Differential Equation Model Research Articles

Solving a non-local linear differential equation model of the Newtonian-type

Solving a non-local linear differential equation model of the Newtonian-type

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences

This study presents the first comprehensive numerical simulation of heat and mass transfer in fractal-like mixed convective nanofluid flows. The flow of non-Newtonian nanofluids over flat and oscillating sheets is modelled mathematically, and a finite difference scheme is used to solve this model. The two-stage scheme can tackle fractal and fractal stochastic mathematical models of partial differential equations. The consistency in the mean square is proved, and Fourier series stability analysis is adopted to find stability conditions for fractal stochastic partial differential equation. The scheme is applied to solve the unsteady Casson nanofluid flow over the flat and oscillatory sheet, which affects thermal radiation, heat source, and chemical reaction. The existence of the solution is also provided for the Navier-Stokes equation of the considered flow model using fractal time derivative. The graph illustrates that the proposed fractal technique achieves faster convergence than the Crank-Nicolson approach. Applications in energy systems, materials science, and environmental engineering are just a few of the domains that could benefit from a better understanding of mixed convective nanofluid flows with fractal features, and that is what this research study hopes to accomplish. Scientists and engineers may better develop efficient and environmentally friendly systems by simulating and analyzing these complicated processes with the suggested finite difference technique.

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion.

Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.

RBF neural network disturbance observer-based backstepping boundary vibration control for Euler–Bernoulli beam model with input saturation

The main objective of this paper is to address the issue of vibration control for a class of Euler–Bernoulli beam systems that are subject to external disturbances and input saturation. The proposed controller differs from other backstepping methods in that it employs a radial basis function (RBF) neural network to accurately estimate boundary disturbances and incorporates the hyperbolic tangent function to ensure input constraints. The nonlinear partial differential equation (PDE) model is initially derived based on Hamilton’s principle to capture the dominant dynamic characteristics of the flexible beam. In the framework of the Lyapunov direct approach, an adaptive RBF neural network-based law is subsequently designed to estimate the state-related boundary disturbances. The backstepping approach is then developed to propose sufficient conditions for ensuring the stability and convergence of closed-loop systems subject to input saturation. Finally, the effectiveness and superiority of the proposed methodology are further demonstrated by comparing the simulation results of constrained backstepping controllers.

African swine fever in wild boar: investigating model assumptions and structure.

African swine fever (ASF) is a highly virulent viral disease that affects domestic pigs and wild boar. Current ASF transmission in Europe is in part driven by wild boar populations, which act as a disease reservoir. Wild boar are abundant throughout Europe and are highly social animals with complex social organization. Despite the known importance of wild boar in ASF spread and persistence, knowledge gaps remain surrounding wild boar transmission. We developed a wild boar modelling framework to investigate the influence of contact-density functions and wild boar social structure on disease dynamics. The framework included an ordinary differential equation model, a homogeneous stochastic model and various network-based stochastic models that explicitly included wild boar social grouping. We found that power-law functions (transmission density0.5) and frequency-based contact-density functions were best able to reproduce recent Baltic outbreaks; however, power-law function models predicted considerable carcass transmission, while frequency-based models had negligible carcass transmission. Furthermore, increased model heterogeneity caused a decrease in the relative importance of carcass-based transmission. The transmission pathways predicted by each model type affected the efficacy of idealized interventions, which highlights the importance of evaluating model type and structure when modelling systems with significant uncertainties.

영재 사사과정에서 공학도구를 활용한 미적분 및 미분방정식 탐구 지도 사례

The purpose of this study is to present a year-long instructional case study of middle school mathematically gifted students in Science Gifted Education Center using technological devices to explore mathematical models based oncalculus and differential equation, and to draw implications for gifted education. In the classes conducted in this study, technological devices were used to compute expressions and visualize graphs. The use of technological devices reduced the amount of time spent on calculations and graphing, allowing more time for interpretation and discussion of models. As a result, students understood the calculus and differential equation models and interpreted the models in various ways using graphs based on abundant discussion time. The results of this study show that it is possible to explore calculus and differential equation models at the middle school level by using technological devices. Furthermore, visualization with technological devices provides opportunities for observation and discovery of mathematics and allows for plentiful discussion and interpretation by securing time for interaction among students, so there is a need for more active use of technological devices in gifted classes.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Multilevel Latent Differential Structural Equation Model with Short Time Series and Time-Varying Covariates: A Comparison of Frequentist and Bayesian Estimators

Continuous-time modeling using differential equations is a promising technique to model change processes with longitudinal data. Among ways to fit this model, the Latent Differential Structural Equation Modeling (LDSEM) approach defines latent derivative variables within a structural equation modeling (SEM) framework, thereby allowing researchers to leverage advantages of the SEM framework for model building, estimation, inference, and comparison purposes. Still, a few issues remain unresolved, including performance of multilevel variations of the LDSEM under short time lengths (e.g., 14 time points), particularly when coupled multivariate processes and time-varying covariates are involved. Additionally, the possibility of using Bayesian estimation to facilitate the estimation of multilevel LDSEM (M-LDSEM) models with complex and higher-dimensional random effect structures has not been investigated. We present a series of Monte Carlo simulations to evaluate three possible approaches to fitting M-LDSEM, including: frequentist single-level and two-level robust estimators and Bayesian two-level estimator. Our findings suggested that the Bayesian approach outperformed other frequentist approaches. The effects of time-varying covariates are well recovered, and coupling parameters are the least biased especially using higher-order derivative information with the Bayesian estimator. Finally, an empirical example is provided to show the applicability of the approach.

Application of Grey Prediction Models for Efficient Traffic Flow Prediction

Traffic flow prediction is crucial in transportation management and optimizing control strategies and infrastructure planning. However, traditional prediction methods often need help handling traffic data's inherent uncertainty and complexity. This paper explores the application of Grey prediction models as a promising approach for efficient traffic flow prediction. Grey prediction models, rooted in Grey systems theory, offer a robust framework for modelling systems with incomplete and uncertain information. The paper provides a comprehensive overview of Grey prediction models, including the Grey Verhulst, Grey-Markov, and Grey differential equation models. It discusses the process of data collection, pre-processing, and model application in traffic flow prediction. Case studies and experimental results demonstrate the effectiveness of Grey prediction models in accurately forecasting traffic flow patterns. Despite challenges such as data sparsity and model complexity, Grey prediction models offer significant advantages over traditional methods, including improved prediction accuracy and adaptability to dynamic traffic conditions. The paper concludes with recommendations for practitioners and researchers interested in leveraging Grey prediction models for efficient traffic management.

Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition

POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

$$(\mu _{1},\mu _{2})$$-Pseudo-asymptotically $$\tau $$-periodic function applied to models for differential equations with delays

$$(\mu _{1},\mu _{2})$$-Pseudo-asymptotically $$\tau $$-periodic function applied to models for differential equations with delays

Network model of skeletal muscle cell signalling predicts differential responses to endurance and resistance exercise training.

Exercise-induced muscle adaptations vary based on exercise modality and intensity. We constructed a signalling network model from 87 published studies of human or rodent skeletal muscle cell responses to endurance or resistance exercise in vivo or simulated exercise in vitro. The network comprises 259 signalling interactions between 120 nodes, representing eight membrane receptors and eight canonical signalling pathways regulating 14 transcriptional regulators, 28 target genes and 12 exercise-induced phenotypes. Using this network, we formulated a logic-based ordinary differential equation model predicting time-dependent molecular and phenotypic alterations following acute endurance and resistance exercises. Compared with nine independent studies, the model accurately predicted 18/21 (85%) acute responses to resistance exercise and 12/16 (75%) acute responses to endurance exercise. Detailed sensitivity analysis of differential phenotypic responses to resistance and endurance training showed that, in the model, exercise regulates cell growth and protein synthesis primarily by signalling via mechanistic target of rapamycin, which is activated by Akt and inhibited in endurance exercise by AMP-activated protein kinase. Endurance exercise preferentially activates inflammation via reactive oxygen species and nuclear factor κB signalling. Furthermore, the expected preferential activation of mitochondrial biogenesis by endurance exercise was counterbalanced in the model by protein kinase C in response to resistance training. This model provides a new tool for investigating cross-talk between skeletal muscle signalling pathways activated by endurance and resistance exercise, and the mechanisms of interactions such as the interference effects of endurance training on resistance exercise outcomes.

