# Linear Multistep Methods for Impulsive Differential Equations

Authors: X. Liu; M. H. Song; M. Z. Liu
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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 orderp=0 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.


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## Body

## 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 [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(1.1)x′(t)=f(t,x),t>0,t≠τd,d∈N,Δx=Ik(x),t=τd,x(t0+)=x0,
where f:R+×ℂn→ℂn and τd<τd+1 with limk→+∞τk=+∞. 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:(1.2)x′(t)=ax,t>0,t≠d,Δx=bx,t=d,x(0+)=x0,
where a, b, x0∈ℂ, b≠-1, d∈N.Remark 1.1.
Ifb=-1, then we obtain that x(t)≡0 for t>1. Therefore we omit this case in the paper.Definition 1.2 (see [4]).
x(t) is said to be the solution of (1.2), if(1)
limt→0+x(t)=x0;(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,+∞)(1.3)x(t)={x0eat(1+b)[t],t>0,t≠d,d∈N,x0ead(1+b)d-1,t=d,
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(2.1)∑i=0kαixn+i=h∑i=0kβifn+i,
where αi and βi are constants subject to the conditions:(2.2)αk=1,|α0|+|β0|≠0,
and fn+i:=f(tn+i,xn+i), i=0,1,…,k.Theorem 2.1 (see [10]).
The linear multistep methods (2.1) are convergent of order p for ODEs  if and only if the following conditions are satisfied:
(2.3)∑i=0kαi=0,∑i=0kαiiq=q∑i=0kβiiq-1forq=1,…,p,∑i=0kαiip+1≠(p+1)∑i=0kβiip.


### 2.2. Linear Multistep Methods for Impulsive Differential Equations

Leth=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
(2.4)x0,0=x0,xwm,l=-∑i=0k-1αi-haβiαk-haβkxwm,l+i-k,w=0,1,…,l=k,…,m,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,l=-∑i=0k-l-1αi-haβiαk-haβkxwm,m+l+i-k-∑i=k-lk-1αi-haβiαk-haβkx(w+1)m,l+i-k,l=1,…,k-1,
where xwm,l is an approximation of x(twm+l), and xwm,0 denotes an approximation of x(w+). Here, we assume that the other starting value besides x0, that is, x0,1,…,x0,k-1, has been calculated by a one-step method of order 2.Remark 2.2.
As a special case, whenk=1 the corresponding consistent process (2.4) takes the form:
(2.5)x0,0=x0,xwm,l=(1+ha1-haβ1)xwm,l-1,l=1,…,m,w=0,1,…,x(w+1)m,0=(1+b)xwm,m,
which is consistent with process (2.2) in [9].Remark 2.3.
Whenk=2, the corresponding process (2.4) takes the form:
(2.6)x0,0=x0,xwm,l=-α0-haβ0α2-haβ2xwm,l-2-α1-haβ1α2-haβ2xwm,l-1,w=0,1,…,l=2,…,m,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,1=-α0-haβ0α2-haβ2xwm,m-1-α1-haβ1α2-haβ2x(w+1)m,0,
where we assume that x0,1 has been calculated by a one-step method of order p≥2.In order to test the convergence, we consider the following equations:(2.7)x′(t)=x,t>0,t≠d,Δx=2x,t=d,x(0+)=2,(2.8)x′(t)=-x,t>0,t≠d,Δx=2x,t=d,x(0+)=1.
We use the process (2.5) in case of β1=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 Table1 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.Table 1
The explicit Euler method for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)105.025613608106683e+0031.283179151329931e-001202.739728003516164e+0036.621084007886913e-002401.433580942834451e+0033.361075658131119e-002807.337102544190930e+0021.693073363701148e-0021603.712172178908295e+0028.496581963094774e-0033201.867162056329107e+0024.256079227844101e-003Ratio1.988136041178199e+0001.996340177952628e+000Table 2
The mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.734606333614340e+0042.611636966771883e+002201.785542792421782e+0042.362243809005279e+002401.811021056819620e+0042.241871891262895e+002801.823713002067653e+0042.182840114478902e+0021601.830041179075918e+0042.153620314650900e+0023201.833200086044313e+0042.139085286186221e+002Ratio9.982768345951743e-0011.006794973794894e+000Table 3
The 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)104.051557443305367e+0041.585857685173612e + 000204.538988757794288e+0041.365280016628375e+000404.842123370314762e+0041.268609538412996e+000805.011596302177681e+0041.223047705616427e+0001605.101279019662662e+0041.200895589468623e+0003205.147422493980224e+0041.189969480734175e+000Ratio9.910356155198985e-0011.009181839460040e+000


