Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Neutral Delay Differential Equations

Hongliang Yuan ,Cheng Song,Peichen Wang

doi:10.1155/2013/683137

Abstract

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

Highlights

Neutral delay differential equations (NDDEs) arise in a variety of fields as biology, economy, control theory, and electrodynamics
Jackiewicz [20,21,22] systematically investigated the convergence of various numerical methods for more general neutral functional differential equations (NFDEs)
We proved that if a two-step RungeKutta (TSRK) is algebraically stable and diagonally stable and its generalized stage order is p, the method with interpolation procedure is D-convergent of order at least min{p, μ + V + 1}

Summary

Introduction

Neutral delay differential equations (NDDEs) arise in a variety of fields as biology, economy, control theory, and electrodynamics (see, e.g., [1,2,3,4,5]). Bellen et al [19] gave a discussion of the stability of continuous numerical methods for a special class of nonlinear neutral delay differential equations. In 2009, Yang et al gave a novel robust stability criteria for stochastic Hopfield neural networks with time delays in [23]. Tanikawa studied the values of random zero-sum games in [26], and in [27] Basin and Calderon-Alyarez gave the delaydependent stability studies for vector nonlinear stochastic systems with multiple delays. These important convergence results are based on the classical Lipschitz conditions. Thanks to the one-sided nature of the Lipschitz condition, the error bounds obtained in the present paper are sharper than those given in the references mentioned

Two-Step Runge-Kutta Methods for NDDEs

The Nonlinear Stability Analysis

Some Concepts

The Convergence of TSRK Method for NDDEs

Numerical Experiments

Conclusions