Second-order Differential-difference Equations Research Articles

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.

Second-order difference equation for Sobolev-type orthogonal polynomials: Part I: theoretical results

We consider a general Sobolev-type inner product involving the Hahn difference operator, so this includes the well-known difference operators and Δ and, as a limit case, the derivative operator. The objective is to construct the ladder operators for the corresponding nonstandard orthogonal polynomials and deduce the second-order differential-difference equation satisfied by these polynomials. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.

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.

Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

We consider a second-order differential-difference equation in a bounded domain Q ⊂ ℝn. We assume that the differential-difference operator contains difference operators with degenerations corresponding to the differential operators. Also, it is assumed that the considered differential-difference operator cannot be expressed by a composition of a difference operator and a strongly elliptic differential operator. The presence of degenerate difference operators does not allow us to obtain the Gårding inequality. We prove a priori estimates implying that the considered differential-difference operator is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space $$ {W}_2^1(Q) $$ . However, using the obtained estimates, we prove smoothness of solutions at least in subdomains Qr generated by translations of the boundary, where $$ \underset{r}{\cup}\overline{Q_r}=\overline{Q} $$ .

On Generalized Solutions of the Second Boundary-Value Problem for Differential-Difference Equations with Variable Coefficients

We consider the second boundary-value problem for a second-order differential-difference equation with variable coefficients on the interval (0,d). We investigate the existence of a generalized solution and obtain conditions on the right-hand side of the equation which ensure the smoothness of generalized solutions on the entire interval (0,d).

A differential equation for varying Krall-type orthogonal polynomials

In this contribution, we consider varying Krall-type polynomials which are orthogonal with respect to a varying discrete Krall-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as to find a second-order differential-difference equation that these polynomials satisfy. We generalize some results that appeared recently in the literature.

Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

We consider a second-order differential-difference equation in a bounded domain Q ⊂ Rn. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.

Local dynamics of a second-order differential-difference equation with large delay at the first derivative

Local dynamics of a second-order differential-difference equation with large delay at the first derivative

A Convection-Siffusion Type Boundary Value Problem to Determine the Expected Time for the Generation of Action Potentials in Nerve Cells by Random Synaptic Inputs in the Dendrites

According to Lange and Miura [1], the determination of the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites can be modeled as general boundary value problem for the linear second order differential difference equation with prescribed boundary condition. This biological problem motivates the study of boundary value problems for singularly perturbed differential difference equation with delay. In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation which models the above biological problem is considered on [0,2]. The solution of this problem exhibits an initial layer at 0 and an interior layer at 1. A numerical method composed of a classical finite difference scheme applied on a piecewise-uniform Shishkin mesh is suggested to solve the problem. The biological implications are studied. [1] C.G.Lange and R.M.Miura, \textit{ Singular perturbation analysis of oundary-value problems for differential-difference equations.V. small shifts with layer behavior }, SIAM J.Appl.Math, Vol.54 1 249--272 (1994).

On smoothness of solutions of some elliptic functional-differential equations with degeneration

We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain.

A parameter uniform method for singularly perturbed differential-difference equations with small shifts

This paper is devoted to the numerical study for a class of boundary value problems of singularly perturbed linear second-order differential-difference equations with small shifts (i. e., containing both terms having a negative shift and terms having a positive shift). In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter e and the shifts depend on the small parameter have been considered. To obtain a parameter-uniform convergence, a piecewise-uniform mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small shifts on boundary layers have also been discussed. To demonstrate the efficiency of the proposed scheme several examples having boundary layers have been carried out.

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential–difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential–difference equations with a turning point, we use El-Mistikawy–Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the performance of the proposed scheme and effect of the size of the delay/advance arguments and coefficients of the delay/advance term on the layer behaviour of the solution.

Polynomial quasisolutions of linear second-order differential-difference equations

We consider the scalar linear second-order differential-difference equation with delay {fx159-01}. This equation is investigated by the method of polynomial quasisolutions based on the representation of an unknown function in the form of a polynomial {ie159-01}. Upon the substitution of this polynomial in the original equation, the residual Δ(t) = O(tN−1) appears. An exact analytic representation of this residual is obtained. We show the close connection between a linear differential-difference equation with variable coefficients and a model equation with constant coefficients, the structure of whose solution is determined by the roots of the characteristic quasipolynomial.

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.

Oscillation Types and Bifurcations of a Nonlinear Second-Order Differential-Difference Equation

This paper considers the second-order differential difference equation $$X\left( t \right) = f\left( {X\left( {t - \tau } \right)} \right) - X\left( t \right)$$ with the constant delay τ > 0 and the piecewise constant function \(f:\mathbb{R} \to \{ a,b\} \) with $$\begin{gathered} f:\mathbb{R} \to \{ a,b\} \hfill \\ f\left( \xi \right): = \left\{ {_a^a } \right.{\text{ }}_{{\text{if }}}^{{\text{if }}} {\text{ }}_\xi ^\xi {\text{ }}_ \geqslant ^{\text{ < }} {\text{ }}_\Theta ^\Theta \hfill \\ \end{gathered} $$ Differential equations of this type occur in control systems, e.g., in heating systems and the pupil light reflex, if the controlling function is determined by a constant delay τ > 0 and the switch recognizes only the positions “on” [f(>) = a] and “off” [f(>) = b], depending on a constant threshold value Θ. By the nonsmooth nonlinearity the differential equation allows detailed analysis. It turns out that there is a rich solution structure. For a fixed set of parameters a, b, Θ, τ, infinitely many different periodic orbits of different minimal periods exist. There may be coexistence of three asymptotically stable periodic orbits (“multistability of limit cycles”). Stability or instability of orbits can be proven.

Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations. V. Small Shifts with Layer Behavior

In this paper, an investigation is initiated of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts, i.e., where the second-order derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary-value problems are associated with expected first-exit times of the membrane potential in models for neurons. In particular, this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries. The analyses of the layer equations using Laplace transforms lead to novel results. It is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small. In the companion paper [SIAM J. Appi. Math., 54 (1994), pp. 273–283], similar boundary-value problems with solutions that exhibit rapid oscillations are studied.

Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Rapid Oscillations

This paper continues the authors’ study of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts. This study was initiated in the companion paper [SIAM J. Appl. Math., 54 (1994), pp. 249–272]. In this paper, the study is extended to problems that have solutions that exhibit rapid oscillations. Restrictions on the sizes of the shifts in terms of the small parameter are found such that, generally, the shifted terms cannot be replaced with truncated Taylor series. In particular, it is shown that, even when the shifts are small relative to the width of an oscillation, they can affect the solution to leading order. The conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions. It is shown that a suitably modified version of the standard WKB method can be used to obtain leading-order oscillatory solutions of these differential-difference equations. These preliminary studies of differential-difference equations with small shifts provide techniques for treating expected first-exit time problems associated with the membrane potential of neurons for generation of action potentials.

Explicit solution of systems of second-order differential-difference equations

In this paper, systems of second-order differential-difference equations are studied. By using the concept of co-solution of certain associated algebraic matrix equations, an explicit solution of initial-value problems for second-order differential-difference systems is constructed.

Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations III. Turning Point Problems

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531; 45 (1985), pp. 687–707). The previous papers focused on problems involving boundary and interior layer phenomena, rapid oscillations, and resonance behavior. The problems studied here have solutions which exhibit turning point behavior, i.e., transition regions between rapid oscillations and exponential behavior. The presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies.

Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Quations II. Rapid Oscillations and Resonances

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531). The previous paper focused on problems involving boundary and interior layer phenomena. Here the problems studied have solutions exhibiting rapid oscillations. The presence of the shift terms can induce large amplitudes, multiphase behavior, and resonance phenomena.n particular, we study two types of resonance phenomena, namely “global” and “local” resonance. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies. In a companion paper (SIAM J. Appl. Math., 45 (1985), pp. 708–734) we study problems with solutions which exhibit turning point behavior.