Order Delay Differential Equations Research Articles

A new technique to solve the initial value problems for fractional fuzzy delay differential equations

Using some recent results of fixed point of weakly contractive mappings on the partially ordered space, the existence and uniqueness of solution for interval fractional delay differential equations (IFDDEs) in the setting of the Caputo generalized Hukuhara fractional differentiability are studied. The dependence of the solution on the order and the initial condition of IFDDE is shown. A new technique is proposed to find the exact solutions of IFDDE by using the solutions of interval integer order delay differential equation. Finally, some examples are given to illustrate the applications of our results.

Stability, Boundedness and periodic solutions to certain second order delay differential equations

Stability, boundedness and existence of a unique periodic solution to certain second order nonlinear delay differential equations is discussed. By employing Lyapunov's direct (or second) method, a complete Lyapunov functional is constructed and used to establish sufficient conditions, on the nonlinear terms, that guarantee uniform asymptotic stability, uniform ultimate boundedness and existence of a unique periodic solution. Obtained results complement many outstanding recent results in the literature. Finally, examples are given to show the effectiveness of our method and correctness of our results.

English

English

Delay-dependent stability of symmetric Runge–Kutta methods for second order delay differential equations with three parameters

The paper is concerned with the delay-dependent stability analysis of symmetric Runge–Kutta methods, which include the Gauss methods and the Lobatto IIIA, IIIB and IIIS methods, for the second order delay differential equations with three parameters. By using the root locus technique, the root locus curve is given and the numerical stability region of symmetric Runge–Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Numerical examples confirming the theoretical results are presented.

W-transform for exponential stability of second order delay differential equations without damping terms

In this paper a method for studying stability of the equation x^{prime prime }(t)+sum_{i=1}^{m}a_{i}(t)x(t- tau_{i}(t))=0 not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation x^{prime prime}(t)+sum_{i=1}^{m}a_{i}(t)x(t)=0 is not exponentially stable, the delay equation can be exponentially stable.

Positive Periodic Solutions for Singular Higher Order Delay Differential Equations

In this article, we consider the singular higher order delay differential equations. Under certain assumptions, some new criteria for guaranteeing the existence of positive periodic solutions are presented. Moreover, two examples are provided to demonstrate the effectiveness and applications of the main results.

Instability for a class of second order delay differential equations

Instability for a class of second order delay differential equations

Bernstein Operational Matrix with Error Analysis for Solving High Order Delay Differential Equations

In this paper, we propose a numerical method based on Bernstein polynomials and their operational matrices for solving both linear and non-linear delay differential equations (DDEs). The Bernstein operational matrices for differentiation and delay functions are introduced. The aim is to reduce the DDEs to a set of algebraic equations. Moreover If the exact solution is polynomial, then the exact solution set will be obtained. Even the exact solution is unknown, an upper bound based on the regularity of the exact solution will be obtained. By using the residual correction procedure and the error analysis of the four- and five-order Runge–Kutta method (RK45) the absolute errors may be estimated and the approximate solution can be corrected. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results.

Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations via Cubic Spline in Tension

This paper deals with the singularly perturbed boundary value problem for a second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A difference scheme on a uniform mesh is accomplished by the method based on cubic spline in tension. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

On asymptotic stability of solutions to third order nonlinear delay differential equation

In this article, we study the asymptotic stability of solutions for the non-autonomous third order delay differential equation by constructing Lyapunov functionals.

Periodicity, Stability, and Boundedness of Solutions to Certain Second Order Delay Differential Equations

The behaviour of solutions to certain second order nonlinear delay differential equations with variable deviating arguments is discussed. The main procedure lies in the properties of a complete Lyapunov functional which is used to obtain suitable criteria to guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results are new and also complement related ones that have appeared in the literature. Moreover, examples are given to illustrate the feasibility and correctness of the main results.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

Stability in the class of first order delay differential equations

The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay differential equation is fulfilled only approximately. Is it true that in this case this function is close to an exact solution of this delay differential equation?

New oscillation results to fourth order delay differential equations with damping

New oscillation results to fourth order delay differential equations with damping

On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.

New oscillation criterion for delay differential equations with non-monotone arguments

We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones.

Solving second order delay differential equations using direct two-point block method

This paper will consider the implementation of direct two-point fourth and fifth order multistep block method in the form of Adams–Moulton method to solve second order delay differential equations (DDEs) directly without transforming the equations into system of first order DDEs. The proposed methods will compute the numerical solutions at two points simultaneously which are implemented in predictor–corrector (PECE) mode. The formulation and stability analysis of the block method are discussed. The block method will approximate the computing solutions using constant step size. Numerical results are presented to show that the proposed methods are suitable for solving second order DDEs compared with the existing methods.

An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays

Abstract This paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory.

A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression

This paper deals with the singularly perturbed boundary value problem for the second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. An exponentially fitted difference scheme on a uniform mesh is accomplished by the method based on cubic spline in compression. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

Periodic solutions for nonautonomous first order delay differential systems via Hamiltonian systems

The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation $x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$ is investigated, where $f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$ is 2N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems.