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We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L 2 -norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t = 0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results.

In this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated θ-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated θ-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated θ-Milstein methods with different θ values. Numerical results are given to verify our theoretical results.

We propose a new family of functionally-fitted block \begin{document}$ \theta $\end{document} -methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block \begin{document}$ \theta $\end{document} -methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of \begin{document}$ r $\end{document} -dimensional functionally-fitted block \begin{document}$ \theta $\end{document} -method is at least \begin{document}$ r $\end{document} for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block \begin{document}$ \theta $\end{document} -methods.

In this paper, we consider highly oscillatory second-order differential equations $$\ddot{x}(t)+\Omega ^2x(t)=g(x(t))$$ with a single frequency confined to the linear part, and $$\Omega $$ is singular. It is known that the asymptotic-numerical solvers are an effective approach to numerically solve the highly oscillatory problems. Unfortunately, however, the existing asymptotic-numerical solvers fail to apply to the highly oscillatory second-order differential equations when $$\Omega $$ is singular. We propose an efficient improvement on the existing asymptotic-numerical solvers, so that the asymptotic-numerical solvers can be able to solve this class of highly oscillatory ordinary differential equations. The error estimation of the asymptotic-numerical solver is analyzed and nearly conservation of the energy in the Hamiltonian case is proved. Two numerical examples including the Fermi–Pasta–Ulam problem are implemented to show the efficiency of our proposed methods.

We present an alternative approach proposed by Albrecht to derive general order conditions for Runge-Kutta-Nystrom methods and relate it to the classical RKN-theory. The RKN-methods are treated as composite linear methods to yield the general order conditions as orthogonal relations. We then exploit the orthogonal structure of the order conditions and obtain a simple recursion to generate the order conditions. Implications of this approach on the classical RKN-theory are discussed and it may be worthwhile to generalize the approach to other discretizations.

The main aim of this study is to propose a class of generalized two-step Milstein methods for solving Ito stochastic differential equations. We study their mean-square consistency and mean-square c...

In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time τ = ω t . The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems.

We propose a class of s-stage hybrid Runge–Kutta methods for solving ordinary differential equations and study their order conditions and numerical stability property. Examples of second-, third-, and fifth-order hybrid Runge–Kutta methods are given. Numerical experiments are conducted to demonstrate the efficiency of the proposed methods.

In this paper, we propose generalized two-step Maruyama methods for solving Ito stochastic differential equations. Numerical analysis concerning consistency, convergence and numerical stability in the mean-square sense is presented. We derive sufficient and necessary conditions for linear mean-square stability of the generalized two-step Maruyama methods. We compare the stability region of the generalized two-step Maruyama methods of Adams type with that of the corresponding two-step Maruyama methods of Adams type and show that our proposed methods have better linear mean-square stability. A numerical example is given to confirm our theoretical results.

In this paper, we propose a class of deterministic implicit two-step Milstein methods for solving Ito stochastic differential equations. Theoretical analysis is conducted for the convergence and stability properties of the proposed methods. We derive sufficient conditions such that these methods have the mean-square(M-S) convergence of order one, as well as sufficient and necessary conditions for linear M-S stability of the implicit two-step Milstein methods. Stability analysis shows that our proposed implicit two-step Milstein methods have much better stability property than those of the corresponding two-step explicit or semi-implicit Milstein methods. Numerical results using two test equations confirm our theoretical analysis results.

In this paper, we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second-order ordinary differential equations. The systems are reformulated as a first-order system, which is then transformed to adiabatic variables. The solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system. We develop Filon-type methods for linear systems by approximating the integral as a linear combination of function values and derivatives. We then present a special combination of Filon-type methods and waveform relaxation methods for nonlinear systems. Both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows, as are illustrated by numerical experiments.

We propose a numerical scheme to obtain an approximate solution of a boundary value problem for fourth order nonlinear integro-differential equation of Kirchhoff type. We first reduce the problem to a nonlinear finite dimensional system based on the Bernoulli polynomial approximation and then solve it by an iterative process together with a collocation method. Convergence of the iterative process and error estimates of the approximate solution are provided. Numerical experiments are conducted to illustrate the performance of the proposed method.

We propose a numerical scheme to obtain an approximate solution of a nonlocal elliptic Kirchhof-type problem. We first reduce the problem to a nonlinear finite dimensional system by a Legendre–Galerkin spectral method and then solve it by an iterative process. Convergence of the iterative process and an error estimation of the approximate solution is provided. Numerical experiments are conducted to illustrate the performance of the proposed method.

The so-called delayed recruitment/renewal equation provides the mathematical model in a diverse spectrum of practical applications and may become singularly perturbed when the time-lag is large relative to the reciprocal of the decay rate. In order to accurately capture its solution features numerically, we design a family of exponential fitting Runge-Kutta methods of collocation type to obtain the numerical approximation. The exponential fitting approximations are proved to have higher order of uniform accuracy. We demonstrate the efficiency of this family of exponential fitting Runge-Kutta methods for the delayed recruitment/renewal equation via application to some important problems.

Functionally-fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions. In this paper, we propose a new type of functionally-fitted block methods for initial value problems of second order ordinary differential equations. The basic theory for the proposed methods is established. First, we present a sufficient condition for the existence of the functionally-fitted block methods. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the r -point functionally-fitted block method is convergent of order at least r for second order ordinary differential equations. Numerical experiments are conducted to demonstrate the validity and efficiency of the functionally-fitted block methods by comparison with some existing numerical methods.

In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.

