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In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving pa...

This paper is concerned with the stability of the one-leg methods for nonlinear Volterra functional differential equations (VFDEs). The contractivity and asymptotic stability properties are first analyzed for quasi-equivalent and A-stable one-leg methods by use of two lemmas proven in this paper. To extend the analysis to the case of strongly A-stable one-leg methods, Nevanlinna and Liniger’s technique of introducing new norm is used to obtain the conditions for contractivity and asymptotic stability of these methods. As a consequence, it is shown that one-leg θ-methods (θ ∈ (1/2, 1]) with linear interpolation are unconditionally contractive and asymptotically stable, and 2-step Adams type method and 2-step BDF method are conditionally contractive and asymptotically stable. The bounded stability of the midpoint rule is also proved with the help of the concept of semi-equivalent one-leg methods.

This paper deals with dissipativity of the variable-stepsize Runge-Kutta methods applied to nonlinear Volterra functional differential equations in Hilbert spaces. The conditional dissipativity of variable-stepsize Runge-Kutta methods is analyzed and hence some new numerical dissipative criteria are derived. The resulting dissipative criteria extend and improve the existing results. Especially, for the algebraically stable variable-stepsize Runge-Kutta methods, we obtain a sharper dissipative result. In the end, we apply some concrete variable-stepsize Runge-Kutta methods to the three classes of nonlinear Nicholson’s blowflies models. The presented numerical examples further illustrate the theoretical results.

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H = O ( h r − 1 r ) . Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.

This paper concerns the numerical stability of one-leg methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay. G(c, p)-algebraically stable one-leg methods with compound quadrature formulae are considered. Nonlinear stability conditions for the presented methods are derived. As an illustration of the application of these investigations, the stability results of one-leg methods for Volterra delay-integro-differential equations are obtained, which are more general than the related results in the previous literature. Two numerical experiments are given to confirm our results.

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.

In the present paper, a family of predictor–corrector (PC) schemes are developed for the numerical solution of nonlinear parabolic differential equations. Iterative processes are avoided by use of the implicit–explicit (IMEX) methods. Moreover, compared to the predictor schemes, the proposed methods usually have superior accuracy and stability properties. Some confirmation of these are illustrated by using the schemes on the well-known Fisher’s equation.

This paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro-differential equations (NDIDEs). We first give a generalization of the Halanay inequality which plays an important role in the study of dissipativity and stability of differential equations. Then, we apply the generalization of the Halanay inequality to NDIDEs and the dissipativity and the asymptotic stability results of the theoretical solution of NDIDEs are obtained. From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions. Therefore, the results, presented in this paper, provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems.

This paper is concerned with the contractivity and asymptotic stability properties of the implicit Euler method (IEM) for nonlinear functional differential equations (FDEs). These properties are first analyzed for Volterra FDEs and then the analysis is extended to the case of neutral FDEs (NFDEs). Such an extension is particularly important since NFDEs are more general and have received little attention in the literature. The main result we establish is that the IEM with linear interpolation can completely preserve these stability properties of the analytical solution to such FDEs.

This paper provides results on the correct simulation, when using continuous Runge–Kutta methods, of certain stability properties of nonlinear neutral delay-differential equations (NDDEs) y ′ ( t ) = f ( t , y ( t ) , y ( t - τ ( t ) ) , y ′ ( t - τ ( t ) ) ) ( t ⩾ t 0 ) . In particular, it is shown that certain continuous Runge–Kutta methods based upon the backward Euler method or the 2-stage Lobatto IIIC method, combined with linear interpolation, are GRN -stable and asymptotically stable for NDDEs.

Some convergence results of one-leg methods for nonlinear neutral delay integro-differential equations (NDIDEs) are obtained. It is proved that a one-leg method is E (or EB) -convergent of order p for nonlinear NDIDEs if and only if it is A-stable and consistent of order p in classical sense for ODEs, where p = 1, 2. A numerical example that confirms the theoretical results is given in the end of this paper.

