System Of Linear Differential Equations Research Articles

An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method’s accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient.

Discontinuous control in problem of disturbance reduction for a linear dynamical system

In the paper, we study the problem of active reduction of the influence of a disturbance on the output of a linear controlled system. The problem is investigated on infinite time intervals. We consider a system of linear differential equations under the action of an unknown disturbance and a control to be formed. Our goal is to design an algorithm for reducing the disturbance by means of an appropriate control on the basis of inaccurate measurements of all (or a part of) phase coordinates of the system. This algorithm should form a feedback control that would guarantee that the trajectory of the given system and the velocity of its changing track the trajectory and the velocity of its changing of some reference system. We present two algorithms for solving this problem. These algorithms forming piece‐wise constant controls are based on the combination of constructions of the theory of dynamical inversion and the theory of feedback control. They are stable with respect to informational noises and computational errors.

Asymptotic Properties of a Class of Systems with Linear Delay

Sufficient conditions for the asymptotic stability of linear systems of differential equations with linear delay are obtained. On the basis of these conditions, some systems of linear differential equations are studied, and one of them is stabilized on an infinite time interval.

Traveling Wave Method

A survey of the development of the traveling wave method for one-dimensional media is presented. The main results and changes in the statement of the problem of representing solutions of linear systems of partial differential equations in terms of “traveling waves” (more precisely, in terms of a system of wave transport equations) are presented. It is shown that as the study of systems becomes more complicated, the problem of representing the solution by the traveling wave method turns out to be applicable not only for hyperbolic systems but also for systems containing (even implicitly) both parabolic and elliptic components and thereby approaches the general problem of decomposition of an arbitrary system of linear equations into a system of first-order equations with a main part of the canonical type and with a subordinate linear part.

Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing thin infinite elastic rods with material coefficients that rapidly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the rod thickness and the period of material oscillations, which are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under the assumption on material symmetries as well as in the general case when these two types of displacements are intertwined.

Game-theoretical problems for fractional-order nonstationary systems

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system’s trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example.

Numerical Methods and Approximations for The Heat Transfer Problem

In many studies of heat transfer problems, many researchers have studied heat transfer problems by solving partial differential equations without using approximations or finding solutions with experimental data. This paper analyzes the heat transfer problem using a heat source in a closed environment and how it transfers in the neighboring sections. Referring to mathematical concepts, this work makes possible the simplification of the complexity that associates with such thermodynamic problems. In this framework, the research group discusses such a problem as a discrete one, easily computable, rather than treating it as a continuous one. The reduction of this problem to the solution of a simple system of linear equations and differential equations gives us the possibility to obtain the desired results regarding heat distribution. In many cases, differential equations are hard and difficult to solve. So, we deal with numerical methods to approximate the differential equations to algebraic equations and solve them. Comparing the different algorithms used and showing which of them works best in our testing conditions gives us the possibility of testing and comparing the results and the proper performance. The program will simulate the heat transfer of a single heat source in a closed environment. The results of the simulations will be presented in graphs and demonstrated in visual settings. In the end, our research will provide conclusions on the performance of the numerical methods. The purpose of this article is to study and analyze heat generation and heat transport in three-dimensional space with respect to neighboring sides of a closed environment.

Counting inversion sequences by parity successions and runs

Let In denote the set of inversion sequences of length n. In this paper, we enumerate members of In represented using positive integers according to parameters which track the number of runs and successions of letters of a given parity. To this end, we consider a four-parameter joint distribution on In and derive several general results for this distribution, providing further treatment of some particular cases. In order to prove our results, we make use of a non-standard generating function which allows for the system of recurrences satisfied by various refinements of the aforementioned joint distribution to be translated into a system of linear differential equations that can be solved explicitly. Finally, the more general problem of enumerating the members of In according to adjacencies involving two elements that are equivalent modulo m is considered.

On the exploitation of a population given by a system of linear equations with random parameters

We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.

GKZ-system of the 2-loop self energy with 4 propagators

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

Boundedness and stability in nonlinear systems of differential equations using a modified variation of parameters formula

In this research we introduce a new variation of parameters for systems of linear and nonlinear ordinary differential equations. We use known mathematical methods and techniques including Gronwall’s inequality and fixed point theory to obtain boundedness on all solutions and stability results on the zero solution.

On Some Properties of Solutions of Systems of Linear Difference Equations with Periodic Right-Hand Sides

We consider homogeneous and inhomogeneous systems of linear difference equations with coefficients that are-periodic functions of discrete time. For homogeneous systems, sufficient conditions for the existence of periodic and almost periodic solutions are obtained. For inhomogeneous systems, it is shown that a necessary and sufficient condition for the existence of an N-periodic solution is the existence of a bounded solution. Necessary and sufficient conditions for theN orthogonality of the fundamental matrix of the homogeneous system are established. Illustrative examples are given.

On the use of Kolmogorov equations for determining ion channels characteristics

The paper proposes a method for determination of the dynamic properties of conduction within voltage-gated ion channels with several sequentially located energy barriers named gate particles using a theory of Markov random processes in continuous time and with a discrete state space. The number of states was taken equal to the number of energy barriers in the channel plus one. If the hypothesis that random flows switching gate particles between an open and a closed are Poisson process, is true, then mathematical description of the system’s state of similar channels can be a system of Kolmogorov linear differential equations for state probabilities. Using this model and based on Volt-Clamp test results, published in open access journals, it is possible to find out how fixed values of membrane potential depend on the intensities of Poisson process for different types of potassium channels. Function parameters, that describe the intensity of ion transition across the channel, in the Kolmogorov equation can be restored using the generalized least squares method. This paper contains the examples of determining the intensities of transitions for two types of voltage-dependent potassium channels known as «delayed rectifiier potassium channels» (IKdr, two identical activation gate particles) and channels with fast activation and inactivation processes (IKa, three identical activation and one inactivation gate particles). It is shown that channel activation and deactivation is described by solving the general Kolmogorov equation.

Analysis and rationale of TOW units for tractors with massive towed objects

The most difficult stage in the operation of a tractor with massive towed objects is the starting mode. This is due to the need to overcome the static friction force, which significantly exceeds the motion friction force. As a solution to this problem, we can consider the use of the initial kinetic energy of the tractor, which can be developed when using limited elastically deformable tow units. To optimize the mathematical model, the following assumptions are made: traction force F on the hook of the tractor is a constant value; the inertial masses of the tractor and towed objects are the same and equal m. The dynamics of a tractor and massive towed objects is described by means of theoretical mechanics, taking into account the elastic deformation of tow units. The composed systems of linear homogeneous differential equations with right-hand sides are solved by the method of elimination of functions with increasing of exponential order, as a result of which differential equations of the fourth and sixth power are formed. Initial conditions at t = 0: since the tow unit is not deformed and no force is applied to the towed object. The periods t2 and t3 are determined, during which the tow units will undergo maximum deformation. To assess the effectiveness of the use of elastically deformable tow units, the results obtained should be compared with similar results corresponding to absolutely rigid tow units. Comparison of displacements, speeds and energy testifies to the high efficiency of the elastically deformable tow units. The use of elastically deformable tow units makes it possible to accumulate the initial kinetic energy of an aircraft tow tractor, which makes it possible to overcome the static friction force and ensure the starting of heavy towed objects. Comparison of the kinematic and dynamic parameters of the tractor with towed objects for options with absolutely rigid and elastically deformable tow units shows that the efficiency of using the latter increases with an increase in the number of towed objects. Elastically deformable tow units can cause oscillations of the tractor-towed objects system. To prevent them, the tow units must be blocked at the moment of their maximum deformation.

Representation of the solution of a system of fractional linear differential equations with Caputo derivatives for modeling processes in fractal media

An increasing amount of attention is currently being paid to the fundamentals of mathematical modeling of nonlocal heat and mass transfer processes in fractal media. Such physical processes are described by a system of fractional linear differential equations with Caputo derivatives. In this article we consider a linear Cauchy problem described by differential equations with fractional Caputo derivatives. Using an analogue of the Cauchy matrix we find an explicit representation of the solution to the problem in question.

On Some Properties of Solutions of Systems of Linear Difference Equations with Periodic Right-Hand Sides

On Some Properties of Solutions of Systems of Linear Difference Equations with Periodic Right-Hand Sides

On asymptotic properties of solutions for differential equations of neutral type

The stability of systems of linear autonomous functional differential equations of neutral type is studied. The study is based on the well-known representation of the solution in the form of an integral operator, the kernel of which is the Cauchy function of the equation under study. The definitions of Lyapunov, asymptotic, and exponential stability are formulated in terms of the corresponding properties of the Cauchy function, which allows us to clarify a number of traditional concepts without loss of generality. Along with the concept of asymptotic stability, a new concept of strong asymptotic stability is introduced. The main results are related to the stability with respect to the initial function from the spaces of summable functions. In particular, it is established that strong asymptotic stability with initial data from the space \(L_1\) is equivalent to the exponential estimate of the Cauchy function and, moreover, exponential stability with respect to initial data from the spaces \(L_p\) for any \(p\ge1.\)

Switched hyperbolic balance laws and differential algebraic equations

Motivated by several applications, we investigate the well-posedness of a switched system composed by a system of linear hyperbolic balance laws and by a system of linear algebraic differential equations. This setting includes networks and looped systems of hyperbolic balance laws. The obtained results are globally in time, provided that the inputs have finite (but not necessarily small) total variation.

Stokes phenomenon and Poisson brackets in r-matrix form

In this paper, we explain that how Poisson brackets in r-matrix form naturally arise from solving linear systems of ordinary differential equations around irregular singularities.

Multistep Methods for Second-Order Differential-Algebraic Equations

We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.