Functionally-fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions. In this paper, we propose a new type of functionally-fitted block methods for ordinary differential equations. The basic theory for the proposed methods is established. First, we derive two sufficient conditions to ensure the existence of the functionally-fitted block methods, discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of the r -point functionally-fitted block method is at least r for general ordinary differential equations. Experimental results are conducted to demonstrate the validity of the functionally-fitted block methods.

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations, delay differential equations, delay differential-algebraic equations, neutral delay differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied. As applications, we adopt the continuous parallel block methods to solve delay differential equations, neutral delay differential equations and delay differential-algebraic equations, and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable. Several numerical experiments are given to illustrate the performance of the continuous parallel block methods.

This paper is concerned with numerical stability of general linear methods (GLMs) for a system of linear neutral delay differential-algebraic equations. A sufficient and necessary condition for asymptotic stability of GLMs solving such system is derived. Based on this main result, we further investigate the asymptotic stability of linear multistep methods, Runge–Kutta methods, and block θ-methods, respectively. Numerical experiments confirm our theoretical result.

This paper is concerned with asymptotic stability of linear neutral delay differential-algebraic equations and Runge--Kutta methods. First, we give a new equivalent sufficient condition for the neutral delay differential-algebraic equations to be delay-independent asymptotically stable. Then we investigate the asymptotic stability of the numerical solutions generated by the Runge--Kutta methods combined with Lagrange interpolation. Some results on the asymptotic stability of Runge--Kutta methods of high order are given. Finally, numerical examples of index 1 and 2 are conducted to confirm our numerical stability result.

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and A"@w-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.

We are concerned with delay-independent asymptotic stability of linear system of neutral differential equations. We first establish a sufficient and necessary condition for the system to be delay-independently asymptotically stable, and then give some equivalent stability conditions. This paper improves many recent results on the asymptotic stability in the literature. One example is given to show that the sufficient and necessary condition is easy to verify.

This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result.

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations (ODEs), delay differential equations, neutral differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ODEs generated by a class of block $\theta$-methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous block $\theta$-methods for ODEs are studied. As an application, we adopt the continuous block $\theta$-methods to solve delay differential equations and prove that the continuous block $\theta$-methods are $GP$-stable if and only if they are $A_{\omega}$-stable for ODEs. Several numerical experiments are given to illustrate the performance of the continuous block $\theta$-methods.

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.

This paper is concerned with asymptotic stability property of linear θ-method for partial functional differential equations with delay. A sufficient condition for the underlying partial functional differential equations to be asymptotically stable is presented. We investigate numerical stability of the linear θ-method by using the spectral radius condition. When θ ∈ [0, 1/2), a sufficient and necessary condition for the linear θ-method to be asymptotically stable is established. When θ ∈ [1/2, 1], the linear θ-method is unconditionally asymptotically stable. The behaviour of the norm of the iteration matrix when the linear θ-method is asymptotically stable is studied by using Kreiss resolvent condition. Numerical experiments have been implemented to confirm the derived stability properties of the numerical method.

We are concerned with the asymptotic stability of a system of linear neutral differential equations with many delays in the form y^'(t)=Ly(t)[email protected]?i=1dM"iy([email protected]"i)[email protected]?i=1dN"iy^'([email protected]"i), where L,M"i,N"[email protected]?C^N^x^N(i=1,2,...,d) are constant complex matrices, @t"i>0(i=1,2,...,d) are constant delays and y(t)=(y"1(t),y"2(t)...y"N(t))^T is an unknown vector-valued function for t>0. We first establish a new result for the distribution of the roots of its characteristic function, next we obtain a sufficient condition for its asymptotic stability and then we investigate the corresponding numerical stability of linear multistep methods applied to such systems. One numerical example is given to testify our numerical analysis.

In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.

The existence of periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay is obtained by using stability properties of a bounded solution.

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space R m and vector-valued functions in a weakly compact subset of a Banach vector space generated by m L μ p -spaces for 1 ⩽ p + ∞ . Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m L μ ∞ -spaces instead.

Dissipative differential equations have frequently appeared in the fields of physics, engineering, and biology. In this paper we investigate numerical dissipativity of linear multistep and one-leg methods applied to a class of dissipative delay differential equations. We show that for such class of dissipative systems these numerical methods are dissipative if and only if they are A-stable for ordinary differential equations. One numerical experiment is given to illustrate our result.

This paper focuses on the analytic and numerical dissipativity of θ-method for delay differential equations with a bounded variable lag. A generalized Halanay inequality is derived, and a sufficient condition is presented to ensure that delay differential equations with a bounded variable lag are dissipative. We then apply θ-method to such delay differential equations, and investigate the numerical dissipativity of the θ-method.

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.

In this paper, we apply accelerated overrelaxation (AOR) methods to find the least square solution of minimal norm to the linear systemAy=bwhere A@?C^m^x^n is a matrix of rank r and b@?C^m. We first augment the system to a block 4x4 consistent system, and then split the augmented coefficient matrix by AOR subproper splitting. Intervals for the two relaxation parameters where the AOR iteration matrix is semiconvergent are presented. Also, we provide a method to compute the least square solution of minimal norm to the system.

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley–Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.

This paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded (state-independent) lag. A generalized Halanay inequality is derived in Section 2, and in Section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is exponentially stable uniformly for sufficiently small e >0. This type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag.

In this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.

This paper deals with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equationsx′=Ax′(t−τ)+Bx(t)+Cx(t−τ), whereA, B, andC are constant complexN ×N matrices, and τ>0. A necessary and sufficient condition such that the differential equations are asymptotically stable is derived. We also focus on the numerical stability properties of adaptations of one-parameter methods. Further, we investigate carefully the characterization of the stability region.

This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank–Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods.