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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 652928, 14 pages
http://dx.doi.org/10.1155/2012/652928
Research Article

Linear Multistep Methods for Impulsive Differential Equations

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 18 January 2012; Accepted 9 March 2012

Academic Editor: Leonid Shaikhet

Copyright © 2012 X. Liu 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 deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and two-step BDF method are of order when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

1. Introduction

Impulsive differential equations provide a natural framework for mathematical modeling in ecology, population dynamic, optimal control, and so on. The studies focus on the theory of impulsive differential equations initiated in [1, 2]. In recent years many researches on the theory of impulsive differential equations are published (see [37]). And the numerical properties of impulsive differential equations begin to attract the authors’ interest (see [8, 9]). But there are still few papers focus on the numerical properties of linear multistep methods for impulsive differential equations. In this paper, we will study the convergence and stability of linear multistep methods.

This paper focuses on the numerical solutions of impulsive differential equations as follows where and with . We assume , where is the right limit of .

In this paper, we consider the following equation: where , , , , .

Remark 1.1. If , then we obtain that for . Therefore we omit this case in the paper.

Definition 1.2 (see [4]). is said to be the solution of (1.2), if(1);(2) is differentiable and for ;(3) is left continuous in and .

Theorem 1.3 (see [8]). Equation (1.2) has a unique solution in where denotes the greatest integer function towards minus infinity.

2. Linear Multistep Methods

2.1. Linear Multistep Methods for ODEs

The standard form of linear multistep methods can be defined by where and are constants subject to the conditions: and , .

Theorem 2.1 (see [10]). The linear multistep methods (2.1) are convergent of order for ODEs if and only if the following conditions are satisfied:

2.2. Linear Multistep Methods for Impulsive Differential Equations

Let be a given stepsize with integer . In this subsection, we consider the case when . The application of the linear multistep methods (2.1) in case of (1.2) yields where is an approximation of , and denotes an approximation of . Here, we assume that the other starting value besides , that is, , has been calculated by a one-step method of order 2.

Remark 2.2. As a special case, when the corresponding consistent process (2.4) takes the form: which is consistent with process (2.2) in [9].

Remark 2.3. When , the corresponding process (2.4) takes the form: where we assume that has been calculated by a one-step method of order .

In order to test the convergence, we consider the following equations: We use the process (2.5) in case of (i.e., the explicit Euler method) and the process (2.6) in case of the mid-point rule and 2-step BDF methods to get numerical solutions at , where the corresponding analytic solution can be calculated by Theorem 1.3. We have listed the absolute errors and the ratio of the errors of the case over that in the following tables.

We can conclude from Table 1 that the explicit Euler method is of order 1 which means that the process (2.4) is defined reasonable. Tables 2 and 3 imply that both methods are of order 0, when applied to the given impulsive differential equations.

tab1
Table 1: The explicit Euler method for (2.7) and (2.8).
tab2
Table 2: The mid-point rule for (2.7) and (2.8).
tab3
Table 3: The 2-step BDF methods for (2.7) and (2.8).

3. The Improved Linear Multistep Methods

In this section, we will consider the improved linear multistep methods: where , and . The application of method (3.1) in case of (1.2), yields In the rest section of this section, we will propose a convergence condition of the method (3.1) for (1.2). Firstly we give a definition about the residual of (3.2), which is essentially the local truncation error.

Definition 3.1. Assume that is the analytic solution of (1.2). Then the residual of process (3.2) is defined by

Definition 3.2. The improved linear multistep methods (3.2) is said to be of order for (1.2), if the residual defined by (3.3) satisfies for arbitrary .

The following theorem gives a condition under which the improved linear multistep methods can preserve their original order for ODEs when applied to (1.2). Without loss of generality, we assume for , where denotes the fractional part of .

Theorem 3.3. Assume (2.3) holds, and there exists a function together with a constant such that and in (3.2) satisfy Then, the improved linear multistep methods (3.2) are of order for (1.2).

Proof. It follows from Definition 3.1 that By Theorem 1.3, we have Therefore, We can express the residual as a power series in : collecting terms in to obtain Then, By (2.3), Therefore, . The proof is complete.

3.1. An Example

Denote , then we can define the coefficients of (3.2) as follows:

Theorem 3.4. Assume that (2.3) holds. Then the improved linear multistep methods formed by (3.11) are of order for (1.2).

Proof. We only need to verify that the condition in Theorem 3.3 holds. Note that Thus, if and only if , that is, , . Therefore, Hence, Thus, the conditions in Theorem 3.3 are satisfied with and . Thus, the proof is complete.

Remark 3.5. If in (3.11), that is, the impulsive differential equations reduce to ODEs, we have , , that is, the improved linear multistep methods (3.2) reduce to the classical linear multistep methods.

