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Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 683137, 14 pages

http://dx.doi.org/10.1155/2013/683137

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

^{1}Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China^{2}School of Management, Harbin University of Commerce, Harbin 150028, China

Received 29 January 2013; Revised 4 April 2013; Accepted 4 April 2013

Academic Editor: Hamid Reza Karimi

Copyright © 2013 Haiyan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 the -stability, -stability, and the weak -stability on the basis of -algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

#### 1. Introduction

Neutral delay differential equations (NDDEs) arise in a variety of fields as biology, economy, control theory, and electrodynamics (see, e.g., [1–5]). The stability and convergence properties of numerical methods for linear NDDEs have been widely researched by many authors (see, e.g., [6–11]). For the case of nonlinear delay differential equations, this kind of methodology had been first introduced by Bellen and Zennaro [12] and Torelli [13] and then developed by Torelli [14], Bellen [15], and Zennaro [16, 17]. In 1997, Koto proved the asymptotic stability of natural Runge-Kutta method for a class of nonlinear delay differential equations in [18]. 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 particular, Jackiewicz [20–22] systematically investigated the convergence of various numerical methods for more general neutral functional differential equations (NFDEs). In 2009, Yang et al. gave a novel robust stability criteria for stochastic Hopfield neural networks with time delays in [23]. Yang et al. [24] studied the exponential stability on stochastic neural networks with discrete interval and distributed delays in 2010. In 2011, Liu [25] gave the robust stability for Neutral time-varying delay systems with non-linear peturbations. On the stability, Tanikawa studied the values of random zero-sum games in [26], and in [27] Basin and Calderon-Alyarez gave the delay-dependent stability studies for vector nonlinear stochastic systems with multiple delays.

However, these important convergence results are based on the classical Lipschitz conditions. The studies focusing on the stability and convergence of the numerical method for nonlinear NDDEs based on a one-sided Lipschitz condition have not yet been seen in literature until now. By means of a one-sided Lipschitz condition, in the present paper we discuss the stability and convergence of two-step Runge-Kutta (TSRK) methods for nonlinear NDDEs. 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.

#### 2. Two-Step Runge-Kutta Methods for NDDEs

It is the purpose of this paper to investigate the nonlinear stability and convergence properties of the following NDDEs: where is a given mapping, is a positive delay term, and is a continuous function. Moreover, we assume that there exist some inner product and the induced norm in , such that where , , , and , , , are constants.

In order to make the error analysis feasible, we always assume that problem (1) has a unique solution which is sufficiently differentiable and satisfies and denotes the problem class that consists of all NDDEs with (2).

Many numerical methods have been proposed for the numerical solution of problem (1).

In this paper, we are concerned with two-step Runge-Kutta (TSRK) method of the formwhere , , is the numerical approximation at to the analytic solution , is a step size, and . The above methods are studied in [11]. Now we consider the adaptation of the two-step Runge-Kutta method to (1):where is the numerical approximation to the analytic solution with .

In particular, . The argument denotes an approximation to and the argument denotes an approximation to which are obtained by a specific interpolation procedure at the point . Using values with , .

Let with integer and , be integers. Define where We assume is to guarantee that no (unknown) values with are used in the interpolation procedure.

It should be pointed out that the adopted interpolation procedures (6) is only a class of interpolation procedure for ; there also exist some other types of interpolation procedures, such as numerical schemes which use Hermite interpolation between grid points (see [28–30]). It is the aim of our future research to investigate the future adaptation of two-step Runge-Kutta methods to NDDEs by means of other interpolation procedures.

#### 3. The Nonlinear Stability Analysis

In this section, we will investigate the stability of the two-step Runge-Kutta methods for NDDEs.

In order to consider the stability property, we also need to consider the perturbed problem of (1): where is a continuous function. The unique exact solution of the problem (8) is denoted as .

Applying the two-step Runge-Kutta method (4a), (4b), and (4c) to (8) leads to

##### 3.1. Some Concepts

*Definition 1. *Let be a real constant, a two-step Runge-Kutta method with an interpolation procedure is said to be -stable if there exists a constant dependent only on the method and such that and
with step size satisfying , where is a positive integer.

-stability is defined by dropping the restriction .

*Definition 2. *Let be a real constant, a two-step Runge-Kutta method with an interpolation procedure is said to be -stable if
with step size satisfying and , where is a positive integer.

-stability is defined by dropping the restriction .

*Definition 3. *Let be a real constant; a two-step Runge-Kutta method with an interpolation procedure is said to be weak -stable if, under the conditions of Definition 2, (11) holds when further satisfies
with being a positive real number and being a nonnegative real number.

Weak -stability is defined by dropping the restriction .

##### 3.2. The Stability of TSRK Methods

Let , , , , , and .

It follows from (5a), (5b), (5c), (9a), (9b), and (9c) that

Now we will write the s-stage TSRK methods (4a), (4b), and (4c) as a general linear method.

Let be the internal stages, the external vectors, and . Then we have a -stage partitioned general linear method:where

*Definition 4. *Let be real constants; a TSRK method is said to be -algebraically stable if there exists a diagonal nonnegative matrix and such that is nonnegative, where

In this paper, we use the linear interpolation procedure. Let with integer and .

Definewhere and for . When the step size satisfies , we have

Theorem 5. *Assume that a TSRK method is -algebraically stable. Then
*

*Proof. *It is well known that
where
By means of -algebraical stability of the method, we have
It follows from (2) and (6) that

Substitution into (20) gives (19).

Theorem 6. *Assume that a TSRK method is -algebraically stable and . Then the method with linear interpolation procedure is -stable.*

*Proof. *The inequality and Theorem 5 lead to
By induction, we have
Because , so we have

Therefore, it is -stable, where .

Theorem 7. *Assume that a TSRK method is -algebraically stable and . Then the method with linear interpolation procedure is -stable.*

*Proof. *Let and .

Then, when , we have and .

The application of Theorem 5 yields
By induction, we have
On the other hand,
Considering , and , we have
Because , so we have

which shows that the method is -stable.

Theorem 8. *Assume that a TSRK method is -algebraically stable, , and , . Then the method with linear interpolation procedure is weak -stable.*

*Proof. *It follows from Theorem 5 that
where . When , we have . Analogous to Theorem 6, we can easily obtain
which shows
On the other hand,
In view of (12) and (35), we have
Considering (9a), (9b), (9c), (22), and (37) with , we have
Because , so we have

which shows that the method is weak -stable.

#### 4. The Convergence of TSRK Method for NDDEs

##### 4.1. Some Concepts

In order to study the convergence of the method, we define

Thus, process (5a), (5b), and (5c) can be written in the more compact form

*Definition 9. *Method (4a), (4b), and (4c) with an interpolation procedure is said to be -convergent of order if the global error satisfies a bound of the form
where is defined by
and depend on , , , , and .

*Definition 10. *TSRK method (4a), (4b), and (4c) is said to be algebraically stable if there exist a real symmetric, positive definite matrix and a nonnegative diagonal matrix such that the matrix
is nonnegative definite.

*Definition 11. *TSRK method (4a), (4b), and (4c) is said to be diagonally stable if there exist an diagonal matrix such that the matrix is positive definite.

*Remark 12. *The concepts of algebraic stability and diagonal stability of TSRK method are the generalizations of corresponding concepts of Runge-Kutta methods. Although it is difficult to examine these conditions, many results have been found; in particular, there exist algebraically stable and diagonally stable multistep formulas of arbitrarily high order (cf. [31]).

*Definition 13. *TSRK method (4a), (4b), and (4c) is said to have generalized stage order if is the largest integer which possesses the following properties.

For any given problem (1) and , there exists an abstract function ,
such that
where the maximum step size and the constant depend only on the method and the bounds , , and ; they are defined by;

The function is defined by

In particular when , generalized stage order is called stage order.

##### 4.2. D-Convergence and Proofs

In this section, we focus on the error analysis of TSRK method for (1). For the sake of simplicity, we always assume that all constants , , , and are dependent only on the method, some of the bounds , and the parameters , , , and .

First, we give a preliminary result which will later be used several times. For any , , , , , for , , , , , , and , for , where . Define and by

Theorem 14. *Suppose that method (4a), (4b), and (4c) is diagonally stable. Then there exist constants , , and such that*

*Proof. *Since the method (4a), (4b), and (4c) is diagonally stable, there exists a positive definite diagonal matrix such that the matrix is positive definite. Therefore, the matrix is obviously nonsingular, and there exists an which depends only on the method such that the matrix
is also positive definite.

Define

Then

Using (2), (6), (53a), and (53b), we have, for ,
where is the minimum eigenvalue of . Therefore,
whereFrom (50a), (56a), and (56b), it follows that
where , which completes the proof of Theorem 14.

Consider the compact form of (9a), (9b), and (9c):

where

Theorem 15. *Suppose that the method (4a), (4b), and (4c) is algebraically stable for the matrices and . Then for (41a), (41b), (58a), and (58b) we have
**
where , is a norm on defined by
*

*Proof. *Define . We get from (52a), (52b), (52c), and (52d) that
With algebraic stability, the matrix
is nonnegative definite. As in [32], we have
Using (2), we further obtain

which gives (60). The proof is completed.

In the following, we assume that the method (4a), (4b), and (4c) has generalized stage order ; that is, there exists a function such that (46) holds. For any , we define and by

where

Theorem 16. *Suppose that the method (4a), (4b), and (4c) is diagonally stable and its generalized stage order is . Then there exist constants and such that
*

*Proof. *It follows from (6) that
From the remainder estimation of Lagrange interpolation formula, we have
Using Cauchy inequality, we further obtain
where . Hence, there exists a constant such that
On the other hand, a combination of (41a) and (47a) leads to
It follows from Theorem 14 that
which on substitution into (72) gives

Therefore, there exist and such that (68) holds. The proof of Theorem 16 is completed.

Theorem 17. *Suppose that method (4a), (4b), and (4c) is algebraically stable and diagonally stable and its generalized stage order is . Then the method with interpolation procedure (6) is -convergent of order at least .*

*Proof. *In view of (41a), (41b), (66a), and (66b), it follows from Theorem 15 that
Using Theorem 14, we have
which on substitution into (76) gives