Asymptotic Stability Analysis of ...-Methods for Functional Differential Equations.
Global error estimation in the numerical solution of retarded differential equations by Euler's method
In this paper Zadunaisky's technique is used to estimate the global error propagated in the numerical solution of the system of retarded differential equations by Euler's method. Some numerical examples are given.
A note on existence and uniqueness of solutions of neutral functional-differential equations with state-dependent delays
Existence and uniqueness theorem for state-dependent delay-differential equations of neutral type is given. This theorem generalizes previous results by Grimm and the author.
Frequency analysis of preconditioned waveform relaxation iterations
The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting...
Toward a two-step Runge-Kutta code for nonstiff differential systems
Various issues related to the development of a new code for nonstiff differential equations are discussed. This code is based on two-step Runge-Kutta methods of order five and stage order five. Numerical experiments are presented which demonstrate that the new code is competitive with the Matlab ode45 program for all tolerances.
A note on the stability of -methods for Volterra integral equations of the second kind
Order conditions for partitioned Runge-Kutta methods
We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.
On numerical integration of implicit ordinary differential equations
In this paper it is shown how the numerical methods for ordinary differential equations can be adapted to implicit ordinary differential equations. The resulting methods are of the same order as the corresponding methods for ordinary differential equations. The convergence theorem is proved and some numerical examples are given.
Stability analysis of reducible quadrature methods for Volterra integro-differential equations
Stability analysis for numerical solutions of Voltera integro-differential equations based on linear multistep methods combined with reducible quadrature rules is presented. The results given are based on the test equation and absolute stability is deffined in terms of the real parameters and . Sufficient conditions are illustrated for - methods and for combinations of Adams-Moulton and backward differentiation methods.
The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments
A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for...
Explicit two-step Runge-Kutta methods
The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order the minimal number of stages for explicit TSRK method of order is equal to the minimal number of stages for explicit Runge-Kutta method of order . Numerical results are presented which...
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