Remark 3.6. In the improved linear multistep method defined by (3.11), the stepsize can be chosen with arbitrary positive integer without any restriction.

Remark 3.7. If , then , . In other words, if all the mesh points are in the same integer interval, then the process (3.2) defined by (3.11) reduces to the classical linear multistep methods (2.1).

4. Stability Analysis

In this section, we will investigate the stability of the improved linear multistep methods (3.1) for (1.2). The following theorem is an extention of Theorem  1.4 in [9], and the proof is obvious.

Theorem 4.1 (see [9]). The solution of (1.2) is asymptotically stable if and only if .

The corresponding property of the numerical solution is described as follows.

Definition 4.2. The numerical solution is called asymptotically stable for (1.2) with if for arbitrary stepsize .

Theorem 4.3. Assume there exist constants , and a consequence of functions , such that for arbitrary and , and the following equality holds Then, if the corresponding linear multistep methods (2.1) are -stable and .

Proof. Denote Then by (4.1), the process (3.2) becomes Therefore, can be viewed as the numerical solution of the equation calculated by linear multistep methods (2.1).
On the other hand, we know that and the methods are -stable. Therefore, . The conclusion is obvious in view of that

Corollary 4.4. The improved linear multistep method (3.11) is asymptotically stable for (1.2) when the corresponding linear multistep methods (2.1) are -stable, and hold.

Proof. It is obvious that for (3.11): where . Note that , , and are all bounded when the method and the stepsize are given. Therefore, are uniformly bounded. Thus the proof is complete.

Remark 4.5. In fact, the improved linear multistep methods (3.11) cannot preserve the asymptotical stability of all equation (1.2). To illustrate this, we consider the following equation: Theorem 4.1 implies that . We have drawn the numerical solution calculated by method (3.11) in case of 2-step BDF methods, which is -stable as we know, on in Figure 1. Figure 1 indicates that the numerical solutions are not asymptotically stable. Hence, we will give another improved linear multistep method in the next section.

652928.fig.001
Figure 1: The numerical solution obtained by (3.11) in case of the 2-step BDF method to (4.6).
4.1. Another Improved Linear Multistep Methods

In this section, we give another improved linear multistep methods. We define the coefficients as follows: where we define , when .

Theorem 4.6. Assume that (2.3) holds. Then, the improved linear multistep methods formed by (4.7) are of order for (1.2).

Proof. It is obvious that Thus, the conditions in Theorem 3.3 are satisfied with , and . The proof is compete.

Theorem 4.7. Assume that . Then, if the corresponding linear multistep methods (2.1) are -stable, where is obtained by the improved linear multistep method (4.7).

Proof. Define Then (4.1) is satisfied, and . Therefore the conditions of Theorem 4.3 are satisfied, and the conclusion follows.

4.2. Another Way to View the Improved Linear Multistep Method (4.7)

In fact, the improved linear multistep methods (4.7) can be viewed as the application of the classical linear multistep methods to the modified form of (1.2).

Denote Then, it is easy to see that is continuous for .

Lemma 4.8. Assume that (4.10) holds. Then is the solution of (1.2) if and only if is the solution of

Proof. Necessity. In view of Theorem 1.3 and (4.10), we obtain that in the case : and , which is coincided with the solution of (4.11). The necessity can be proved in the same way, and the proof is complete.

It follows from (4.10) that the numerical solutions of (1.2) can be approximated by means of as follows: It is obvious that the methods (4.7) and (4.13) are the same.

5. Numerical Experiment

In this section, some numerical experiments are given to illustrate the conclusion in the paper.

5.1. Convergence

The improved 2-step linear multistep methods (3.11) takes the form: where we assume that has been calculated by a one-step method of order . We use the methods (5.1) and (4.13) in case of the mid-point rule and 2-step BDF method. We consider (2.7) and (2.8) and calculate the numerical solutions at with stepsize . We have listed the absolute errors and the ratio of the errors of the case over the error in the case , from which we can estimate the convergent order. We can see from Tables 4, 5, 6, and 7, that all methods can preserve their original order for ODEs.

tab4
Table 4: Equation (5.1) in case of mid-point rule for (2.7) and (2.8).
tab5
Table 5: Equation (5.1) in case of 2-step BDF methods for (2.7) and (2.8).
tab6
Table 6: Equation (4.13) in case of mid-point rule for (2.7) and (2.8).
tab7
Table 7: Equation (4.13) in case of 2-step BDF methods for (2.7) and (2.8).
5.2. Stability

To illustrate the stability, we consider (4.6). We use method (4.13) in case of the 2-step BDF method. We draw the module of numerical solution on in Figure 2. We can see from the figure that the method can preserve the stability of the analytic solution.

652928.fig.002
Figure 2: The numerical solution obtained by (4.13) in case of the 2-step BDF method to (4.6).

Acknowledgment

This work is supported by the NSF of China (no. 11071050).

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