Convergence of multistep methods for retarded differential equations with parameters

A class of numerical methods for nonlinear boundary-value problems for retarded differential equations with a parameter is considered. Sufficient conditions for convergence and error estimates are given     

An algorithm for the computation of consistent initial values for differential–algebraic equations

We use boundary value methods to compute consistent initial values for fully implicit nonlinear differential-algebraic equations. The obtained algorithm uses variable order formulae and a deferred correction technique to evaluate the error. A rigorous theory is stated for nonlinear index 1, 2 and 3 DAEs of Hessenberg form. Numerical tests on classical index 1, 2 and 3 DAE problems are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.     

非线性中立型泛函微分方程Runge-Kutta 法的稳定性和收敛性

本文涉及Runge-Kutta 法变步长求解非线性中立型泛函微分方程(NFDEs) 的稳定性和收敛性.为此, 基于Volterra 泛函微分方程Runge-Kutta 方法的B- 理论, 引入了中立型泛函微分方程Runge-Kutta 方法的EB (expanded B-theory)-稳定性和EB-收敛性概念. 之后获得了Runge-Kutta 方法变步长求解此类方程的EB - 稳定性和EB- 收敛性. 这些结果对中立型延迟微分方程和中立型延迟积分微分方程也是新的.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

Towards explicit methods for differential algebraic equations

Explicit methods have previously been proposed for parabolic PDEs and for stiff ODEs with widely separated time constants. We discuss ways in which Differential Algebraic Equations (DAEs) might be regularized so that they can be efficiently integrated by explicit methods. The effectiveness of this approach is illustrated for some simple index three problems. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65-L80, 34-04     

Boundedness and Asymptotic Stability of Multistep Methods for Generalized Pantograph Equations

In this paper, we deal with the boundedness and the asymptotic stability of linear and one-leg multistep methods for generalized pantograph equations of neutral type, which arise from some fields of engineering. Some criteria of the boundedness and the asymptotic stability for the methods are obtained.     

Difference methods for solving boundary value problems for fractional differential equations

Difference schemes for second-order ordinary and partial differential equations with a fractional time derivative are considered. Stationary and nonstationary problems for the diffusion equation in one-and multidimensional domains are examined separately. The stability and convergence of the difference schemes for these equations are proved.     

On the similarity of some three-point methods for solving nonlinear equations

In this short note we discuss certain similarities between some three-point methods for solving nonlinear equations. In particular, we show that the recent three-point method published in [R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 222-229] is a special case of the family of three-point methods proposed previously in [R. Thukral, M.S. Petkovi?, Family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278-2284].     

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.     

Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

Extended method of quasilinearization for systems of nonlinear differential equations

The extended quasilinearization method of Lakshmikantham et al. (J. Optim. Theory Appl. 87 (1995), 379-401) for the first order initial value problems is applied to the nonlinear systems. It is shown that there exist monotone sequences which converge uniformly to the unique solution of the system and the convergence is quadratic. Futhermore, a variety of results are obtained by splitting the functions involved into the difference of two convex or two concave functions, each of which is interesting by itself, with the same conclusion. Moreover, new results are extracted, as a byproduct, from the present results which offer simultaneous bounds for the cases where there is no splitting involved     

Monotone methods and stability results for nonlocal reaction-diffusion equations with time delay

In this paper, we study the applications of the monotone iteration method for investigating the existence and stability of solutions to nonlocal reaction-diffusion equations with time delay. We emphasize the importance of the idea of monotone iteration schemes for investigating the stability of solutions to such equations. We show that every steady state of such equations obtained by using the monotone iteration method is priori stable and all stable steady states can be obtained by using such method. Finally, we apply our main results to three population models.     

A note on some quadrature based three-step iterative methods for non-linear equations

In a recent paper [N.A. Mir, T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 193 (2007) 366-373], some new three-step iterative methods for non-linear equations have been proposed. In this note, we show that the Algorithm 2.2 and Algorithm 2.3 given by the authors have twelfth-order and ninth-order convergence respectively, not seventh-order one as claimed in their work.     

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

This paper is concerned with the numerical solution of nonlinear functional differential and functional equations. The adaptation of one-leg methods is considered. It is proved that an A-stable one-leg method is globally stable and a strongly A-stable one-leg method is asymptotically stable under suitable conditions. A numerical test is given to confirm the theoretical results.     

Nonlinear differential algebraic equations

We consider a system of nonlinear ordinary differential equations that are not solved with respect to the derivative of the unknown vector function and degenerate identically in the domain of definition. We obtain conditions for the existence of an operator transforming the original system to the normal form and prove a general theorem on the solvability of the Cauchy problem.     

线性Volterra多延迟积分微分方程线性多步法的GPm稳定性

讨论了多步法求解线性Volterra多延迟积分微分方程数值方法的GPm稳定.证明了对任给的步长h>0,A-稳定的线性多步法保持原线性系统的渐近稳定性,从而是GPm稳定.     

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.