System Of Ordinary Differential Equations Research Articles

Analytical simulation of Darcy–Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy–Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy–Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy’s formula for porous media flow: two nanofluid particles, molybdenum disulphide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

A mathematical model to study the role of dystrophin protein in tumor micro-environment.

In this research work, the authors have developed a mathematical model to examine the interaction between dystrophin protein and tumor. The authors formulated a system of ordinary differential equations to describe the dynamics of the dystrophin-tumor interaction system. Jacobian matrix and Routh-Hurwitz stability techniques were used to determine equilibrium points, perform stability and bifurcation analysis, and establish the conditions required for the stability of the proposed model. Numerical simulations are performed using Euler's method to investigate the temporal evolution of the proposed model under different parameter values, such as tumor growth rate and feedback strength of dystrophin protein. The numerical results are presented in tables, and corresponding to each table, a graphical analysis is done. The graphical analysis includes creating phase portraits to visually represent stability regions around the equilibrium points, bifurcation diagrams to identify critical points, and time series analysis to highlight the behavior of the proposed model. The authors explore how variations in dystrophin expression impact tumor progression, identifying potential therapeutic implications of maintaining higher dystrophin levels. This comprehensive analysis enhances our understanding of the dystrophin-tumor interaction, providing a basis for further experimental validation and potential therapeutic strategies.

Vector-borne disease outbreak control via instant releases.

This paper is devoted to the study of optimal release strategies to control vector-borne diseases, such as dengue, Zika, chikungunya and malaria. Two techniques are considered: the sterile insect one (SIT), which consists in releasing sterilized males among wild vectors in order to perturb their reproduction, and the Wolbachia one (presently used mainly for mosquitoes), which consists in releasing vectors, that are infected with a bacterium limiting their vectorial capacity, in order to replace the wild population by one with reduced vectorial capacity. In each case, the time dynamics of the vector population is modeled by a system of ordinary differential equations in which the releases are represented by linear combinations of Dirac measures with positive coefficients determining their intensity. We introduce optimal control problems that we solve numerically using ad-hoc algorithms, based on writing first-order optimality conditions characterizing the best combination of Dirac measures. We then discuss the results obtained, focusing in particular on the complexity and efficiency of optimal controls and comparing the strategies obtained. Mathematical modeling can help testing a great number of scenarios that are potentially interesting in future interventions (even those that are orthogonal to the present strategies) but that would be hard, costly or even impossible to test in the field in present conditions.

Numerical Simulation Study of Thermal Performance in Hot Water Storage Tanks with External and Internal Heat Exchangers

This paper presents a numerical analysis of two hot water storage tank configurations—one equipped with an external heat exchanger (Tank-1) and the other with an internal heat exchanger (Tank-2). The objective is to evaluate and compare their thermal performance during charging and discharging processes. The numerical model is developed by solving a system of ordinary differential equations using the 4th-order Runge–Kutta method, implemented in the Python programming language. The results indicate that Tank-1 demonstrated a higher charging efficiency of 94.6%, achieving full charge in approximately 2 h and 20 min. In comparison, Tank-2 required 3 h and 47 min to reach full charge, with a charging efficiency of 85.9%. During discharge, both configurations exhibited similar behavior, with an efficiency of 13.63% over approximately 33 min. The analysis showed that the external heat exchanger configuration led to more effective thermal stratification, supported by the Richardson number analysis, which indicated a significant effect of buoyancy during charging. This design advantage makes Tank-1 particularly suitable for applications requiring rapid heating and minimal heat loss, such as in cold climates or intermittent demand systems. The numerical model demonstrated reliable predictive accuracy, achieving an RMSE of 6.1% for the charging process and 6.8% for the discharging process, thereby validating the model’s reliability. These findings highlight the superior performance of the external heat exchanger configuration for fast and efficient energy storage, particularly for applications in cold climates.

Numerical solutions and stability analysis of hybrid Casson nanofluid flow with MHD and heat transfer effects

AbstractThis research addresses the complex dynamics of hybrid Casson nanoliquids in flow and heat transfer applications, focusing on the interaction between fluid dynamics and thermal phenomena in the presence of magnetohydrodynamics (MHD), viscous dissipation, and Joule heating. The motivation behind this study stems from the need to enhance the efficiency of heat transfer processes in various engineering applications, such as cooling systems and electronic devices, where hybrid nanofluids can offer superior thermal performance compared to conventional fluids. The novelty of this work lies in its comprehensive numerical investigation of a hybrid Casson nanoliquid over a moving permeable surface, incorporating a detailed analysis of MHD effects, viscous dissipation, and Joule heating. Using the Tiwari and Das model to formulate the governing equations, the study employs similarity transformations to convert these equations into a system of ordinary differential equations (ODEs). MATLAB is then used to derive numerical solutions, with a stability analysis ensuring the physical dependability of these solutions. Key findings reveal that the temperature distribution of the hybrid nanoliquid shows a positive correlation with both Prandtl and Hartmann numbers. Additionally, a positive relationship between temperature and the Eckert number is observed. These insights offer valuable guidance for engineers aiming to optimize heat transfer processes using hybrid nanofluids, highlighting their potential for improved thermal management in practical applications.

The influence of lymphatic vessels on nanoparticle distribution and heat transfer within tissue

AbstractThis study analytically investigates the dynamics of nanoparticle transport within a three‐dimensional porous cylinder simulating a lymphatic vessel, without external heat sources. The governing equations and boundary conditions are transformed to yield a system of ordinary differential equations, which are solved numerically using MATLAB built‐in function, bvp4c. Key parameters are visually examined and physically interpreted in relation to temperature, velocity, concentration, and Nusselt number profiles. The study reveals that the distribution of temperature and Nusselt number are maximized by increasing the heat transfer coefficient, whereas NP concentration is increased by decreasing it. Furthermore, the Brownian motion parameter enhances both heat transmission and NP concentration. It is also observed that simpler extravasation into lymphatics decreases tissue nanoparticle levels and heat conduction. Ultimately, optimal intra‐lymphatic nanoparticle distribution pathways are achieved by specifically varying heat transfer and interstitial mass flux patterns. By simulating biological barriers and lymphatic drainage, this model enhances our understanding of the underlying transport mechanisms controlling nanoparticle mobilization.

From interacting agents to Boltzmann-Gibbs distribution of money

Abstract We investigate the unbiased model for money exchanges: agents give at random time a dollar to one another (if they have one). Surprisingly, this dynamics eventually leads to a geometric distribution of wealth (shown empirically by Dragulescu and Yakovenko, and rigorously by several follow-up papers). We prove a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. This deterministic description is then analyzed by taking advantage of several entropy–entropy dissipation inequalities and we provide a quantitative almost-exponential rate of convergence toward the equilibrium (geometric distribution) in relative entropy.

Unraveling the role of exercise in cancer suppression: insights from a mathematical model

Recent experimental studies have shown that physical exercise has the potential to suppress tumor progression. Such suppression has been reported to be mediated by the exercise-induced activation of natural killer (NK) cells through the release of IL-6, a cytokine. Aimed at shedding light on how exercise-induced NK cell activation helps in the suppression of cancer, we developed a coarse-grained mathematical model based on a system of ordinary differential equations describing the interaction between IL-6, NK-cells, and tumor cells. The model is then used to study how exercise duration and exercise intensity affect tumor suppression. Our results show that increasing exercise intensity or increasing exercise duration leads to greater and sustained tumor suppression. Furthermore, multi-bout exercise patterns hold promise for improving cancer treatment strategies by adjusting exercise intensity and frequency. Thus, the proposed mathematical model provides insights into the role of exercise in tumor suppression and can be instrumental in guiding future experimental studies, potentially leading to more effective exercise interventions.

Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

The paper discusses the computational technology for constructing one type of small-scale magnetohydrodynamic turbulence models – shell models. Any such model is a system of ordinary quadratic nonlinear differential equations with constant coefficients. Each phase variable is interpreted in absolute value as a measure of the intensity of one of the fields of the turbulent system in a certain range of spatial scales (scale shell). The equations of any shell model must have several quadratic invariants, which are analogues of conservation laws in ideal magnetohydrodynamics. The derivation of the model equations consists in obtaining such expressions for constant coefficients for which the predetermined quadratic expressions will indeed be invariants. Derivation of these expressions «manually» is quite cumbersome and the likelihood of errors in formula transformations is high. This is especially true for non-local models in which large-scale shells that are distant in size can interact. The novelty and originality of the work lie in the fact that the authors proposed a computational technology that allows one to automate the process of deriving equations for shell models. The technology was implemented using computer algebra methods, which made it possible to obtain parametric classes of models in which the invariance of given quadratic forms is carried out absolutely accurately – in formula form. The determination of the parameter values in the resulting parametric class of models is further carried out by agreement with the measures of the interaction of shells in the model with the probabilities of their interaction in a real physical system. The idea of the described technology and its implementation belong to the authors. Some of its elements were published by the authors earlier, but in this work, for the first time, its systematic description is given for models with complex phase variables and agreement of measures of interaction of shells with probabilities. There have been no similar works by other authors previously. The technology allows you to quickly and accurately generate equations for new non-local turbulence shell models and can be useful to researchers involved in modeling turbulent systems.

Small-sample Bayesian error estimation for ergodic, chaotic systems of ordinary differential equations

The discretization of chaotic systems introduces statistical errors in addition to discretization errors into the estimation of quantities of interest. In order to efficiently arrive at estimates of quantities of interest, these two forms of error should be balanced; however, simulations are run without knowledge of the true/asymptotic outputs of interest or their error behaviors. In this work, we develop a framework for error modeling and identification using small-sample Bayesian inference that allows approximation of the optimal balance between sampling time and discretization precision without the computation of high-cost libraries of reference solutions. The result enables the possibility of running chaotic and turbulent simulations in a way that minimizes the total error between sampling and discretization without prior knowledge of the error behavior of the system.

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

Rapid estimation of left ventricular contractility with a physics-informed neural network inverse modeling approach

Physics-based computer models based on numerical solutions of the governing equations generally cannot make rapid predictions, which in turn limits their applications in the clinic. To address this issue, we developed a physics-informed neural network (PINN) model that encodes the physics of a closed-loop blood circulation system embedding a left ventricle (LV). The PINN model is trained to satisfy a system of ordinary differential equations (ODEs) associated with a lumped parameter description of the circulatory system. The model predictions have a maximum error of less than 5% when compared to those obtained by solving the ODEs numerically. An inverse modeling approach using the PINN model is also developed to rapidly estimate model parameters (in ∼ 3 min) from single-beat LV pressure and volume waveforms. Using synthetic LV pressure and volume waveforms generated by the PINN model with different model parameter values, we show that the inverse modeling approach can recover the corresponding ground truth values for LV contractility indexed by the end-systolic elastance Ees with a 1% error, which suggests that this parameter is unique. The PINN inverse modeling approach is then applied to estimate Ees using waveforms acquired from 11 swine models, including waveforms acquired before and after administration of dobutamine (an inotropic agent) in 3 animals. The estimated Ees is about 58% to 284% higher for the data associated with dobutamine compared to those without, which implies that this approach can be used to estimate LV contractility using single-beat measurements. The PINN inverse modeling can potentially be used in the clinic to simultaneously estimate LV contractility and other physiological parameters from single-beat measurements.

Unraveling autonomic cardiovascular control complexity during orthostatic stress: Insights from a mathematical model

Understanding cardiovascular control mediated by the autonomic system remains challenging due to its inherent complexity. Consequently, syndromes such as orthostatic intolerance continue to evoke debates regarding the underlying pathophysiological mechanisms. This study develops a comprehensive mathematical model simulating the control of the sympathetic branch of the cardiovascular system in individuals with normal and abnormal responses to the head-up-tilt test. We recruited four young women: one control, one with vasovagal syncope, one with orthostatic hypertension, and one with orthostatic hypotension, exposing them to an orthostatic head-up tilt test (HUTT) employing non-invasive methods to measure electrocardiography and continuous blood pressure.Our work encompasses a compartmental model formulated using a system of ordinary differential equations. Using heart rate as input, we predict blood pressure, flow, and volume in compartments representing the veins, arteries, heart, and the sympathetic branch of the baroreflex control system. The latter is modulated by high- and low-pressure baroreceptor afferents activated by changes in blood pressure induced by the HUTT. Sensitivity analysis, parameter subset selection, and optimization are employed to estimate patient-specific parameters associated with autonomic performance. The model has seven sensitive and identifiable parameters with significant physiological relevance that can serve as biomarkers for patient classification.Results show that the model can reproduce a spectrum of blood pressure responses successfully, fitting the trajectory displayed by the experimental data. The controller exhibits behavior that emulates the operation of the sympathetic system. These encouraging findings underscore the potential of computational methods in evaluating pathologies associated with autonomic nervous system control, warranting further exploration and novel approaches.

Optimization of heat transfer in a channel with stretching walls using a magnetized tetra-hybrid nanofluid

Nanoparticles have numerous applications and are used frequently in different cooling, heating, treatment of cancer cells and manufacturing processes. The current investigation covers the utilization of tetra hybrid nanofluid (aluminium oxide, iron dioxide, titanium dioxide and copper) for Crossflow model over a vertical disk by considering the shape effects (bricks, cylindrical and platelet) of nanoparticles, electro-magneto-hydrodynamic effect and quadratic thermal radiation. The present study is devoted to present the mathematical formulation of the tetra-hybrid nanofluid flow in a porous channel with stretching/shrinking walls. The present inspection model is derived from given partial differential equations (PDEs) and then transformed into a system of ordinary differential equations (ODEs) by incorporating similarity variables. The transformed ODEs are solved using the bvp4c methodology, which yields numerical results. From the obtained results it is observed that the maximum amount of [Formula: see text] happens when there is slightly increase in [Formula: see text] with stretched wall that is [Formula: see text]. A high radiation value indicates a considerable contribution from radiative heat transfer, whereas stretching and shrinking increase and reduce the size of the boundary. The combined impact caused an increase in the Nusselt number. The graphical and tabular representations are used to explain the physical behaviour of various model parameters. Previous outcomes are also contrasted with the current outcomes.

Invariant solutions, lie symmetry analysis, bifurcations and nonlinear dynamics of the Kraenkel-Manna-Merle system with and without damping effect

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics.

Динамічна нестійкість при взаємодії композитної оболонки, виготовленої за адитивними технологіями, з газовим потоком

The dynamic instability of a composite cylindrical-conical thin-walled structure interacting with a supersonic gas flow is analyzed. This structure consists of three layers. The middle layer is manufactured by FDM additive technologies from ULTEM material. The top and bottom layers are manufactured from carbon-filled plastic. Free linear vibrations of the thin-walled structure are studied by Rayleigh?Ritz semi-analytical method to obtain a model of dynamic instability. The free linear vibrations are analyzed numerically. The obtained eigenfrequencies and eigenmodes are in close agreement with the data obtained by the ANSYS commercial software. The calculated eigenmodes were used to construct a model of composite cylindrical-conical shell instability. This model of instability is a system of ordinary linear differential equations in the generalized coordinates of the thin-walled structure. The supersonic gas flow is described by a piston theory, which accounts for the angle of attack. The study of the dynamic instability of the composite cylindrical-conical shell reduces to analyzing the trivial equilibrium instability of the system of ordinary differential equations. Characteristic exponents are calculated to analyze the stability of the trivial solution. These characteristic exponents are calculated from an eigenvalue problem. If the angle of attack is 12° and the Mach number is small, the minimal value of the critical pressure is observed for three circular waves. If the Mach number is increased, the minimal critical pressure is observed for four and five circular waves. If the angle of attack is 6° and the Mach number is small, the minimal critical pressure is observed for two circular waves. If the Mach number is increased, the minimal critical pressure is observed for three and four circular waves. The dynamic stability is lost for eigenmodes with a small number of circular waves

Identification of moment equations via data-driven approaches in nonlinear Schrödinger models

IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments.MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically.Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

Mathematical Model to Study the Effect of Refuge on Cannibalism in Atractosteus tropicus

Cannibalism is a behavior that different species of fish exhibit in the early stages of their life, and it has been widely reported. In Tabasco, Mexico, the ancestral species Atractosteus tropicus is farmed, which is a freshwater fish with a high nutritional and economic value. This species exhibits high cannibalistic behavior both in its larval and juvenile stages, which considerably decreases its production. Therefore, strategies have been developed to mitigate the effects of this behavior. One of them is the placement of shelters (rocks and artificial vegetation), which allow the vulnerable population to protect themselves from cannibals. The goal of this work is to study the effect of shelters on the cannibalistic behavior of the A. tropicus population through a mathematical model. The population is divided into two classes, the vulnerable population (prey) and the cannibal population (predator). Moreover, a system of ordinary differential equations is established, which is analyzed, and sufficient conditions for the coexistence of the two species are shown. Numerical simulations show coexistence by varying levels of refuge. The results obtained in this work can be applied to other populations that exhibit cannibalistic behavior.

Exploring nanoparticle dynamics in binary chemical reactions within magnetized porous media: a computational analysis

Artificial Neural Networks are incredibly efficient at handling complicated and nonlinear mathematical problems, making them very useful for tackling these challenges. Artificial neural networks offer a special computational architecture that is extremely valuable in disciplines like biotechnology, biological computing, and computational fluid dynamics. The present work investigates the applicability of back-propagation artificial neural networks in conjunction with the Levenberg-Marquardt algorithm for evaluating heat transmission in hybrid nanofluids. This work focuses on the computational analysis of a MgO + GO/EG hybrid nanofluid’s steady mixed convection flow over an exponentially stretched sheet, considering multiple slip boundary conditions, thermal conductivity, heat generation, and thermal radiation. A nonlinear system of ordinary differential equations is produced from the basic associated partial differential system by performing the proper exponential similarities modifications. For generating benchmark datasets, the resulting ordinary differential equations are processed employing the bvp4c method. Considering benchmark datasets set aside for training (70%), testing (15%), and validation (15%), the Levenberg-Marquardt algorithm, which employs back-propagation in artificial neural networks, is implemented. The accuracy of the suggested strategy for handling nonlinear problems is verified utilizing mean squared error, error histograms, and regression analysis, which are all used to evaluate the methodology. Outstanding agreement is seen when ANN outputs are compared to numerical results. The flow properties, including temperature, velocity, and concentration profiles, are shown graphically and numerically. For practical purposes, it is therefore essential to analyze the flow and heat transfer in hybrid nanofluids over exponentially extending and shrinking surfaces under mixed convection and heat source scenarios. Hybrid nanofluid problems have a wide range of practical and industrial applications, such as medication delivery, manufacturing, microelectronics, nuclear plant cooling, and marine engineering.

A mathematical model of glomerular fibrosis in diabetic kidney disease to predict therapeutic efficacy.

Glomerular fibrosis is a tissue damage that occurs within the kidneys of chronic and diabetic kidney disease patients. Effective treatments are lacking, and the mechanism of glomerular damage reversal is poorly understood. A mathematical model suitable for hypothesis-driven systems pharmacology of glomerular fibrosis in diabetes was developed from a previous model of interstitial fibrosis. The adapted model consists of a system of ordinary differential equations that models the complex disease etiology and progression of glomerular fibrosis in diabetes. Within the scope of the mechanism incorporated, advanced glycation end products (AGE)-matrix proteins that are modified due to high blood glucose-were identified as major contributors to the delay in the recovery from glomerular fibrosis after glucose control. The model predicted that inhibition of AGE production is not an effective approach for accelerating the recovery from glomerular fibrosis. Further, the model predicted that treatment breaking down accumulated AGE is the most productive at reversing glomerular fibrosis. The use of the model led to the identification that glucose control and aminoguanidine are ineffective treatments for reversing advanced glomerular fibrosis because they do not remove accumulated AGE. Additionally, using the model, a potential explanation was generated for the lack of efficacy of alagebrium in treating advanced glomerular fibrosis, which is due to the inability of alagebrium to reduce TGF- . Using the mathematical model, a mechanistic understanding of disease etiology and complexity of glomerular fibrosis in diabetes was illuminated, and then hypothesis-driven explanations for the lack of efficacy of different pharmacological agents for treating glomerular fibrosis were provided. This understanding can enable the development of more efficacious therapeutics for treating kidney damage in diabetes.