Mathematical Problems in Engineering

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

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

Research Article

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

Haiyan Yuan,¹ Cheng Song,² and Peichen Wang¹

¹Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

²School 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 form

where , , 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 .

Define

where 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,

where

From (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

where is the minimum characteristic value of . On the other hand,

In view of (47a), (47b), (48a), (66a), and (66b), the application of Theorem 14 leads to

which gives

where denotes the maximum eigenvalue of the matrix . A combination of (46) and (76)–(81) leads to

where

Therefore,

Considering Theorem 16, we further obtain

where . Using discrete Bellman inequality, we have

Considering , we obtain

Considering (87) and (88), we can easily conclude that method (4a), (4b), and (4c) with interpolation procedure (6) is -convergent of order at least . The proof is completed.

5. Numerical Experiments

Consider the following nonlinear neutral delay differential equations:

and its perturbed problem

We can calculate , , , and . Equation (89) has a unique true solution:

Apply the two-step Runge-Kutta method induced by the GL method in [12]

(92)

to (89) and (90), is computed by the Lagrange interpolation procedure with (see Figures 1 and 2 and Table 1).

Table 1: Convergence order of two-step Runge-Kutta methods for system (89).

5
10
20
40

Convergence order
4.0012
3.9880
3.9864
3.9860

Figure 1: The numerical solution of (92) for (89) with .

Figure 2: The numerical solution of (92) for (90) with .

It is obvious that the corresponding method for NDDEs is stable and convergent, and the convergence order is .

6. Conclusions

In this paper we gave the stability and convergence results of two-step Runge-Kutta methods with linear interpolation procedure for solving nonlinear NDDEs (1). First, we gave the definitions of -algebraically stable, algebraically stable and diagonally stable. Then we proved that if a TSRK method is -algebraically stable, and   , then the method with linear interpolation procedure is weak -stable. We also proved that if a TSRK is algebraically stable and diagonally stable and its generalized stage order is , then the method with interpolation procedure is -convergent of order at least .

We believe that the results presented in this paper can be extended to other general NDDEs. However, it is difficult to extend these results to more general neutral functional differential equations. Results extending the results presented in this paper to more general neutral functional differential equations and other delay differential equations such as delay integral differential equations will be discussed elsewhere.

Acknowledgments

This work was supported by the Education Foundation of Heilongjiang Province of China (12523039), the Natural Science Foundation of Heilongjiang Province of China (QC2011C020), and the Doctor Foundation of Heilongjiang Institute of Technology (2012BJ27).

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