System Of Ordinary Differential Equations Research Articles

A Mathematical Model of Pressure Ulcer Formation to Facilitate Prevention and Management.

Pressure ulcers are a frequent issue involving localized damage to the skin and underlying tissues, commonly arising from prolonged hospitalization and immobilization. This paper introduces a mathematical model designed to elucidate the mechanics behind pressure ulcer formation, aiming to predict its occurrence and assist in its prevention. Utilizing differential geometry and elasticity theory, the model represents human skin and simulates its deformation under pressure. Additionally, a system of ordinary differential equations is employed to predict the outcomes of these deformations, estimating the cellular death rate in skin tissues and underlying layers. The model also incorporates changes in blood flow resulting from alterations in skin geometry. This comprehensive approach provides new insights into the optimal bed surfaces required to prevent pressure ulcers and offers a general predictive method to aid healthcare personnel in making informed decisions for at-risk patients. Compared to existing models in the literature, our model delivers a more thorough prediction method that aligns well with current data. It can forecast the time required for an immobilized individual to develop an ulcer in various body parts, considering different initial health conditions and treatment strategies.

Simulation of coupled groundwater flow and contaminant transport using quintic B-spline collocation method

PurposeThe purpose of this study is to introduce a new numerical scheme with no stability condition and high-order accuracy for the solution of two-dimensional coupled groundwater flow and transport simulation problems with regular and irregular geometries and compare the results with widely acceptable programs such as Modular Three-Dimensional Finite-Difference Ground-Water Flow Model (MODFLOW) and Modular Three-Dimensional Multispecies Transport Model (MT3DMS).Design/methodology/approachThe newly proposed numerical scheme is based on the method of lines (MOL) approach and uses high-order approximations both in space and time. Quintic B-spline (QBS) functions are used in space to transform partial differential equations, representing the relevant physical phenomena in the system of ordinary differential equations. Then this system is solved with the DOPRI5 algorithm that requires no stability condition. The obtained results are compared with the results of the MODFLOW and MT3DMS programs to verify the accuracy of the proposed scheme.FindingsThe results indicate that the proposed numerical scheme can successfully simulate the two-dimensional coupled groundwater flow and transport problems with complex geometry and parameter structures. All the results are in good agreement with the reference solutions.Originality/valueTo the best of the authors' knowledge, the QBS-DOPRI5 method is used for the first time for solving two-dimensional coupled groundwater flow and transport problems with complex geometries and can be extended to high-dimensional problems. In the future, considering the success of the proposed numerical scheme, it can be used successfully for the identification of groundwater contaminant source characteristics.

Statistical investigation of heat transfer efficiency on a ternary nanoflow over a stretching surface

Ternary hybrid nanofluids offer significant advantages in heat transfer due to their enhanced thermal conductivity compared to traditional fluids. Investigating their flow behavior over expanding surfaces can contribute to the development of more efficient heat exchangers and cooling systems in the electronics, energy production, and transportation industries. This paper presents a detailed analysis of the flow and thermal transfer characteristics of a ternary hybrid nanofluid (SiO2–Cu–Al2O3/H2O) over a radially stretching surface, utilizing both numerical techniques (Keller Box method) and statistical techniques. Furthermore, the model includes factors like convective heat transport, radiation, internal fluid friction, and suction. Similarity transformations transform the governing equations for fluid flow and heat transfer into a system of nonlinear ordinary differential equations. Then the equations are solved numerically using the Keller Box method with the aid of MATLAB software. Additionally, we have employed statistical techniques such as correlation analysis, probable error estimation, and multivariate regression to validate and ensure the accuracy of the numerical outcomes. Parameter ranges for Biot number, radiation, unsteadiness, Eckert number, and energy generation parameter are 0.1 ≤ Mp ≤ 0.7, 0.1 ≤ Ks ≤ 1.5, 0.2 ≤ Ps ≤ 0.8, 0.1 ≤ EN ≤ 0.4, 0.5 ≤ RN ≤ 2, 1≤ BT ≤ 2.5, 0.1 ≤ Hsc ≤ 0.7 and found to significantly impact heat transfer behavior within these bounds. This study utilizes 3D surface plots to visualize the influence of key parameters on engineering quantities. R squared values show that the data strongly match the regression model. The findings demonstrate excellent concordance between the predicted values and the actual Nusselt number and skin friction measurements. Biot number has a very strong correlation with heat transfer, with minimal chance of error. And the same is evidenced by the correlation matrix and multiple regression analysis. Eckert number also exhibits a negative correlation with the Nusselt number. This translates to a situation where increasing Eckert number directly leads to a decrease in the rate of heat transfer, with no margin for error. Energy generation parameter demonstrates a negative correlation with heat transfer. However, there's a slight possibility of error, with an estimated error percentage of around 0.0017.

Implication of electromagnetohydrodynamic flow of a non‐Newtonian hybrid nanofluid in a converging and diverging channel with velocity slip effects: A comparative investigation using numerical and ADM approaches

AbstractThis study delves into the intricate interplay of magnetic and electric fields (EMHD) on the flow characteristics of a non‐Newtonian bio‐hybrid nanofluid, consisting of Ag+Graphene/blood, within converging and diverging geometries. The investigation takes into account the effects of velocity slip at the walls, offering a comprehensive examination of this complex fluid system. A novel bio‐hybrid nanofluid model was introduced, featuring a unique combination of Ag+Graphene/blood nanoparticles. To address this multifaceted problem, the research employed mathematical modeling based on nonlinear partial differential equations (PDEs), encompassing continuity and momentum equations. These PDEs were then transformed into a system of nonlinear ordinary differential equations (ODEs) through similarity transformations. The study explored both numerical and analytical solutions, with a particular focus on the application of the Adomian decomposition method (ADM). To validate the findings, the study compared the analytical results with those obtained using the HAM‐based Mathematica package and the Runge–Kutta Fehlberg 4th–5th order (RKF‐45) method in specific scenarios. Active parameters, including nanofluid volume fraction, slip factors, and the influence of magnetic and electric fields, were systematically examined to unveil their impacts on velocity and skin friction within this multifaceted nanofluid system. It is found that the skin friction coefficient decreases with the Increasing both the nanoparticle volume fraction, Hartmann number and the angle in both channels. Results obtained also reveal an in the converging section, higher Casson parameters lead to increased yield stress but are offset by the higher shear rates, resulting in a higher velocity profile. In the diverging section, the fluid resists flow due to the reduced shear stress, leading to a decreased velocity profile.

Numerical solution of optimal control problems for linear systems of ordinary differential equations

Abstract An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.

Computational analysis of nanoparticles and waste discharge concentration past a rotating sphere with Lorentz forces

Abstract As industries rely more and more on magnetohydrodynamic (MHD) systems for different uses in power, production, and management of the environment, it becomes essential to optimize these operations. The study seeks to improve the effectiveness and productivity of cooling structures, chemical reaction reactors, and contaminant control methods by investigating these intricate interconnections. Because of this, the work scrutinizes the endothermic/exothermic (EN/EX) chemical processes, convective boundary conditions, and pollutant concentration impacts on MHD nanofluid circulation around a rotating sphere. The governing equations based on the above assumptions are reduced into a system of ordinary differential equations and solved numerically with Runge–Kutta Fehlberg’s fourth- and fifth- order schemes. The obtained numerical outcomes from the numerical scheme are presented with the aid of graphs, and the results show that the rate of mass transfer decreases with an increase in the external pollutant local source and solid volume percentage. For changes in the values of the activation energy parameter and solid fraction, the rate of thermal dispersion drops for the EN case and upsurges for the EX case. The concentration profile shows increment with the addition of the external pollutant source variation parameter and local pollutant external source parameter. The outcomes of the present work can be helpful in cooling equipment, developing advanced methods for controlling pollution, environmental management, MHD generators, and various industrial contexts.

Fold-Fold Singularity in a Piecewise Smooth Mathematical Model Describing the Dynamics of a Stockless Market

Fold-fold singularities are critical points or singularities in piecewise smooth dynamical systems (PWS) where both the stability and the structure of the system change. These singularities are of great importance in the study of specific dynamics, such as those in markets, as they indicate significant transformations in their evolution, including sudden variability in prices or changes in the behavior of offers and demand. Despite the substantial increase in the use of mathematical and computational tools applied to market dynamics, the current literature does not thoroughly address the study of the existence of fold-fold singularities in piecewise smooth systems within this context. Therefore, due to the importance of markets as economic activities, this paper proves the existence of such a singularity in a mathematical model that describes the dynamics of a stockless market, which is represented by a system of ordinary differential equations defined with piecewise smooth functions.

Innovative thermal management in the presence of ferromagnetic hybrid nanoparticles

In the present work, a simple intelligence-based computation of artificial neural networks with the Levenberg-Marquardt backpropagation algorithm is developed to analyze the new ferromagnetic hybrid nanofluid flow model in the presence of a magnetic dipole within the context of flow over a stretching sheet. A combination of cobalt and iron (III) oxide (Co-Fe2O3) is strategically selected as ferromagnetic hybrid nanoparticles within the base fluid, water. The initial representation of the developed ferromagnetic hybrid nanofluid flow model, which is a system of highly nonlinear partial differential equations, is transformed into a system of nonlinear ordinary differential equations using appropriate similarity transformations. The reference data set of the possible outcomes is obtained from bvp4c for varying the parameters of the ferromagnetic hybrid nanofluid flow model. The estimated solutions of the proposed model are described during the testing, training, and validation phases of the backpropagated neural network. The performance evaluation and comparative study of the algorithm are carried out by regression analysis, error histograms, function fitting graphs, and mean squared error results. The findings of our study analyze the increasing effect of the ferrohydrodynamic interaction parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} to enhance the temperature and velocity profiles, while increasing the thermal relaxation parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} decreases the temperature profile. The performance on MSE was shown for the temperature and velocity profiles of the developed model about 9.1703e−10, 7.1313ee−10, 3.1462e−10, and 4.8747e−10. The accuracy of the artificial neural networks with the Levenberg-Marquardt algorithm method is confirmed through various analyses and comparative results with the reference data. The purpose of this study is to enhance understanding of ferromagnetic hybrid nanofluid flow models using artificial neural networks with the Levenberg-Marquardt algorithm, offering precise analysis of key parameter effects on temperature and velocity profiles. Future studies will provide novel soft computing methods that leverage artificial neural networks to effectively solve problems in fluid mechanics and expand to engineering applications, improving their usefulness in tackling real-world problems.

A novel ionic model for matured and paced atrial-like human iPSC-CMs integrating [formula omitted] and [formula omitted] currents

This work introduces the first atrial-specific in-silico human induced pluripotent stem cells-derived cardiomyocytes (hiPSC-CMs) model, based on a set of phenotype-specific IKur,IKCa and IK1 membrane currents. This model is built on novel in-vitro experimental data recently published by some of the co-authors to simulate the paced action potential of matured atrial-like hiPSC-CMs. The model consists of a system of stiff ordinary differential equations depending on several parameters, which have been tuned by automatic optimization techniques to closely match selected experimental biomarkers. The new model effectively simulates the electronic in-vitro hiPSC-CMs maturation process, transitioning from an unstable depolarized membrane diastolic potential to a stable hyperpolarized resting potential, and exhibits spontaneous firing activity in unpaced conditions. Moreover, our model accurately reflects the experimental rate dependence data at different cycle length and demonstrates the expected response to a specific current blocker. This atrial-specific in-silico model provides a novel computational tool for electrophysiological studies of cardiac stem cells and their applications to drug evaluation and atrial fibrillation treatment.

Fast permeability measurement for tight reservoir cores using only initial data of the one chamber pressure pulse decay test

In this study, a mathematical model for fast determination of the permeabilities of tight rocks using measurements taken from the initial period of the One Chamber Pressure Pulse Decay (OC-PPD) test is presented. The model applies to measurements taken both before and after the pressure pulse front has reached the downstream end of the specimen. The analytical solutions for the pressure decay in the upstream chamber are derived based on a parabolic arc approximation of pore pressure distribution along the test specimen. This approximation allows converting the initial–boundary value problem of fluid diffusion in the specimen, governed by partial differential equations, to a system of ordinary differential equations that can be easily solved by explicit formulae. Thus, an explicit formula for the pressure decay rate is obtained, which enables inverse analysis of the initial experimental data to estimate the rock permeability. The proposed method expedites the pulse decay test as it does not require the system to reach equilibrium. The method is validated with three sets of experimental data of the OC-PPD test using helium as the diffusing fluid, for which the relative error of the permeability is found to be less than 6%. This method is particularly useful if the equilibrium time of the pulse decay test for rock specimens with permeabilities in the range of nano-Darcy takes hours or days.

Integrated artificial intelligence and non‐similar analysis for forced convection of radially magnetized ternary hybrid nanofluid of Carreau‐Yasuda fluid model over a curved stretching surface

