System Of ODEs Research Articles

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ2. The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley–Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.

Modeling Study of Factors Determining Efficacy of Biological Control of Adventive Weeds

We model the spatiotemporal dynamics of a community consisting of competing weed and cultivated plant species and a population of specialized phytophagous insects used as the weed biocontrol agent. The model is formulated as a PDE system of taxis–diffusion–reaction type and computer-implemented for one-dimensional and two-dimensional cases of spatial habitat for the Neumann zero-flux boundary condition. In order to discretize the original continuous system, we applied the method of lines. The obtained system of ODEs is integrated using the Runge–Kutta method with a variable time step and control of the integration accuracy. The numerical simulations provide insights into the mechanism of formation of solitary population waves (SPWs) of the phytophage, revealing the factors that determine the efficacy of combined application of the phytophagous insect (classical biological method) and cultivated plant (phytocenotic method) to suppress weed foci. In particular, the presented results illustrate the stabilizing action of cultivated plants, which fix the SPW effect by occupying the free area behind the wave front so that the weed remains suppressed in the absence of a phytophage.

Scrutinization using both numerical and analytical techniques for Darcy Forchheimer flow in the gyrotactic microorganism of nanofluid over a rotating disk

The current investigation explores the impact of Darcy Forchheimer’s flow of nanofluid due to a spinning disk with slip conditions and microorganisms. The mechanism of fluid flow over a rotating disk incorporates the theoretical as well as the practical interest of it in applied sciences and engineering. Energy and mass transport equations are produced with thermal radiation and chemical reactions. The modeled couple of nonlinear PDEs are simplified to the system of ODEs using the self-similar solution. Analytical solutions are found for the resulting nonlinear ODEs using the homotopy analysis method (HAM) technique, while numerical solutions are found using ND-solve and bvp4c. The convergent of the series solution is investigated. The graphical evaluation of prospective parameters encompasses the examination of their consequences on several factors, including the velocity profile, temperature profile, volume percentage of nanoparticles, and profile of microorganisms. The occurrence of radiation parameter, thermophoresis, and Brownian motion effects is commonly observed to contribute to the augmentation of heat transfer. We computed the percentage result in numerical as well as analytical study. The utilization of the provided observations can effectively contribute to the enhancement of heat transfer devices, centrifugal compressors, internal combustion engines, and microbial fuel cells.

Numerical Computations of Time-Dependent Auto-Catalytic Glycolysis Chemical Reaction-Diffusion System

A combined semi-discretized spectral matrix collocation algorithm based on (new) family of Krawtchouk polynomials is proposed to solve the time-dependent nonlinear auto-catalytic glycolysis reaction-diffusion system arising in mathematical chemistry. The first stage of the numerical algorithm is devoted to the Taylor series time advancement procedure yielding to a (linear and steady) system of ODEs. In the second stage and in each time frame, a matrix collocation technique based on the Krawtchouk polynomials is utilized to the resulting system of ODEs in an iterative manner. The results of the performed numerical experiments with Neumann boundary conditions are given to show the utility and applicability of the combined Taylor-Krawtchouk spectral collocation algorithm. The positive property of the glycolysis chemical model is sustained by the proposed algorithm and verified through comparisons with existing numerical methods in the literature. The combined technique is simple and flexible enough to easily produce the approximate solutions of diverse physical and applied models in engineering and science.

Group sparse Bayesian learning for data-driven discovery of explicit model forms with multiple parametric datasets

Extracting the explicit governing equations of a dynamic system from limited data has attracted increasing attention in the data-driven modeling community. Compared to black-box learning approaches, the sparse-regression-based learning method enables discovering an analytical model form from data, which is more appealing due to its white-box nature. However, distilling explicit equations from real-world measurements with data uncertainty is challenging, where many existing methods are less robust. Moreover, it is unclear how to efficiently learn a parametric system from multiple data sets with different parameters. This paper presents a group sparse Bayesian learning approaches to uncover the explicit model forms of a parametric dynamical system with estimated uncertainties. A deep neural network is constructed to improve the calculation of derivatives from noisy measurements. Group sparsity is leveraged to enable synchronous learning from a group of parametric datasets governed by the equations with the same functional form but different parameter settings. The proposed approach has been studied over a few linear/nonlinear ODE systems in explicit and implicit settings. In particular, a simplified parametric model of intracranial dynamics was identified from multiple synthetic datasets with different patient-specific parameters. The numerical results demonstrated the effectiveness of the proposed approach and the merit of synchronous learning from multiple datasets in a group sparsifying Bayesian setting.

On weak solvability of fractional models of viscoelastic high order fluid

In this paper, we establish the existence of a weak solution to the initial boundary value problem for the motion equations of viscoelastic incompressible fluid with constitutive law containing high-order fractional derivatives and with memory along the trajectories of the velocity field. The proof is by approximation of the original problem by a sequence of regularized problems followed by a passage to the limit based on appropriate a priori estimates. Methods of the theory of fractional derivatives calculus and the theory of regular Lagrangian flows (generalization of the classical solution of ODE systems) are used.

