing

An exploration of diffusion-thermo and radiation absorption impacts on non-Newtonian MHD flow towards two distinct geometries with biot number

Temperature and liquid flow monitoring are required in many industrial applications in order to maximize machine performance. When exploring polymers, fabricating synthetic films and transparent materials, and manufacturing metal-based equipment, frictional forces and thermal flow rates need to be controlled. The significance of the study of prominent characteristics of Diffusion-Thermo, Quadratic thermal radiation and radiation absorption over a permeable stretching geometry are discussed. Similarity transformations reduced the boundary layer partial differential to an ODE. The acquired system of ODEs is solved in numerical-manner using MATLAB bvp4c. The distributions f '(η), θ(η) and φ(η) are described graphical aspect, with specified governed parameters. As Diffusion thermo parameter raises, temperature field enhances. As radiation absorption raises, the temperature and velocity enhance and concentration decreases. A rise in radiation absorption parameter causes a raise in the buoyancy, which in turn raises the flow rate, and that this is reflected in the velocity profiles. The radiation absorption parameter raises temperatures nearer porous boundary layer and lowers concentration. Rising temperature ratio parameter θw values were accompanied by rising temperatures. Temperature is enhancing when improving of Biot parameter. The skin-friction is heightened with raising ‘n’ values. Nusselt number is declined, raising values of Df. The Sherwood number is enhanced with raising values of ‘Sc’. An incorporation of Diffusion thermo, Quadratic thermal radiation and radiation absorption effects in a nano-Carreau flow across Wedge and Cone is the novelty. The flow of a boundary-layer across a wedge gets made growingly significant, not long ago, owing to its many practical applications, including those in medical, bioengineering, crude oil extraction, nuclear power plants, a polymer extrusion process, production of plastic sheet, metal spinning, friction drag of a ship, and electronic gadgets, etc. This study will be extended by incorporating micro-polar, tangent hyperbolic, Sutterby, with distinct non-Newtonian fluids.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Modelling the evolution of human postmenopausal longevity using ordinary differential equations

We present two simple ODE systems based on the mathematical model in Le et al. (2022) investigating how human postmenopausal longevity could have evolved. Specifically, our ODE systems address a fairly recent and provocative hypothesis that ancestral males choosing younger mates drove the evolution of menopause (Morton et al., 2013). We analyse one system in two ways: (1) determining the explicit solution and (2) reducing it to one ODE and drawing a bifurcation diagram. We analyse the other system by reducing it to two ODEs and using a phase plane. The concepts in this article are suitable for a late second-year or third-year course in nonlinear ODEs. Our models demonstrate that undergraduate students of dynamical systems can critically engage with a highly cited and internationally recognised hypothesis of human evolution.

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.

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.