AbstractThe current study investigates the boundary layer flow of Carreau‐Yasuda (C‐Y) ternary hybrid nanofluid model in a porous medium across curved surface stretching at linear rate under the influence of applied radial magnetic field. , and are nanoparticles and ethylene glycol is considered as base fluid. The effects of viscous dissipation and ohmic heating are present in the energy equation. The governing partial differential equation (PDEs) is nondimensionalized using non‐similarity transformations. They can be treated as ordinary differential equations (ODEs) using local non‐similarity method and solutions are obtained via bvp4c MATLAB tools. The results are evaluated by introducing computational intelligence approach utilizing the AI‐based Levenberg–Marquardt scheme with a backpropagation neural network (LMS‐BPNN) to investigate flow stability. The authors intend to use AI‐based LMS‐BPNN is to optimize the behavior of the hybrid nanofluid (HNF) flow of Carreau‐Yasuda fluid across a stretching curved sheet. Initial/reference solutions are obtained through bvp4c function (an embedded MATLAB function designed to solve systems of ODEs) by systematically adjusting input parameters as demonstrated in Scenarios 1–5. There are three options to divide the numerical data: 80% for training, 10% for testing, and an additional 10% for validation. The LMS‐BPNN is used for approximate solutions of Scenario 1–5. The efficiency and reliability of LMS‐BPNN are validated through fitness curves based on correlation index (R), error, and regression analysis. The velocity and temperature profiles asymptotically satisfy boundary conditions of Scenario 1–5 with LMS‐BPNN.

Exact solutions, conservation laws, and shock wave propagation of two-lanes traffic flow model via Lie symmetry

In this study, we consider a hyperbolic system of quasi-linear partial differential equations, governed by the traffic flow model on two lanes. We employ symmetry analysis and establish one-dimensional optimal subalgebras. Subsequently, we reduce the model into a system of ordinary differential equations for each optimal subalgebra and construct some new exact solutions; some of them are presented graphically. Further, by imposing the traveling wave transformation, we derive solutions including peakon-type solitons and upward parabola solitons. Furthermore, we demonstrate the existence of the nonlinear self-adjointness property of the model and formulate conservation laws. Finally, we discussed the evolutionary behavior of C1-waves, characteristic shock, and their interactions through one of the obtained exact solutions.

Magnetohydrodynamics bio-convection flow at Casson fluid stagnation point in porous medium: Cross-diffusion effect and heat production

This study examines the effect of heat production and radiation absorption on the magnetohydrodynamic Casson fluid flow at the stagnation point in a porous medium. We convert the group of fluid flow equations, which are non-linear partial differential equations with suitable boundary constraints, into a set of non-linear ordinary differential equations using similarity transformations. The homotopy analysis method (HAM) solves the converted system of ordinary differential equations. We draw graphs for numerous values of non-dimensional parameters and tables of surface drag force, rates of heat transfer, and mass transfer to analyze the relationship between velocity field, temperature field, concentration field, and other essential parameters involved in the study. We have proven that the Dufour number, radiation parameter, and heat generation parameter elevate the fluid temperature, whereas the magnetic parameter lowers it. The Casson fluid parameter, buoyancy force parameter, and mixed convection parameter all promote fluid movement throughout the flow field. The presented tabular data allows us to see the trend of heat and mass transfer rates, as well as drag force rates, against important parameters, enhancing our understanding of these rates.

Asymptotic Representations of Solutions of $$n\times n$$ Systems of Ordinary Differential Equations with a Large Parameter

Asymptotic Representations of Solutions of $$n\times n$$ Systems of Ordinary Differential Equations with a Large Parameter

Sensor placement considering the observability of traffic dynamics: On the algebraic and graphical perspectives

In this paper, we present a new sensor location model that aims to maximize observability of link densities in a dynamic traffic network described using a piecewise linear system of ordinary differential equations. We develop an algebraic approach based on the eigenstructure to determine the sensor location for achieving full observability with a minimal number of sensors. Additionally, a graphical approach based on the concept of structural observability is developed. By exploiting the special property of flow conservation in traffic networks, we derive a simple analytical result that can be used to identify observable components in a partially observable system, which is the main contribution of this paper. The graphical and algebraic properties of observability are then integrated into a sensor location optimization model considering a wide range of traffic conditions. Through numerical experiments, we demonstrate the good performance of our sensor deployment strategies in terms of the average observability and estimation errors.

Influence of heat source/sink on a rotating cone in a rotating nanofluid with magnetic field impact: Application of Hosoya polynomial‐based collocation method

AbstractThe present analysis considers the angular velocities of the free flow and the arbitrarily fluctuating cone over time, leading to an unsteady stream over a rotating cone in a rotating nanofluid. The effects of heat source/sink and magnetic field on an unsteady flow past a rotating cone in a rotating nanoliquid are considered in this examination. The dimensional governing equations are transformed into nondimensional ordinary differential equations (ODEs) using the similarity variables. The nonlinear system of ODEs has been solved using the Hosoya polynomial‐based collocation method (HPBCM), and the obtained values are compared with the numerical method Runge Kutta Fehlberg's fourth‐fifth order (RKF‐45) scheme. The effects of numerous factors on the momentum and thermal distributions are shown graphically. Results reveal that the ratio of the cone angular velocity to the free stream angular velocity increases the velocity profile but converse trend is seen for the thermal profile. The upsurge in the values of the magnetic parameter intensifies the velocity profile. The rise in the values of the heat source/sink parameter upsurges the thermal profile. As the unsteady parameter increases temperature profile declines.

Chemically reactive and mixed convective hybrid nanofluid flow between two parallel rotating disks with Joule heating: A thermal computational study

An electrically conducting and magnetically influenced three-dimensional flow of modified nanofluid between two rotating disks is investigated in this study. Nanometer-size particles of two distinct materials (Al2O3, Ag) suspended in water in the hybrid nanofluid are considered. The Joule heating effects, mixed convection, chemical factor, and exponential heating are considered in this study. The physical problem under these assumptions is transformed into a system of equations. To convert partial differential equations (PDEs) into systems of ordinary differential equations (ODEs), the suitable similarity variables are introduced. The reduced system is numerically solved using bvp4c, a well-known higher-order algorithm. The effects of significant variables on the profiles of velocity, temperature, and concentration are depicted graphically. We have chosen the pertinent parameters in the specific intervals; [Formula: see text][Formula: see text][Formula: see text][Formula: see text][Formula: see text], [Formula: see text][Formula: see text][Formula: see text] and [Formula: see text]. The interactions between a number of significant factors, including skin friction and Nusselt and Sherwood numbers at the higher and lower disks, are shown in tables. The results demonstrate that the local skin fraction decreases as the mixed convection factor is increased, which physically increases the heat transmission rate between the two discs. Furthermore, as the magnetic field and radiation number rise, the rate of heat transfer at the lower and top discs decreases. The numerical results in the form of a table are compared with the available literature, and our results beat the previously published work. The results described here are desirable choices for thermal uses such as automotive cooling systems, solar thermal systems, engineering, medical areas, heat sinks, or current energy storage since they have demonstrated higher thermal characteristics and stability than simple nanofluids (NFs). In Table 6, a comparative numerical study is presented for the various of state variables with the available literature, where the present study is validated.

A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves' movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-dependent terms. The stability analysis of this system was examined using the von-Neumann Fourier series method. L₂, L_{∞} and absolute error norms were used to measure the numerical results' convergence to the real solution. The numerical results calculated were compared with the exact solution and some studies in the literature.

Modeling Rugose Spiraling Whitefly Infestation on Coconut Trees Using Delay Differential Equations: Analysis via HPM

In this paper, a delay - induced pest control model is proposed. We have introduced a time delay in healthy trees and whitefly population in the infected tree density of the proposed system of equations to reduce the probability of healthy trees becoming infected, as well as the level of infection. We have analyzed the impact of time delay on the stability of the equilibrium and establish requirements to verify its asymptotic stability over all delays. The solutions of this system of non-linear ordinary differential equations(ODEs) and delay differential equations(DDEs) are presented by using homotopy perturbation method(HPM). Numerical simulation is also obtained for the same model of both ODE and DDE by using MATLAB software. The proposed method works very well and is easy to use, as shown by these findings. Our aim is to control the spread of whitefly such that harvest is not affected.

Rayleigh-Benard convection and sensitivity analysis of magnetized couple stress water conveying bionanofluid flow with thermal diffusivities effect

Bionanofluids containing biological nanoparticles suspended in base fluids exhibit altered dynamics due to coupled effects between the nanoparticles and fluid properties. More efficient cooling can enhance medical technology performance, reduce energy usage, and provide safer, more sustainable healthcare solutions. This work overcomes previous research limitations by elucidating the combined impacts of microorganisms and couple stress properties on the behavior of water-based bionanofluids containing copper nanoparticles, under inclined magnetic fields. Governing equations for momentum, energy, concentration, and microorganisms are transformed into ordinary differential equations (ODEs) via similarity methods. The resulting ODE system is then solved semi-analytically using the Homotopy Analysis Method, revealing distinct profile behaviors. Additionally, quantitative indicators including skin friction for different values of M, λ,β, and Nr increased by 7.19%, 62.81%, 31.27%, and 21.32% respectively. Similarly Nusselt number for different values Qe , Ec, and Df decrease by 3.502 %, 2.5705%, and 2.447% respectively. Furthermore, Sherwood Numbers for different values of Sc, Kc, increased by 16.41%, and 4.133% respectively. Finally, Microorganisms for different values of Lb, Pe, and σ1 increase by 2.934%, 2.61%, and 1.172% respectively.