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
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 [
This paper focuses on the numerical solutions of impulsive differential equations as follows
In this paper, we consider the following equation:
If
Equation (
The standard form of linear multistep methods can be defined by
The linear multistep methods (
Let
As a special case, when
When
In order to test the convergence, we consider the following equations:
We can conclude from Table
The explicit Euler method for (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 5.025613608106683 | 1.283179151329931 |
20 | 2.739728003516164 | 6.621084007886913 |
40 | 1.433580942834451 | 3.361075658131119 |
80 | 7.337102544190930 | 1.693073363701148 |
160 | 3.712172178908295 | 8.496581963094774 |
320 | 1.867162056329107 | 4.256079227844101 |
1.988136041178199 | 1.996340177952628 |
The mid-point rule for (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 1.734606333614340 | 2.611636966771883 |
20 | 1.785542792421782 | 2.362243809005279 |
40 | 1.811021056819620 | 2.241871891262895 |
80 | 1.823713002067653 | 2.182840114478902 |
160 | 1.830041179075918 | 2.153620314650900 |
320 | 1.833200086044313 | 2.139085286186221 |
9.982768345951743 | 1.006794973794894 |
The 2-step BDF methods for (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 4.051557443305367 | 1.585857685173612e + 000 |
20 | 4.538988757794288 | 1.365280016628375 |
40 | 4.842123370314762 | 1.268609538412996 |
80 | 5.011596302177681 | 1.223047705616427 |
160 | 5.101279019662662 | 1.200895589468623 |
320 | 5.147422493980224 | 1.189969480734175 |
9.910356155198985 | 1.009181839460040 |
In this section, we will consider the improved linear multistep methods:
Assume that
The improved linear multistep methods (
The following theorem gives a condition under which the improved linear multistep methods can preserve their original order for ODEs when applied to (
Assume (
It follows from Definition
Denote
Assume that (
We only need to verify that the condition in Theorem
If
In the improved linear multistep method defined by (
If
In this section, we will investigate the stability of the improved linear multistep methods (
The solution
The corresponding property of the numerical solution is described as follows.
The numerical solution
Assume there exist constants
Denote
On the other hand, we know that
The improved linear multistep method (
It is obvious that for (
In fact, the improved linear multistep methods (
The numerical solution obtained by (
In this section, we give another improved linear multistep methods. We define the coefficients as follows:
Assume that (
It is obvious that
Assume that
Define
In fact, the improved linear multistep methods (
Denote
Assume that (
Necessity. In view of Theorem
It follows from (
In this section, some numerical experiments are given to illustrate the conclusion in the paper.
The improved 2-step linear multistep methods (
Equation (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | ||
20 | ||
40 | ||
80 | ||
160 | ||
320 | ||
Equation (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 3.675982365982200e + 002 | 9.516633893408510e − 003 |
20 | 9.552174783206283e + 001 | 2.325825408788229e − 003 |
40 | 2.442645505222754e + 001 | 5.749450957563962e − 004 |
80 | 6.181500463695556e + 000 | 1.429301543146577e − 004 |
160 | 1.555180585986818e + 000 | 3.563220614588580e − 005 |
320 | 3.900496418718831e − 001 | 8.895535536623811e − 006 |
3.987135018310406e + 000 | 4.005627991612696e + 000 |
Equation (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 1.533815775376497e + 003 | 4.656258034474448e − 006 |
20 | 4.273673528511026e + 002 | 1.126844488719137e − 006 |
40 | 1.116157821124107e + 002 | 2.771258855727155e − 007 |
80 | 2.844156657094209e + 001 | 6.871261282181962e − 008 |
160 | 7.173499151609576e + 000 | 1.710731889481565e − 008 |
320 | 1.800995925495954e + 000 | 4.267991404738325e − 009 |
3.983073503974838e + 000 | 4.008283352169618e + 000 |
Equation (
Absolute errors for ( | Absolute errors for ( | |
---|---|---|
10 | 3.034120865054534e + 003 | 9.093526915804340e − 006 |
20 | 8.173504073904187e + 002 | 2.227110126540310e − 006 |
40 | 2.159842038101851e + 002 | 5.509781502155420e − 007 |
80 | 5.579637010861916e + 001 | 1.370190298999319e − 007 |
160 | 1.419913379106947e + 001 | 3.416407701184454e − 008 |
320 | 3.582733454190020e + 000 | 8.529607353757740e − 009 |
3.963212438944776e + 000 | 4.005351664492912e + 0 00 |
To illustrate the stability, we consider (
The numerical solution obtained by (
This work is supported by the NSF of China (no. 11071050).