System Of Linear Differential Equations Research Articles

A global higher regularity result for the static relaxed micromorphic model on smooth domains

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\operatorname {div};\Omega )\cap H_0(\operatorname {curl};\Omega )$ into $H^1(\Omega )$ .

ОЦІНЮВАННЯ ВПЛИВУ ЗАСОБІВ ПІДКЛЮЧЕННЯ РЕЗЕРВУ НА ПОКАЗНИКИ НАДІЙНОСТІ СИСТЕМИ ПЕРЕДАВАННЯ ДАНИХ

Structural redundancy is included in modern data transmission systems to ensure the specified reliability indicators. Structural redundancy gives the system fault tolerance and significantly increases its reliability. To manage the redundant resource, it is necessary to have additional tools that will connect the redundant equipment to the main configuration of the data transmission system. The redundant equipment is a separate technical means and, from the point of view of reliability, it is connected in series with other modules of the data transmission system. Thus, the inclusion of redundant connection facilities in the data transmission system reduces its reliability. In addition, even with perfect reliability, redundant connections may fail or malfunction and, under certain conditions, cause the transmission system to fail. To determine the impact of reserve connection facilities on the reliability of the transmission system, the article applies an original technology for building models of discrete-continuous stochastic systems. ASNA software was used as a software tool for modeling the reliable behavior of the data transmission system. This software provides automated construction of the model in the form of a graph of states and transitions based on a structural automaton model. Based on the graph of states and transitions, an analytical model is automatically generated in the form of a system of linear differential Kolmogorov-Chapman equations. Using the developed model, the article quantitatively analyzes the impact of redundant connection means (switches) on the reliability indicators of the data transmission system. In the study, the proposed models, unlike the existing ones, take into account both reliability indicators (probability of failure) and functionality indicators (probability of switching, speed) of the reserve connection means. On the basis of the developed model, the limit values of reliability and functionality of the redundant connection means (switches) were determined, at which they will minimize the reliability of the fault-tolerant data transmission system. Based on the analysis of the obtained results, practical recommendations for the selection of reliability indicators of the backup connection means, their speed and probability of switching are formed.

Volatility estimation of hidden Markov processes and adaptive filtration

The partially observed linear Gaussian system of stochastic differential equations with low noise in observations is considered. The coefficients of this system are supposed to depend on some unknown parameter. The problem of estimation of these parameters is considered and the possibility of the approximation of the filtering equations is discussed. An estimators are used for estimation of the quadratic variation of the derivative of the limit of the observed process. Then this estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component. This estimator is also used for construction of method of moments estimators in the case where the drift in observable component and the volatility of the state component depend on some unknown parameter. Then this method of moments estimator and Fisher-score device allow us to introduce the One-step MLE-process and adaptive Kalman–Bucy filter. The asymptotic efficiency of the proposed filter is discussed.

Reduced-order modeling on a near-term quantum computer

Quantum computing is an advancing area of research in which computer hardware and algorithms are developed to take advantage of quantum mechanical phenomena. In recent studies, quantum algorithms have shown promise in solving linear systems of equations as well as systems of linear ordinary differential equations (ODEs) and partial differential equations (PDEs). Reduced-order modeling (ROM) algorithms for studying fluid dynamics have shown success in identifying linear operators that can describe flowfields, where dynamic mode decomposition (DMD) is a particularly useful method in which a linear operator is identified from data. In this work, DMD is reformulated as an optimization problem to propagate the state of the linearized dynamical system on a quantum computer. This reformulation was chosen as a means of facilitating implementation on a near-term quantum computer. Quadratic unconstrained binary optimization (QUBO), a technique for optimizing quadratic polynomials in binary variables, allows for quantum annealing algorithms to be applied. A quantum circuit model (quantum approximation optimization algorithm, QAOA) is utilized to obtain predictions of the state trajectories. Results are shown for the quantum-ROM predictions for flow over a 2D cylinder at Re = 220 and flow over a NACA0009 airfoil at Re = 500 and α=15∘. The quantum-ROM predictions are found to depend on the number of bits utilized for a fixed point representation and the truncation level of the DMD model. Comparisons with DMD predictions from a classical computer algorithm are made, as well as an analysis of the computational complexity and prospects for future, more fault-tolerant quantum computers.

