Linear Multistep Methods for Impulsive Differential Equations

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 twostep BDFmethod are of order p 0when 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.


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 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 x t f t, x , t > 0, t / τ d , d ∈ N, where f : R × C n → C n and τ d < τ d 1 with lim k → ∞ τ k ∞.We assume Δx x t 0 − x t , where x t 0 is the right limit of x t .
In this paper, we consider the following equation: x

Linear Multistep Methods for ODEs
The standard form of linear multistep methods can be defined by where α i and β i are constants subject to the conditions: and f n i : f t n i , x n i , i 0, 1, . . ., k. 2.3

Linear Multistep Methods for Impulsive Differential Equations
Let h 1/m be a given stepsize with integer m.In this subsection, we consider the case when m ≥ k.The application of the linear multistep methods 2.1 in case of 1.2 yields where x wm,l is an approximation of x t wm l , and x wm,0 denotes an approximation of x w .
Here, we assume that the other starting value besides x 0 , that is, x 0,1 , . . ., x 0,k−1 , has been calculated by a one-step method of order 2.
where we assume that x 0,1 has been calculated by a one-step method of order p ≥ 2.
In order to test the convergence, we consider the following equations:

2.8
We use the process 2.5 in case of β 1 0 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 t 5, 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 m 160 over that m 320 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.

The Improved Linear Multistep Methods
In this section, we will consider the improved linear multistep methods: where 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 x t is the analytic solution of 1.2 .Then the residual of process 3.2 is defined by The improved linear multistep methods 3.2 is said to be of order p for 1.2 , if the residual defined by 3.3 satisfies R n O h p 1 for arbitrary n.
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 {t n i−k } 0 for i i 0 , i 1 , . . ., i c , 0 ≤ i c < k, where {t} denotes the fractional part of t.Theorem 3.3.Assume 2.3 holds, and there exists a function μ n μ n together with a constant λ such that α n i and β n i in 3.2 satisfy

3.4
Then, the improved linear multistep methods 3.2 are of order p for 1.2 .
Proof.It follows from Definition 3.1 that By Theorem 1.3, we have

3.7
We can express the residual as a power series in h: collecting terms in R n to obtain 3.9 By 2.3 , r q 0, q 0, 1, . . ., p, r p 1 / 0.

3.10
Therefore, R n k O h p 1 .The proof is complete.

An Example
Denote l m{n/m}, 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

8
Discrete Dynamics in Nature and Society Hence,

3.14
Thus, the conditions in Theorem 3.3 are satisfied with μ n 1 b n/m and λ a.Thus, the proof is complete.

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.The corresponding property of the numerical solution is described as follows.
Then, lim n → ∞ x n 0 if the corresponding linear multistep methods 2.1 are A-stable and < 0. Proof.Denote Then by 4.1 , the process 3.2 becomes Therefore, y n can be viewed as the numerical solution of the equation y t y t calculated by linear multistep methods 2.1 .
On the other hand, we know that < 0 and the methods are A-stable.Therefore, lim n → ∞ y n 0. The conclusion is obvious in view of that 4.4 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 A-stable, and a < 0 hold.
Proof.It is obvious that for 3.11 : where γ i 1 b k−l−i /m 1 .Note that k, l, and i are all bounded when the method and the stepsize are given.Therefore, |1/γ i | 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:

4.6
Theorem 4.1 implies that lim t → ∞ x t 0. We have drawn the numerical solution calculated by method 3.11 in case of 2-step BDF methods, which is A-stable as we know, on 0, 500 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.

Another Improved Linear Multistep Methods
In this section, we give another improved linear multistep methods.We define the coefficients as follows: where we define β n i 0, when a 0.
Theorem 4.6.Assume that 2.3 holds.Then, the improved linear multistep methods formed by 4.7 are of order p for 1.2 .
Proof.It is obvious that

4.8
Thus, the conditions in Theorem 3. Proof.Define  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. x 0 e at 1 b t x 0 e a ln 1 b t x 0 .4.12 and y k 1 b x k e a ln 1 b k x 0 , which is coincided with the solution of 4.11 .The necessity can be proved in the same way, and the proof is complete.

4.13
It is obvious that the methods 4.7 and 4.13 are the same.

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

Convergence
The improved 2-step linear multistep methods 3.11 takes the form: where we assume that x 0,1 has been calculated by a one-step method of order p ≥ 2. 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 t 5 with stepsize h 1/m.We have listed the absolute errors and the ratio of the errors of the case m 160 over the error in the case m 320, 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.

Remark 3 . 5 .Remark 3 . 6 .Remark 3 . 7 .
If b 0 in 3.11 , that is, the impulsive differential equations reduce to ODEs, we have α n i α i , β n i β i , that is, the improved linear multistep methods 3.2 reduce to the classical linear multistep methods.In the improved linear multistep method defined by 3.11 , the stepsize h 1/m can be chosen with arbitrary positive integer m without any restriction.If m ≥ l ≥ k, then α n i α i , β n i β i .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 .

Theorem 4 .1 see 9 .
The solution x t ≡ 0 of 1.2 is asymptotically stable if and only if | 1 b e a | < 1.

Definition 4 . 2 .
The numerical solution x n is called asymptotically stable for 1.2 with | 1 b e a | < 1 if lim n → ∞ x n 0 for arbitrary stepsize h 1/m.

Theorem 4 . 3 .
Assume there exist constants , C > 0 and a consequence of functions γ i γ i n, i , i 0, . . ., k such that |1/γ i | < C for arbitrary n and 0 ≤ i ≤ k, and the following equality holds

5 tFigure 1 :
Figure 1: The numerical solution obtained by 3.11 in case of the 2-step BDF method to 4.6 .

Figure 2 :
Figure 2: The numerical solution obtained by 4.13 in case of the 2-step BDF method to 4.6 .
0 ∈ C, b / − 1, d ∈ N.Definition 1.2 see 4 .x t is said to be the solution of 1.2 , if 1 lim t → 0 x t x 0 ; 2 x t is differentiable and x t ax t for t ∈ 0, ∞ , t / d, d ∈ N; 3 x t is left continuous in 0, ∞ and x d 1 b x d , d ∈ N. Theorem 1.3 see 8 .Equation 1.2 has a unique solution in 0, ∞ x t x 0 e at 1 b t , t > 0, t / d, d ∈ N, x 0 e ad 1 b d−1 , t d, 1.3 where • denotes the greatest integer function towards minus infinity.

Table 1 :
The explicit Euler method for 2.7 and 2.8 .

Table 2 :
The mid-point rule for 2.7 and 2.8 .
3 are satisfied with μ n 1, and λ a ln 1 b .The proof is compete.Assume that a ln 1 b < 0.Then, lim n → ∞ x n 0 if the corresponding linear multistep methods 2.1 are A-stable, where x n is obtained by the improved linear multistep method 4.7 .

Table 4 :
Equation 5.1 in case of mid-point rule for 2.7 and 2.8 .

Table 5 :
Equation 5.1 in case of 2-step BDF methods for 2.7 and 2.8 .

Table 6 :
Equation 4.13 in case of mid-point rule for 2.7 and 2.8 .

Table 7 :
Equation 4.13 in case of 2-step BDF methods for 2.7 and 2.8 .