In this paper, we focus on the error behavior of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (NVDIDEs) with constant delay. The convergence properties of the Runge-Kutta methods with two classes of quadrature technique, compound quadrature rule and Pouzet type quadrature technique, are investigated.

This paper is concerned with the numerical solution of neutral delay differential equations (NDDEs). We focus on the stability of general linear methods with piecewise linear interpolation. The new concepts of GS(p)-stability, GAS(p)-stability and weak GAS(p)-stability are introduced. These stability properties for (k,p,0)-algebraically stable general linear methods (GLMs) are further investigated. Some extant results are unified.

This paper is devoted to the investigations into the nonlinear stability properties of explicit and diagonally implicit Runge–Kutta methods for solving neutral delay differential equations in Banach space. Two approaches to numerically treating “neutral term” are considered, which allow us to prove several results on numerical stability and conditional contractivity of explicit and diagonally implicit Runge–Kutta methods. In particular, some new results on stiff ODEs and stiff DDEs, which can be regarded as the generalizations of previous results in the literature, are obtained.

This paper is concerned with the stability properties of theoretical solution and numerical solution to nonlinear neutral differential equations with two delays in Banach space. Some sufficient conditions for the stability and asymptotic stability of the systems are given. The numerical stability of θ -methods when applied to the systems is also considered. A series of new stability results of θ -methods are obtained.

This paper is devoted to investigations into numerical stability properties of one-leg methods for nonlinear neutral delay differential equations. At first, a series of new stability concepts, such as GS-stability, GAS-stability and Weak GS-stability, are introduced. Then it is proved that a strongly A-stable one-leg method with linear interpolation is GAS-stable, and that an A-stable one-leg method with linear interpolation is GS-stable and Weakly GS-stable. Some numerical experiments are given in the last section of this paper which confirm our results.

In this paper the numerical solutions of nonlinear neutral differential equations with constant delay and with proportional delay, two typical examples of neutral functional differential equations, are investigated. The focus is on the stability of numerical solutions produced by $\theta$-methods. It is proved that if $\frac{1}{2}\leq\theta\leq 1$ the numerical solution to neutral differential equations with constant delay is globally stable and asymptotically stable under some conditions which are sufficient for stability and asymptotic stability of the exact solution. For neutral differential equations with proportional delay, two discretization approaches, the approach to transforming it into neutral equations with constant delay and the approach to directly discretizing it based on full-geometric mesh, are discussed. The integration on the first mesh interval $[0, T_0]$, which needs to be considered for both approaches, is also investigated. By taking into account the order of the methods, we construct two algorithms for the integration on $[0, T_0]$. It is also proved that under some assumptions the numerical solutions to the neutral equation with proportional delay produced by the two approaches are globally stable and asymptotically stable if $\frac{1}{2} < \theta\leq 1$.

This paper discusses the one-leg θ-methods for the numerical solution of nonlinear neutral functional differential equations with pantograph delay y ′ ( t ) = f ( t , y ( t ) , y ( pt ) , y ′ ( pt ) ) , t ⩾ 0 , 0 p 1 , y ( 0 ) = y ˆ . In recent years, the numerical techniques for the solution to this kind of equations have been developed by numerous authors who have mainly considered the linear system. Instead, in the present paper we study the nonlinear case. The first-step integration of above systems on [ 0 , T 0 ] by one-leg θ-methods is analyzed. It is proved that the numerical solution preserves the contractivity property for some T0. On the quasi-geometric mesh, the numerical stability of one-leg θ-methods for above systems is investigated. Some sufficient conditions for global stability and asymptotic stability are established. Two numerical examples are also included.

This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.

This paper is concerned with the dissipativity of theoretical solutions to nonlinear Volterra functional differential equations (VFDEs). At first, we give some generalizations of Halanay's inequality which play an important role in study of dissipativity and stability of differential equations. Then, by applying the generalization of Halanay's inequality, the dissipativity results of VFDEs are obtained, which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay-integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice.