Chengming Huang

An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays

This paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays, and with the study of the stability of discretizations of such differential equations. We start with a delay-dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients. We study the exact stability region of the continuous problem as a function of the parameters of the model. Next, it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems. In the second part of the paper, we study delay partial differential equations. The stability region of the fully continuous problem is analyzed first. Then a semidiscretization in space is applied. It is shown that the spatial discretization leads to a reduction of the stability region when the standard second-order central difference operator is employed to approximate the diffusion operator. Finally we consider the delay-dependent stability of the fully discrete problem, where the partial differential equation is discretized both in space and in time. Some numerical examples and further discussions are given.

An energy conservative difference scheme for the nonlinear fractional Schrödinger equations

Abstract In this paper, an energy conservative Crank–Nicolson difference scheme for nonlinear Riesz space-fractional Schrodinger equations is studied. We give a rigorous analysis of the conservation properties, including mass conservation and energy conservation in the discrete sense. Based on Brouwer fixed point theorem, the existence of the difference solution is proved. By virtue of the energy method, the difference solution is shown to be unique and convergent at the order of O ( τ 2 + h 2 ) in the l 2 -norm with time step τ and mesh size h. Finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results.

Stability and error analysis of one-leg methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay dierential equations (DDEs). We focus on the stability behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability, GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable and D-convergent of order s, if it is consistent of order s in the classical sense. c 1999 Elsevier Science B.V. All rights reserved. AMS classication: 65L06; 65L20

Dissipativity of one-leg methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipative initial value problems with delays by one-leg methods. We focus on the dissipativity of numerical methods. Some general results are given. Specially, an irreducible one-leg method is dissipative for finite-dimensional dynamical systems with delays if and only if it is A-stable and it is dissipative for infinite-dimensional systems if and only if it is strongly A-stable.

Exponential mean square stability of numerical methods for systems of stochastic differential equations

This paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split-step theta method for stochastic systems. First, we consider linear autonomous systems. Under a sufficient and necessary condition for exponential mean square stability of the exact solution, it is proved that the two classes of theta methods with @q>=0.5 are exponentially mean square stable for all positive step sizes and the methods with @q 0.5 still unconditionally preserves the exponential mean square stability of the underlying systems, but the stochastic theta method does not have this property. Finally, we consider stochastic differential equations with jumps. Some similar results are derived.

Numerical solution of fractional integro-differential equations by a hybrid collocation method

Fractional integro-differential equations have been recently solved by many methods, such as Adomian decomposition method, differential transform method, collocation method and Taylor expansion approach. In this paper a hybrid collocation method is used which combines a non-polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval. A theoretical analysis for the convergence order of the method is presented. Some numerical examples are given which confirm the theoretical results.

Mean square stability and dissipativity of two classes of theta-methods for systems of stochastic delay differential equations

In this paper, we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients. This condition admits that the diffusion coefficient can be highly nonlinear, i.e., it does not necessarily satisfy a linear growth or global Lipschitz condition. It is proved that, for all positive stepsizes, the classical stochastic theta method with @q>=0.5 is asymptotically mean square stable and the split-step theta method with @q>0.5 is exponentially mean square stable. Conditional stability results for the methods with @q 0.5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data.

Dissipativity of multistep runge-kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipative initial value problems with delays by multistep Runge-Kutta methods. We investigate the dissipativity properties of (k, l)-algebraically stable multistep Runge-Kutta methods with constrained grid and linear interpolation procedure. In particular, it is proved that an algebraically stable, irreducible multistep Runge-Kutta method is dissipative for finite-dimensional dynamical systems with delays, which extends and unifies some extant results. In addition, we obtain dissipativity results of A-stable linear multistep methods by using the relationship between one-leg methods and linear multistep methods.

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations

In this paper, a split-step theta (SST) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). It is proved that the SST method with ? ? 0 , 1 / 2 ] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient, but for ? ? ( 1 / 2 , 1 ] , the SST can reproduce the exponential mean square stability unconditionally. Then, based on the stability results of SST scheme, we examine the exponential mean square stability of the stochastic linear theta (SLT) approximation for NSDDEs and obtain the similar stability results to that of the SST method. Moreover, for sufficiently small stepsize, we show that the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. These results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for NSDDEs.