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This paper is concerned with the long-time behaviour of the numerical solutions to a class of linear non-autonomous neutral delay differential equation with proportional delays. Our purpose is to give some asymptotic estimates of the θ-methods with constant stepsize discretization and formulate their upper bounds. Asymptotic estimate not only describes more accurate than asymptotic stability, but also gives an upper bound estimate of the solution for the long-time behaviour. We also compare the known results and show that our formulae improve and generalize these results. Some numerical examples are given in the end of this paper to confirm the theoretical results.

The aim of this paper is to improve some results obtained in our earlier paper [Z. Yu and M. Liu, Almost surely asymptotic stability of numerical solutions for neutral stochastic delay differential equations , Discrete Dyn. Nat. Soc. 2011 2011, article id 217672, 11 p., doi:10.1155/2011/217672]. In this paper, we establish an improved theorem and show that the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to neutral stochastic delay differential equations. To obtain the desired result, some new proof techniques are adopted.

The second author's work [F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations , Numer. Math. 115 2010, pp. 681–697] and Mao's papers [D.J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations , SIAM J. Numer. Anal. 45 2007, pp. 592–607; X. Mao, Y. Shen, and G. Alison, Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations , J. Comput. Appl. Math. 235 2011, pp. 1213–1226] showed that the backward Euler–Maruyama BEM method may reproduce the almost sure stability of stochastic differential equations SDEs without the linear growth condition of the drift coefficient and the counterexample shows that the Euler–Maruyama EM method cannot. Since the stochastic θ-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic θ-method can capture the stability of exact solutions of SDEs. Without the linear growth condition of the drift term, this paper concludes that the stochastic θ-method can capture the stability for θ∈1/2, 1]. For θ∈[0, 1/2, a counterexample shows that the stochastic θ-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions.

In this paper, we investigate how many convergence orders of discontinuous Galerkin (DG) method for numerically solving neutral delay differential equations (NDDEs). Although discontinuous behaviou...

In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre–Gauss–Radau interpolation and a multi-domai...

In this paper, we are interested in numerical methods with variable step-size for stochastic pantograph differential equations (SPDEs). SPDEs are very special stochastic delay differential equations (SDDEs) with unbounded memory. The problem of computer memory hold, when the numerical methods with constant step-size are applied to the SPDEs. In this work, we construct split-step theta (SSθ) methods with variable step-size for SPDEs. The boundedness and strong convergence of the numerical methods are investigated under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. It is proved that, the SSθ methods with variable step-size for θ∈[12,1] converge strongly to the exact solution. In addition, the strong order 0.5 is given under mild assumptions. The mean-square stability (MS-Stability) of the numerical methods with θ∈(12,1] is given. Finally, some illustrative numerical examples are presented to show the efficiency of the methods, and how MS-Stability of SS...

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.

The purpose of this paper is devoted to studying the implicit–explicit IMEX one-leg methods for stiff delay differential equations DDEs which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods.

In this paper, we introduce ordinary and delay differential equations to describe the interactions between a malignant tumour and the immune system in vivo in the presence of human immunodeficiency virus HIV infection of CD4+ T-cells. In the delay model, we take into account the time lags required by the healthy effector cell components to recognize the pathogens and tumour cells. The models consist of four populations: tumour cells, healthy effector cells CD4+ T-cells, effector cells infected by HIVs and free viral particles. The presence of delay term in the model leads to a notable increase in the complexity of the observed behaviour. We investigate the qualitative behaviour of the models and find the conditions that guarantee the asymptotic stability of the steady states. Numerical simulations are provided to illustrate and extend the theoretical results. The obtained results are consistent with the real phenomena and give a better understanding of cancer immunity and viral oncogenesis.

Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results.

This paper is concerned with the numerical study of singularly perturbed boundary value problems for delay differential equations with a turning point. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighbourhood of the boundary layers. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments are carried out to illustrate, in practice, the result of convergence proved theoretically and demonstrate the effect of the delay argument and the coefficient of the delay term on the layer behaviour of the solution.

In this paper, we propose a global collocation method for the numerical solution of the delay differential equations (DDEs). The method presented is based on sextic C 1-splines s(x) as an approximation to the exact solution y(x) of the DDEs. Convergence results shows that the error bounds ‖ s j −y j ‖=O(h 6), j=0, 1, in the uniform norm. Moreover, the stability analysis properties of these methods have been studied. Numerical experiments will also be considered.

In this paper, we will study oscillations of numerical solutions for the neutral delay differential equation [image omitted] where p∈ and p≠0, τ, q∈(0,+∞), σ≥0. Conditions under which numerical solutions of the above differential equation are oscillatory are obtained. The condition that leads to oscillations of the linear θ-method is also given. To verify our results, we give numerical experiments.

This paper considers the Runge-Kutta methods used on one-dimensional delay differential equations (DDEs), which undergo Hopf bifurcation from the origin. It not only shows the numerical approximations undergo Neimark-Sacker bifurcation, but also these Neimark-Sacker bifurcation points close the Hopf bifurcation point of the DDEs with the order of methods. Meanwhile, compared with another interpolation method, we conclude that the given interpolation method is better in the convergence of Neimark-Sacker bifurcation points. Finally, by using 2-stage Gauss method to delay Logistic equation, the theoretical analysis is supported.

In this paper, the model predictive control (MPC) is developed for linear time-varying systems with distributed time delay in state. The Chebyshev operational matrices of product, integration and delay are utilized to transform the solution of distributed delay differential equation to the solution of algebraic equations. The Chebyshev functions are also applied to derive approximate solution of finite horizon optimal control problem involved in MPC. The proposed method is simple and computationally advantageous. Illustrative example demonstrates the validity and applicability of the technique.

In this paper a class of delay differential equations with a perturbation parameter e is examined. A hybrid finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type is derived. We show that the scheme is almost second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.

This paper presents the use of spline functions of polynomial form to approximate the solution of system of second order delay differential equations. The error analysis and stability of the method are theoretically investigated. A numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

In this paper a study of the existence, uniqueness, stability and convergence of a class of C2-spline collocation methods for solving delay differential equations (DDEs) is introduced. Letting the interior collocation points [image omitted] , j=1(1)3 be dependent on the parameters c1, c2∈(0, 1) and c3=1 it is shown that the proposed methods for DDEs possess a convergence rate of order six if 58-57(c1+c2)+55c1 c2=0, and they are unstable if c1+c2

A numerical method based on the Adomian decomposition method which has been developed by Adomian [Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, MA] is introduced in this paper for the approximate solution of delay differential equations. The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only a few terms are required to obtain an approximate solution which is found to be accurate and efficient.

In this paper we show that the monotone iterative technique provides two monotone sequences that converge uniformly to extremal (periodic) solutions of second order delay differential equations without assuming properties of monotonicity in the nonlinear part. Moreover, we obtain optimal existence conditions with upper and lower solutions in the reverse order. Our results are new even for ordinary differential equations.

In this paper, an ϵ-uniform fitted operator method which solves boundary-value problems for singularly perturbed differential-difference equations containing small delay with boundary layer behavior is presented. Both the cases, i.e., when boundary layer is on the left side and when boundary layer is on the right side are discussed here. It is shown that the scheme is ϵ-uniform by establishing the error estimate. The effect of small delay on the boundary layer solution is shown by considering several numerical experiments. Numerical results in terms of maximum errors are tabulated and plots giving computed and exact solution demonstrate the efficiency of the method.

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.

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 introduce some exponentially fitted methods to overcome the drawbacks of the corresponding classical counterparts. Numerical results are given.

In this paper a matrix separation of the variables method for solving initial-boundary value problems for coupled systems of second order partial differential equations is presented. Finite computable approximate solutions and upper error bounds for them in terms of data are given.