Journal of Applied Mathematics

Volume 2012 (2012), Article ID 978729, 18 pages

doi:10.1155/2012/978729

Research Article

Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions

Yeguo Sun

Department of Mathematics and Computational Science, Huainan Normal University, 238 Dongshan West Road, Huainan 232038, China

Received 30 March 2012; Revised 2 May 2012; Accepted 14 August 2012

Academic Editor: Ram N. Mohapatra

Copyright © 2012 Yeguo Sun. 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.

We propose long-time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

1. Introduction

Time delay systems which are described by delay differential equations (DDEs) or more generally functional differential equations (FDEs) have been studied rather extensively in the past thirty years since time delays are often the sources of instability and encountered in various engineering systems such as chemical processes, economic markets, chemical reactions, and population dynamics [1, 2]. Nakagiri [1] studied the structural properties of linear autonomous functional differential equations in Banach spaces within the framework of linear operator theory. The system stability and the compactness of the operators describing the solution trajectories are investigated in [2]. Depending on whether the existence of time delays or not, stability criteria for time delay systems can be divided into two types: delay-dependent ones and delay-independent ones. De la Sen and Luo [3] deal with the global uniform exponential stability independent of delay of time-delay linear and time-invariant systems. By exploiting appropriate Lyapunov functional candidate, new delay-dependent robust stability criteria of uncertain time-delay systems are proposed in [4].

However, most DDEs do not have analytic solution, so it is generally necessary to resort to numerical methods [5, 6]. It is well known that a numerical method which is convergent in a finite interval does not necessarily yield the same asymptotic behavior as the underlying differential equation. If the numerical solution defines a dynamical system, then we would study whether this dynamical system inherits the dynamics of the underlying system. Hence it is crucial to understand the behavior of numerical solution in order that we may interpret the data and facilitate the design of algorithms which provide correct qualitative information without being unduly expensive. The classical analysis of linear multistep methods [7–9] and Runge-Kutta methods [10–12] for delay differential equations involves assessment of stability and accuracy but has also been supplemented by considerable practical experience and experimentation.

As a basic tool, the Runge-Kutta method plays an important role in numerical integrations of delay differential equations. We usually designed this kind of numerical schemes in two ways. The first way is based on Taylor's expansion coupled with other techniques. The next is to construct numerical schemes by using collocation approximation. For instance, Butcher [13, 14] provided some implicit Runge-Kutta processes based on the Radau Quadrature formulas; also see [15, 16] and the references therein.

Recently, Guo et al. [17] proposed an integration process for ordinary differential equations based on modified Laguerre functions, which are very efficient for long-time numerical simulations of dynamical systems. But so far, to our knowledge, there is no work concerning the applications of Laguerre approximation to integration process for delay differential equations.

In this paper, we construct a new integration processes for delay differential equations based on modified Laguerre functions and establish its global convergence in certain weighted Sobolev space. Numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

2. Modified Laguerre-Radau Function for Delay Differential Equations

Let , , and define the weighted space as usual, with the following inner product and norm:

The modified Laguerre polynomial of degree is defined by (cf. [18])

For example,

The modified Laguerre polynomials satisfy the recurrence relation

The set of Laguerre polynomials is a complete -orthogonal system, namely,

where is the Kronecker function. Thus, for any ,

Let be any positive integer and the set of all algebraic polynomials of degree at most . We denote by the modified Laguerre-Radau interpolation points. Indeed, , and are the distinct zeros of . By using (2.1) and the formula (2.12) of [19], the corresponding Christoffel numbers are as follows:

For any ,

Next, we define the following discrete inner product and norm:

For any ,

For all , the modified Laguerre-Radau interpolant is determined by

By (2.10), for any ,

The interpolant can be expanded as

By virtue of (2.5) and (2.10),

Define the modified Laguerre functions as the base functions. According to (2.4), the functions satisfy the recurrence relation

Denote by and the inner product and the norm of the space , respectively. The set of is a complete -orthogonal system, that is,

We now introduce the new Laguerre-Radau interpolation. Set

Let and be the same as in (2.7), and take the nodes and weights of the new Laguerre-Radau interpolation as

The discrete inner product and norm can be defined similarly as

For any , , we have ,  , and , . Thus by (2.10),

The new Laguerre-Radau interpolant is determined by

Due to the equality (2.20), for any ,

Let

Then, with the aid of (2.16) and (2.22), we derive that

There is a close relation between and . From the previous two equalities, it follows that

This with (2.14) implies

Consider the following delay differential equation:

For any fixed positive integer , we define

and denote

Then system of (2.27) can be transformed into

We suppose that is sufficiently continuously differentiable for . Let

Then we obtain that

Now we derive an explicit expression for the left side of (2.32). Let be the coefficients of in terms of . Due to (2.15), we have

This equality and (2.24) imply that

Denote for and

Then

Denote

Then, we obtain

We now approximate by . Clearly, . Furthermore, we set

By replacing by and neglecting in (2.38), we derive a new integration process by using the modified Laguerre functions. It is to seek such that

The global numerical solution is

with

Let

Then (2.40) is equivalent to the system

3. Convergence Analysis

In this section, we estimate the global error of numerical solution. For any th continuously differentiable function , we set

The following lemmas will play a key role in obtaining our main results.

Lemma 3.1 (see [19]). If , then for an integer ,

Lemma 3.2 (see [19]). If , then for an integer ,

Theorem 3.3. Suppose that there exists a real number such that

and , , and are finite. Then

Proof. Let . Subtracting (2.32) from (2.44), we get

where

We now multiply (3.7) by and sum the result for . Due to , we obtain that

where

Using (2.20) and the Cauchy inequality, we deduce that

Thus (3.9) reads

Since there exists such that

where . Therefore

Hence it suffices to estimate . With the aid of (2.26), Lemmas 3.1 and 3.2, we deduce that, for ,

On the other hand,

It follows from the above result that for

Consequently,

Thus, (3.14) implies that

Furthermore, using (3.15) again, we obtain that, for any ,

This completes the proof.

Remark 3.4. The norm is finite as long as that satisfies certain conditions. If satisfies conditions

and , then , , see Tian [5]. Furthermore, if fulfills some additional conditions, then the norms appearing at the right sides of (3.20) are finite. Therefore, for certain positive constant depending only on ,

Consequently, for , the scheme (2.40) has the global convergence and the spectral accuracy in . Moreover, at the infinity, the numerical solution has the same accuracy. This also indicates that the pointwise numerical error decays rapidly as the mode increases, with the convergence rate as . On the other hand, for any fixed , the norm is bounded, and so the pointwise numerical error decays automatically as , at least less than , being a small number. Hence, it is very efficient for long-time numerical simulations of dynamical systems, especially for stiff problems.

Remark 3.5. The method proposed is also available for solving systems of delay differential equations. Let

We consider the system

We approximate by . We can derive a numerical algorithm which is similar to (2.38). Further, let be the Euclidean norm of . Assume that

Then we can obtain an error estimate similar to (3.20).

4. Refinement and Numerical Results

In the preceding sections, we introduced an integration process for solving delay differential equations. Theoretically, their numerical errors with bigger decrease faster. But in practical computation, it is not convenient to use them with very big . On the other hand, the distance between the adjacent interpolation nodes and increases fast as and increase, especially for the nodes which are located far from the origin point . This is one of advantages of the Laguerre interpolation approximation, since we can use moderate mode to evaluate the unknown function at large , but it is also its shortcoming. In fact, if the exact solution oscillates or changes very rapidly between two large adjacent interpolation nodes, then we may lose information about the structure of exact solution between those nodes. To remedy this deficiency, we may refine the numerical results as follows.

Let be a moderate positive integer, , and the set of nodes . We use (2.40) with the interpolation nodes to obtain the original numerical solution

Then we take and consider the following delay differential equation:

We get the refined numerical solution for . Repeating the above procedure, we obtain the refined numerical solution for .

5. Numerical Results

We consider the following system of four homogeneous delay differential equations:

The initial functions and solutions are given by

For , Figures 1 and 2 show plots of the exact solution and on , which controls the frequency of oscillation in the initial data and solution. Figures 3 and 4 are the exact and numerical solution on with , . The refined numerical results are given in Figures 5 and 6 on .

Figure 1: Exact solution.

Figure 2: Exact solution.

Figure 3: Numerical solution and exact solution.

Figure 4: Numerical solution and exact solution.

Figure 5: Numerical solution of refined method and exact solution.

Figure 6: Numerical solution of refined method and exact solution.

The numerical experiments show that our numerical integration processes are efficient for numerically solving delay differential equations.

6. Conclusions

In this paper, we proposed new integration processes of delay differential equations, which have fascinating advantages. On the one hand, the suggested integration processes are based on the modified Laguerre functions on the half line; they provide the global numerical solution and the global convergence naturally and thus are available for long-time numerical simulations of dynamical systems. On the other hand, benefiting from the rapid convergence of the modified Laguerre functions, these processes possess the spectral accuracy. In particular, the numerical results fit the exact solutions well at the interpolation nodes. Furthermore, We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

Acknowledgments

This work is supported by the Program of Science and Technology of Huainan no. 2011A08005. The authors wish to thank the anonymous referees for carefully correcting the preliminary version of the paper. Special and warm thanks are addressed to Professor Ram N. Mohapatra for his valuable comments and suggestions for improving this paper.

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