Event-Triggered Fuzzy Adaptive Stabilization of Parabolic PDE–ODE Systems

Event-Triggered Fuzzy Adaptive Stabilization of Parabolic PDE–ODE Systems

Characteristic Analysis of Soret and Corolis Forces on a Natural Convection in a Finite Cavity with Isotropic and Anisotropic Permeable Media

Using 3D transmission in a definite cavity with anisotropic and isotropic permeable media rotating at a fixed rotational velocity, the Rayleigh-Benard issue for a viscous, unstable, laminar, incompressible fluid heated from below a horizontal layer is extended in this paper's research. Seven controlling PDEs from the given physical configuration are similarly transformed to produce a system of non-dimensional ODEs. The Rayleigh, Taylor, and Prandtl numbers are examined for their impacts on temperature gradient and velocity in both isotropic and anisotropic conditions using the Fourier series approach. It has been discussed and determined that the results of the stream function and isotherms on a variety of factors are good.

Numerical treatment of the radiated and dissipative power-law nanofluid flow past a nonlinear stretched sheet with non-uniform heat generation

The main aim of this paper is to investigate the effect of non-uniform heat generation and viscous dissipation on the boundary layer flow of a power-law nanofluid over a nonlinear stretching sheet. Within the thermal domain, the analysis considers both thermal radiation and variable thermal conductivity. Through the use of similarity transformations, the governing boundary layer equations are transformed into a system of ODEs. The spectral collocation method (SCM) with shifted Vieta-Lucas polynomials (VLPs) is implemented to give an approximate expression for the derivatives and then use it to numerically solve the proposed system of equations. By employing this technique, the system of ODEs is converted into a system of nonlinear algebraic equations. The dimensionless temperature, concentration, and velocity are graphically presented and analyzed for various values of the relevant governing parameters. Through the presented graphical solutions, we can see that the main outcomes indicate that an increase in the power law index, thermal conductivity parameter, and radiation parameter leads to a noticeable decrease in the local Nusselt number, with reductions of around 0.05 percent, 0.23 percent, and 0.11 percent, respectively. In contrast, the Prandtl parameter demonstrates an opposing effect, elevating the local Nusselt number by about 0.1 percent. We validated the accuracy of the numerical solutions by comparing them in some special cases with existing literature.

Regime-switching affine term structures

We consider an HJM model setting for Markov-chain modulated forward rates. The underlying Markov chain is assumed to induce regime switches on the forward curve dynamics. Our primary focus is on the interest rate and energy futures markets. After deriving HJM-drift conditions for the two markets, we prove under the assumption of affine structure for the term structure that the forward curves are solutions to specific systems of ODEs that can be solved explicitly in many cases. This allows for a tractable model setting, and we present an algorithm for obtaining consistent forward curve models within our framework. We conclude by presenting some numerical examples and an application to real data under a one-factor Gaussian model setting.

Neuro-Heuristic Computational Intelligence Approach for Optimization of Electro-Magneto-Hydrodynamic Influence on a Nano Viscous Fluid Flow

In this investigative study, the electro-magneto hydrodynamic (EMHD) influence on a nano viscous fluid model is scrutinized by designing an artificial neural network (ANN) paradigm using a neuro-heuristic approach (NHA) through the combination of GAs (genetic algorithms) and one of the most efficient locally searching solver SQP (sequential quadratic programming), i.e., NHA-GA-SQP. The fluid flow for the proposed problem is initially interpreted in the form of PDEs and then utilization of suitable similarity transformation on these PDEs yields in terms of a stiff nonlinear system of ODEs. The numerical results of the suggested fluidic model based on the variation of its physically existing parameters are calculated through the NHA-GA-SQP solver to detect the variation in velocity, thermal gradient, and concentration during the fluid flow. A detailed analysis of obtained outcomes through the NHA-GA-SQP algorithm and their comparison with the reference results estimated via the Adams method are presented. The calculation of the proposed solver’s accuracy, stability, and consistency through various statistical operators is also involved in the current inspection.

Intelligent computing technique to study heat and mass transport of Casson nanofluidic flow model on a nonlinear slanted extending sheet

AbstractThis paper explains the importance and benefits of using AI in fluid mechanics, emphasizing its capability of finding the solution of fluid flow systems through iterative optimization techniques. It explores how AI can enhance the understanding of fluid dynamics to improve engineering processes. In the presented studies, the heat and mass transport of the Casson nanofluidic flow model on a nonlinear slanted extending sheet (HMT‐CNFM) via AI‐based Levenberg Marquard methodology with backpropagated trained neural networks (TNN‐BLMM) is studied. By applying an appropriate transformation, the governing PDEs representing HMT‐CNFM are converted into a system of nonlinear ODEs. The dataset for Levenberg Marquard methodology with backpropagated trained neural network (TNN‐BLMM) for all six scenarios of this proposed model via computational power of the Lobatto IIIA scheme using the “bvp4c” package in MATLAB and then graphically illustrate all these six scenarios through nftool to attain mean square error, regression, error histogram, performance, and fit curve. Training, testing, and validation processes of NN‐BLMM are organized for the investigation of the HMT‐CNFM model.

A Functional Approach to Interpreting the Role of the Adjoint Equation in Machine Learning

The connection between numerical methods for solving differential equations and machine learning has been revealed recently. Differential equations have been proposed as continuous analogues of deep neural networks, and then used in handling certain tasks, such as image recognition, where the training of a model includes learning the parameters of systems of ODEs from certain points along their trajectories. Treating this inverse problem of determining the parameters of a dynamical system that minimize the difference between data and trajectory by a gradient-based optimization method presents the solution of the adjoint equation as the continuous analogue of backpropagation that yields the appropriate gradients. The paper explores an abstract approach that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times.

Analytical solution of an Ill-posed system of nonlinear ODE’s

In this paper, we discuss a solution strategy for an overdetermined ODE system that uses elliptic functions. In particular, we show how an apparently ill-posed ODE system can be made amenable to treatment by Jacobian elliptic functions by introducing additional free parameters and still obtain a solution that corresponds well to the physically expected behaviour.

The Traffic Reaction Model: A kinetic compartmental approach to road traffic modeling

In this work, a family of finite volume discretization schemes for LWR-type first order traffic flow models (with possible on- and off-ramps) is proposed: the Traffic Reaction Model (TRM). These schemes yield systems of ODEs that are formally equivalent to the kinetic systems used to model chemical reaction networks. An in-depth numerical analysis of the TRM is performed. On the one hand, the analytical properties of the scheme (nonnegative, conservative, capacity-preserving, monotone) and its relation to more traditional schemes for traffic flow models (Godunov, CTM) are presented. Finally, the link between the TRM and kinetic systems is exploited to offer a novel compartmental interpretation of traffic models. In particular, kinetic theory is used to derive dynamical properties (namely persistence and Lyapunov stability) of the TRM for a specific road configuration. Two extensions of the proposed model, to networks and changing driving conditions, are also described.

The convergence rate of the equilibrium measure for the hybrid LQG Mean Field Game

In this work, we study the convergence rate of the N-player Linear-Quadratic-Gaussian (LQG) game with a Markov chain common noise towards its asymptotic Mean Field Game. By postulating a Markovian structure via two auxiliary processes for the first and second moments of the Mean Field Game equilibrium and applying the fixed point condition in Mean Field Game, we first provide the characterization of the equilibrium measure in Mean Field Game with a finite-dimensional Riccati system of ODEs. Additionally, with an explicit coupling of the optimal trajectory of the N-player game driven by N dimensional Brownian motion and Mean Field Game counterpart driven by one-dimensional Brownian motion, we obtain the convergence rate O(N−1/2) with respect to 2-Wasserstein distance.

Application of machine learning in the investigation of entropy generation incurred during electroosmotic flows of Casson nanofluid

The present research article delves into the application of machine learning techniques in the study of entropy generation arising from the electro‐MHD Casson nanofluid flow in a symmetric microchannel. The inertial effect on fluid flow through a non‐Darcy porous medium is considered where we have taken into account the velocity, thermal, and concentration slips at boundaries. The resulting system of PDEs is converted to system of ODEs using appropriate wave frame transformations. The metamorphosed system is solved using bvp4c routine of MATLAB. The impact of several factors represented by respective parameters on velocity, nanofluid temperature, fraction of nanoparticles, entropy generation, and Bejan number are shown through graphs. This study also employs a machine learning technique in which Particle Swarm Optimization (PSO) algorithm has been used in conjunction with an Artificial Neural Network (ANN) to develop a predictive model capable of optimizing entropy generation within this specific fluidic context. Such an analysis could be useful in determining the optimum entropy generation in pulsatile transport of physiological fluid through intestine. With the use of above mentioned machine learning technique, the considered model offers the minimum entropy generation for specific values of the electroosmotic parameter, dimensionless temperature difference, and velocity slip parameter. It is observed that the flow with minimum electroosmosis effect and maintained at a specific velocity slip gives the optimal entropy generation.

Constructing a Precise Method to Control Non-Linear Systems Employing Special Functions and Machine Learning

This paper is about finding the best way to control certain types of equations that describe heat and diffusion using methods that are not exact but close to the best solution. A new way to control things better is suggested using patterns we've seen before and computer learning. By employing the data gathered from the EEFs are computed using the Karhunen-Loève decompose in the PDE computer. The EEFs are used to convert the PDE system into a high-order ODE system. A smaller model (ROM) that may depict the primary behavior of the PDE subsystem is created using the SP approach. This facilitates understanding of the system. In order to reduce its size, a more basic model (ROM) that demonstrates the primary motions of the PDE system is created using the SP approach. Next, the optimal controller for the ROM is designed using the HJB technique. This ensures that the high-order ODE system is stable using the SP theory. By splitting the best way to control something into two sections, we can solve a math problem to figure out the first part and come up with a new equation for the second part. Third, a method is suggested for updating and controlling based on small improvements to solve a specific equation, and it has been proven to work effectively. Moreover, we use a technique called the NN approach to estimate the cost function. Finally, the simulation results demonstrate the effectiveness of the proposed optimum control approach when applied to a diffusion-reaction process using a unique spatially component.

A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.