Double capture of coordinated evaders in recurrent differential games

In finite-dimensional Euclidean space, a problem of pursuit of two evaders by a group of pursuers, which is described by a linear nonstationary system of differential equations, is considered under the assumption that the fundamental matrix of the homogeneous system is a recurrent function. It is assumed that the evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and the prehistory of the control of the evaders. The set of admissible controls is a strictly convex compact with a smooth boundary, and the goal sets are the origin of coordinates. The goal of the group of pursuers is to catch at least one evader by two pursuers. In terms of initial positions and parameters of the game, a sufficient condition for capture is obtained. This study is based on the method of resolving functions, which makes it possible to obtain sufficient conditions for solvability of the problem of pursuit in some guaranteed time.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

A Data–Physics-Driven Modeling Approach of Key Equipment for Large-Scale Distribution Network Simulation

Fueled by pressing global climate concerns, the integration of large-scale renewable distributed generation sources, including distributed wind power and photovoltaics, along with electricity substitution loads into the distribution network has been accelerated to diminish carbon emissions. This shift introduces significant challenges and necessitates the advanced operation and control of distribution systems to accommodate these changes effectively. Against this backdrop, there is a growing expectation for an open and scalable central control mode, equipped with compatible interfaces, to offer a visionary development platform for the grid. This platform is anticipated to meet the evolving needs of future distribution system development, ensuring adaptability and forward compatibility. The aforementioned platform requires open, scalable, and interface-compatible models of key distribution network equipment as its foundation. To address the challenges presented, this paper proposes a data–physics-driven modeling approach for automating simulations in distribution systems. This method employs a simplified and standardized system of linear differential equations with undetermined coefficients to capture the common physical characteristics of specific device types. The models designed through this approach are notably open, allowing for real-time data to refine undetermined coefficients and accurately depict the dynamic behavior of equipment over various periods. Their scalability also stands out, rendering them apt for large-scale distribution network simulations. The paper elaborates on models for distributed photovoltaic, wind turbine, energy storage, and electric vehicle, and demonstrates their application within an IEEE-33 node distribution network topology built on Python.

A numerical study on fractional order financial system with chaotic and Lyapunov stability analysis

In the last few decades, academic research has focused more on financial problems and poverty levels. These are among the two major challenges of the modern world today. To understand the challenge of financial crisis and poverty in societies. This paper explains a deterministic financial system of non linear differential equations. Which aim to provide recommendations to address the twin challenges. In this fractional chaotic model, memory and chaos are portrayed as effects on a system. Financial systems of fractional order have three compartments: interest rate x(t), investment demand rate y(t) and price indexes rate z(t). For the first time, the Legendre wavelet approximation is applied to fractional order financial systems. A major objective of this process is to convert fractional differential equations (FDEs) into algebraic equations by using Legendre wavelets and their fractional integral operators. For residual analysis, convergence and stability analysis, we have used iteration methods and Legendre wavelets. This study indicates good results between the approximate and numerical solutions and that this method is efficient and accurate. Using the iteration method, we prove that the solution exists, while the contraction theory proves its uniqueness. The findings of this study will be extremely useful for future financial management. We have demonstrated the applicability and effectiveness of these methods by analysing numerical simulations for the fractional order financial model and the numerical simulation has been calculated by MATLAB programming.

Searching for Laurent Solutions of Truncated Systems of Linear Differential Equations with the Use of EG-Eliminations

Searching for Laurent Solutions of Truncated Systems of Linear Differential Equations with the Use of EG-Eliminations

A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion

This paper presents the computational solutions of a time-dependent nonlinear system of partial differential equations (PDEs) known as the Lotka-Volterra competition system with diffusion. We propose a combined semi-discretized spectral matrix collocation algorithm to solve this system of PDEs. The first part of the algorithm deals with the time-marching procedure, which is performed using the well-known Taylor series formula. The resulting linear systems of ordinary differential equations (ODEs) are then solved using the spectral matrix collocation technique based on the novel Touchard family of polynomials. We discuss and establish the error analysis and convergence of the proposed method. Additionally, we examine the stability analysis and the equilibrium points of the model to determine the stability condition for the system. We perform numerical simulations using diverse model parameters and with different Dirichlet and Neumann boundary conditions to demonstrate the utility and applicability of our combined Taylor-Touchard spectral collocation algorithm.

WAYS OF INCREASING OF LOADING CAPACITY OF THE REINFORCED CONCRETE SHALLOW SHELLS WITH A FLAT RECTANGULAR CONTOUR

Shallow shells with variable curvature, prestressing contour and fixed above by horizontal reinforcing of two opposite contour elements are considered in the given article. These two offered ways practically produces non-moment state and increase loading capacity of the shallow shells several times. Method of definition of loading capacity of the shallow shells with a flat rectangular contour was worked out, and it's aim consists of that initial moment state before loss of loading capacity is described by nonlinear differential equations of the theory of the shallow shells. And then accentuating the main part of the solution of equations we receive the linear system of homogeneous differential equations with variable coefficients. The solution of the linearizated equations enables to define the forms of losing of loading capacity of shallow shells.

Formulation of physical fields from systems of first order linear partial differential equations

We present a heuristic approach to formulate physical fields from systems of first order linear partial differential equations by considering the symmetric and antisymmetric properties of the matrices that represent the systems of equations. Particularly, we show that Maxwell and Dirac fields can be established by imposing different types of symmetric and antisymmetric conditions on their associated representing matrices.

Estimation of the Impact of Vaccination Intervention on Recovered Coronavirus Patients

This work estimated the impact of vaccination intervention on coronavirus patients who have recovered from the disease and the vulnerability index of the recovered population due to the impact of vaccine was also investigated. This work adopted a numerical solution to study the continuous dynamical system of linear first order differential equations describing a SEIR (Susceptible, Exposed, Infected, Recovered) model on the spread of Coronavirus Disease – 2019 (COVID-19). To tackle this problem, MATLAB ordinary differential equation of order 45 (ODE45) numerical method was adopted for the analysis. The vulnerability index of the recovered population was low due to the impact of vaccine meaning that the recovered population will gain immunity and they will not be re-infected. The study recommended that coronavirus patients who have recovered from the disease should ensure that they have vaccination administered to them to avoid re-occurrence of the virus attack as an intervention strategy.

Simulation of exchange processes in multi-component environments with account of data uncertainty

The goal of the work. Proposals for methods of solving systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients that defined in interval form and intended for modeling exchange processes in multicomponent environments. Research subject: systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients defined in interval form. Research method: interval analysis. The obtained results. Systems of linear homogeneous and non-homogeneous differential equations, which are used in modeling exchange processes in multicomponent environments, are considered. Such systems can be considered, for example, in problems of chemical kinetics, materials science, and the theory of Markov processes. To obtain the solution of these equations, specialized calculators of analytical transformations were used and tested. The Matlab system (ode15s solver) was used for numerical analysis of systems of differential equations. It is shown that the application of interval methods of numerical analysis at the initial stage of system modeling has some advantages over probabilistic methods because they do not require knowledge of the laws of distribution of the results of the system state parameter measurements and their errors. It is shown that existing methods of solving systems of linear differential equations can be divided into two groups. Common to these groups is the use of interval expansion of classical methods for solving differential equations given in interval form. The difference between these two groups of methods is as follows. The methods of the first group can be used for all types of differential equations but require the creation of special software. The peculiarity of the methods of the second group is that they can be used to solve equations analytically or using numerical analysis packages. The application of the methods of the second group is shown on the example of solving a system of differential equations, the coefficients of which are determined in interval form. The system of these equations is intended for modeling the processes of exchange with the external environment of the elements of the model of a specific physical system. In the case when the coefficients of these equations are variables, their piecewise-constant approximation is applied and a criterion that determines the possibility of its application is given. The technique proposed in the paper can be applied to solve systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients if they are given by slowly varying functions. In the case when the coefficients of the equations are determined in the interval form, the technique allows obtaining their solution also in the interval form and does not require the creation of special software.

Discord in the voter model for complex networks.

Online social networks have become primary means of communication. As they often exhibit undesirable effects such as hostility, polarization, or echo chambers, it is crucial to develop analytical tools that help us better understand them. In this paper we are interested in the evolution of discord in social networks. Formally, we introduce a method to calculate the probability of discord between any two agents in the multistate voter model with and without zealots. Our work applies to any directed, weighted graph with any finite number of possible opinions, allows for various update rates across agents, and does not imply any approximation. Under certain topological conditions, the opinions are independent and the joint distribution can be decoupled. Otherwise, the evolution of discord probabilities is described by a linear system of ordinary differential equations. We prove the existence of a unique equilibrium solution, which can be computed via an iterative algorithm. The classical definition of active links density is generalized to take into account long-range, weighted interactions. We illustrate our findings on real-life and synthetic networks. In particular, we investigate the impact of clustering on discord and uncover a rich landscape of varied behaviors in polarized networks. This sheds lights on the evolution of discord between, and within, antagonistic communities.

Arbitrary coefficient assignment to the characteristic function of the scalar type for linear systems with state delays by linear static delayed output feedback

We consider a control system defined by a system of linear differential equations in real or complex space with multiple lumped and distributed delays in the state variable. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function of the scalar type by linear static delayed output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for solvability of the arbitrary coefficient assignment problem by the static delayed output feedback controller when the system coefficients have a special form. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We prove that the obtained results generalize the corresponding theorem proved earlier for a control system defined by a linear delay-differential equation of nth order. Examples are presented illustrating the results obtained.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Optimal Stabilization in Systems of Linear Differential Equations

Optimal Stabilization in Systems of Linear Differential Equations

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

A falling fluid droplet in an oscillating flow field

We examine the flow in and around a falling fluid droplet in a vertically oscillating flow. We assume axisymmetric Stokes flow, and for small deformations to the droplet, the governing equations can be linearized leading to an infinite system of linear ordinary differential equations. In this study, we have analytically solved the problem in the small-capillary limit. We note that the solution locally breaks down at the poles of the droplet. The drag and center of the mass were also obtained. In the case when only odd modes are present, the droplet shows three-dimensional axisymmetric heart-shaped solutions oscillating vertically in time. When only even modes are present, the droplet exhibits axisymmetric stretching and squeezing.