Finite element model for partial differential equations in optimization of vehicle safety collision problem

Vehicle safety on roadsides is vital for preventing collisions, controlling failures and accidents, and ensuring driver and passenger wellness. Finite Element Analysis (FEA) in vehicle safety relies on the vehicle’s physical attributes for predicting and preventing collisions. This article introduces a Differential FEA (DFEA) model for predicting vehicle collisions regardless of the speed and direction for driver/ passenger safety. The proposed model induces a vehicle’s speed, direction, and displacement from two perspectives: self and approaching vehicle. The collision probability with the trailing or persuading vehicle is calculated by distinguishing the forward and rear features. The differential calculus for the point of deviation and distance metrics are significantly estimated for a vehicle’s front and rear ends. Such calculus generates a maximum and minimum possibility for self and approaching vehicle contact. This contact is further split based on the collision threshold; the threshold is determined using the safe distance between two vehicles for collision-less driving. The threshold exceeding vehicles are alerted for their change in direction/ speed through contact point (rear/front) differential derivatives. This ensures collision detection under fewer contact errors, leveraging the safety of the duo vehicles.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Adaptive composite learning control of a flexible two‐link manipulator with unknown spatiotemporally varying disturbance

AbstractThis article presents a novel adaptive composite learning (ACL) control strategy combining reinforcement learning and a disturbance observer (DOB) to address vibration issues in a flexible two‐link manipulator (FTLM) system affected by unknown spatiotemporally varying disturbances. Based on the assumed mode method, the FTLM system is initially transformed into an ordinary differential equation model, while effectively capturing the elastic deformation and vibration characteristics of the flexible link. A composite learning controller, based on the actor‐critic algorithm and DOB, is then developed to achieve trajectory tracking and vibration suppression in the FTLM system. The DOB in the controller compensates for unknown disturbances resulting in reduced system error. It is noting that the proposed optimal control strategy is continuously gathering system experience and evaluating the current policy's effectiveness. The stability and robustness of the closed‐loop system incorporating the composite controller are analyzed using Lyapunov's direct method, and the semi‐global uniform ultimate boundedness of the tracking and vibration errors are also demonstrated. To validate the effectiveness and superiority of the proposed ACL controller, comparative simulations and experiments are conducted on the Quanser experimental platform.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

The Block-Correlated Pseudo Marginal Sampler for State Space Models

Pseudo Marginal Metropolis-Hastings (PMMH) is a general approach to Bayesian inference when the likelihood is intractable, but can be estimated unbiasedly. Our article develops an efficient PMMH method that scales up better to higher dimensional state vectors than previous approaches. The improvement is achieved by the following innovations. First, a novel block version of PMMH that works with multiple particle filters is proposed. Second, the trimmed mean of the unbiased likelihood estimates of the multiple particle filters is used. Third, the article develops an efficient auxiliary disturbance particle filter, which is necessary when the bootstrap disturbance filter is inefficient, but the state transition density cannot be expressed in closed form. Fourth, a novel sorting algorithm, which is as effective as previous approaches but significantly faster than them, is developed to preserve the correlation between the logs of the likelihood estimates at the current and proposed parameter values. The performance of the sampler is investigated empirically by applying it to nonlinear Dynamic Stochastic General Equilibrium models with relatively high state dimensions and with intractable state transition densities and to multivariate GARCH diffusion-driven volatility in the mean models. Although we only apply the method to state space models, the approach will be useful in a wide range of applications such as large panel data models and stochastic differential equation models with mixed effects.

Building erudition in the wound healing process: an inflammation model analysis

The main four stages of wound healing are hemostasis, inflammation, proliferation, and remodelling. In the inflammation stage, the body rids of pathogens and debris, and prepares for tissue regrowth. An ordinary differential equation model of the inflammation is constructed that expands on previous models and incorporates dynamics reported in the updated literature. The interaction between debris and apoptotic neutrophils, along with classical and alternative macrophages are included. In all, the model incorporates pathogens, debris, apoptotic neutrophils, neutrophil, M1 macrophages, and M2 macrophages. Parameter values are analysed based on 1. experimental data and 2. reported ranges and expected peak times of white blood cells. Next, global sensitivity analyses which is able to test change in parameter values in tandem is conducted to interpret the overall influence each parameter has on white blood cell, debris, and pathogen output.

Advancing real-time drilling trajectory prediction with an efficient nonlinear DDE model and online parameter estimation

This study introduces a streamlined approach to predict planar drilling trajectory by employing an efficient nonlinear Delay Differential Equation (DDE) model with high computational speed and predictive accuracy. The efficient model is validated and compared with other models by utilizing not just data from existing literature but also two sets of experimental data with two distinct rigs. Following the validation, the model is integrated with an online estimation technique—calibrating one parameter in real-time using historical data—empowering it to forecast drill bit paths with remarkable fidelity. The validation of the model and its estimation strategy underscores their potential for application in real-time drilling control systems, setting a new benchmark for future advancements in the field.