Differential-difference Equations Of Neutral Type Research Articles

Pursuit Problem for Differential-Difference Equations of Neutral Type

Pursuit Problem for Differential-Difference Equations of Neutral Type

Smoothness of solutions to the damping problem for nonstationary control system with delay of neutral type on the whole interval

We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coe cients and several delays. This problem is equivalent to the boundary-value problem for a system of second-order differentialdifference equations, which has a unique generalized solution. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals. Su cient conditions for the initial function are obtained to ensure the smoothness of the generalized solution over the entire interval.

On One Damping Problem for a Nonstationary Control System with Aftereffect

We consider a control system described by the system of differential-difference equations of neutral type with variable matrix coefficients and several delays. We establish the relation between the variational problem for the nonlocal functional describing the multidimensional control system with delays and the corresponding boundary-value problem for the system of differential-difference equations. We prove the existence and uniqueness of the generalized solution of this boundary-value problem.

Smoothness of Solutions to the Damping Problem for Nonstationary Control System with Delay

We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coefficients and several delays. This problem is equivalent to the boundary-value problem for a system of second-order differential-difference equations, which has a unique generalized solution. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals. Sufficient conditions for the initial function are obtained to ensure the smoothness of the generalized solution over the entire interval.

On the Modified Third Method in the Pursuit Problem for Differential-Difference Equations of Neutral Type

On the Modified Third Method in the Pursuit Problem for Differential-Difference Equations of Neutral Type

Existence of the Solution to the Cauchy Problem for Nonlinear Stochastic Partial Differential-Difference Equations of Neutral Type

Existence of the Solution to the Cauchy Problem for Nonlinear Stochastic Partial Differential-Difference Equations of Neutral Type

Моделювання крайових задач для лінійних диференціально-різницевих рівнянь нейтрального типу

In space navigation problems, optimal control of systems with aftereffect, ecology and immunology problems, boundary value problems for differential-difference and integro-differential equations with delays arise, which are an important part of modern theory of differential-functional equations. Finding precise solutions of boundary value problems for differential-difference equations is possible only for the simplest classes of such problems. At present, projection-iterative methods, numerical-analytical method and others are suggested for boundary value problems with delay and of neutral type. The spline-collocation method for solving boundary value problems for differential-difference equations is one of the most efficient algorithms that allows building simple computational schemes. In this paper, we investigate the scheme of modeling boundary value problems for linear differential-difference equations of neutral type with many variable deviations of the argument. A functional space is defined to which the solutions of the considered boundary value problems belong, the properties of the solution smoothness are investigated depending on the structure of the argument deviations. Simple and verifiable sufficient conditions for the boundary value problem solution existence are given. For finding the solution of the boundary value problem, an iterative computational scheme based on the spline approximation method is described. In order to take into account possible discontinuities of the boundary value problem solution derivatives, cubic splines of defect two are used for neutral-type equations. Coefficient conditions for the initial equation which ensure the convergence of the iterative process are obtained. An estimate of the approximate solution error is conducted. A model example of a boundary value problem for a neutral type differential-difference equation is presented on which the iterative scheme is demonstrated. Numerical experiments confirm the obtained theoretical results.

Damping Problem for a Neutral Control System with Delay

A linear control system described by a system of differential-difference equations of neutral type with several delays and variable matrix coefficients is considered. The relationship between the variational problem for a nonlocal functional describing the multidimensional control system with delays and the corresponding boundary value problem for the system of differential-difference equations is established. The existence and uniqueness of a generalized solution to this boundary value problem are proved.

On One Damping Problem for a Nonstationary Control System with Aftereffect

We consider a control system described by the system of differential-difference equations of neutral type with variable matrix coefficients and several delays. We establish the relation between the variational problem for the nonlocal functional describing the multidimensional control system with delays and the corresponding boundary-value problem for the system of differential-difference equations. We prove the existence and uniqueness of the generalized solution of this boundary-value problem.

ON STABILITY WITH DERIVATIVE OF A CLASS OF DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

We consider an important class of differential-difference equations of neutral type, for which we study the asymptotic properties of a solution. We present necessary and sufficient conditions of exponential stability. The conditions have a geometric form of a region in the parameter space. The behavior of a solution at the boundaries of the region is analyzed separately. The loss of stability on the boundary can occur in various ways, namely through the appearance of stationary solutions, and through the appearance of periodic modes. Along with the asymptotic properties of a solution, we study analogous properties of its derivative.

On existence of solution of the Cauchy problem for nonlinear diffusion stochastic partial differential-difference equations of neutral type with random external perturbations

The question related to the existence of the Cauchy problem solution in the class of nonlinear diffusion stochastic partial differential-difference equations of a neutral type with random external disturbances which are independent from the Wiener process is considered. Sufficient conditions are obtained for the diffusion coefficients of nonlinear stochastic differential-difference equations of a neutral type (NDSDRRNT) that guarantee the existence of the solution with the probability of 1. The method of the proof is based on the results of O.M. Stanzhitsky and A.O. Tsukanova on the existence and uniqueness of the Cauchy problem solution for the stochastic differential reaction-diffusion equation of a neutral type. In this paper, we prove the existence of a "mild solution" of NDSDRRNT. In some cases, such equations are mathematical models of real processes, the consideration of which is planned in the future work.

Periodic and asymptotically periodic solutions of systems of differential-difference equations of neutral type

We establish sufficient conditions for the existence of a continuously differentiable T -periodic solution of a system of differential-difference equations of neutral type and study some properties of this solution.

Asymptotic behavior of the solution of a linear stochastic differential-difference equation of neutral type

Asymptotic behavior of the solution of a linear stochastic differential-difference equation of neutral type

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.

О связи фундаментального решения и функции Коши линейных дифференциально-разностных уравнений нейтрального типа

О связи фундаментального решения и функции Коши линейных дифференциально-разностных уравнений нейтрального типа

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type $$dx(t)/dt + p(t)dx(t - 1)/dt = a(t)x(t - 1) + b(t)x(t) + f(t)$$ . We study the existence of and methods for finding solutions possessing required smoothness on intervals of length greater than 1.

On approximation of systems of differential-difference equations of neutral type by systems of ordinary differential equations

We construct a scheme of approximation of a system of differential-difference equations of neutral type by systems of ordinary differential equations and investigate the convergence conditions of this scheme.

Exponential stability of linear differential-difference equations of neutral type

Exponential stability of linear differential-difference equations of neutral type

Estimates of the solutions of difference-differential equations of neutral type

In this paper, we study scalar difference-differential equations of neutral type of general form $$\sum\limits_{j = 0}^m {\int_0^h {u^{(j)} (t - \theta )d\sigma _j (\theta ) = 0,t > h,} } $$ where the σj(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: $$\left\| {u(t)} \right\|W_2^m (T,T + h) \leqslant CT^{q - 1} e^{\kappa T} \left\| {u(t)} \right\|W_2^m (0,h),$$ where C is a constant independent of u 0(t) and the values of q and ℋ are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σ j(θ) or for the case in which the function σ m(θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σ m(θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem.

Approximation of periodic solutions of linear differential-difference equations of neutral type by spline-collocation method

We construct a method for the approximation of periodic solutions of linear differential-difference equations of the neutral type by using cubic splines and investigate conditions for the convergence of the proposed approximation scheme.