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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 652928, 14 pages
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.
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.
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 [3–7]). 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.
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 , .
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 2.3. 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.
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.
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 .
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:
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.
4. Stability Analysis
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 .
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
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.
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 .
Proof. It is obvious that Thus, the conditions in Theorem 3.3 are satisfied with , and . The proof is compete.
4.2. Another Way to View the Improved Linear Multistep Method (4.7)
Denote Then, it is easy to see that is continuous for .
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.
5. Numerical Experiment
In this section, some numerical experiments are given to illustrate the conclusion in the paper.
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.
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.
This work is supported by the NSF of China (no. 11071050).
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