## 2.1. Linear Multistep Methods for ODEs

The standard form of linear multistep methods can be defined by(2.1)∑i=0kαixn+i=h∑i=0kβifn+i,
where αi and βi are constants subject to the conditions:(2.2)αk=1,|α0|+|β0|≠0,
and fn+i:=f(tn+i,xn+i), i=0,1,…,k.Theorem 2.1 (see [10]).
The linear multistep methods (2.1) are convergent of order p for ODEs  if and only if the following conditions are satisfied:
(2.3)∑i=0kαi=0,∑i=0kαiiq=q∑i=0kβiiq-1forq=1,…,p,∑i=0kαiip+1≠(p+1)∑i=0kβiip.


## 2.2. Linear Multistep Methods for Impulsive Differential Equations

Leth=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
(2.4)x0,0=x0,xwm,l=-∑i=0k-1αi-haβiαk-haβkxwm,l+i-k,w=0,1,…,l=k,…,m,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,l=-∑i=0k-l-1αi-haβiαk-haβkxwm,m+l+i-k-∑i=k-lk-1αi-haβiαk-haβkx(w+1)m,l+i-k,l=1,…,k-1,
where xwm,l is an approximation of x(twm+l), and xwm,0 denotes an approximation of x(w+). Here, we assume that the other starting value besides x0, that is, x0,1,…,x0,k-1, has been calculated by a one-step method of order 2.Remark 2.2.
As a special case, whenk=1 the corresponding consistent process (2.4) takes the form:
(2.5)x0,0=x0,xwm,l=(1+ha1-haβ1)xwm,l-1,l=1,…,m,w=0,1,…,x(w+1)m,0=(1+b)xwm,m,
which is consistent with process (2.2) in [9].Remark 2.3.
Whenk=2, the corresponding process (2.4) takes the form:
(2.6)x0,0=x0,xwm,l=-α0-haβ0α2-haβ2xwm,l-2-α1-haβ1α2-haβ2xwm,l-1,w=0,1,…,l=2,…,m,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,1=-α0-haβ0α2-haβ2xwm,m-1-α1-haβ1α2-haβ2x(w+1)m,0,
where we assume that x0,1 has been calculated by a one-step method of order p≥2.In order to test the convergence, we consider the following equations:(2.7)x′(t)=x,t>0,t≠d,Δx=2x,t=d,x(0+)=2,(2.8)x′(t)=-x,t>0,t≠d,Δx=2x,t=d,x(0+)=1.
We use the process (2.5) in case of β1=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 Table1 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.Table 1
The explicit Euler method for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)105.025613608106683e+0031.283179151329931e-001202.739728003516164e+0036.621084007886913e-002401.433580942834451e+0033.361075658131119e-002807.337102544190930e+0021.693073363701148e-0021603.712172178908295e+0028.496581963094774e-0033201.867162056329107e+0024.256079227844101e-003Ratio1.988136041178199e+0001.996340177952628e+000Table 2
The mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.734606333614340e+0042.611636966771883e+002201.785542792421782e+0042.362243809005279e+002401.811021056819620e+0042.241871891262895e+002801.823713002067653e+0042.182840114478902e+0021601.830041179075918e+0042.153620314650900e+0023201.833200086044313e+0042.139085286186221e+002Ratio9.982768345951743e-0011.006794973794894e+000Table 3
The 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)104.051557443305367e+0041.585857685173612e + 000204.538988757794288e+0041.365280016628375e+000404.842123370314762e+0041.268609538412996e+000805.011596302177681e+0041.223047705616427e+0001605.101279019662662e+0041.200895589468623e+0003205.147422493980224e+0041.189969480734175e+000Ratio9.910356155198985e-0011.009181839460040e+000


## 3. The Improved Linear Multistep Methods

In this section, we will consider the improved linear multistep methods:(3.1)∑i=0kα̂inxn+i-k=h∑i=0kβ̂inf(tn+i-k,xn+i-k),
where α̂in=α̂in(n,i), β̂in=β̂in(n,i) and h=1/m. The application of method (3.1) in case of  (1.2), yields(3.2)∑i=0kα̂inxn+i-k=h∑i=0kaβ̂inxn+i-k.
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 thatx(t) is the analytic solution of (1.2). Then the residual of process (3.2) is defined by
(3.3)Rn=∑i=0kα̂inx(tn+i-k)-h∑i=0kaβ̂inx(tn+i-k).Definition 3.2.
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 Rn=O(hp+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 {tn+i-k}=0 for i=i0,i1,…,ic,0≤ic<k, where {t} denotes the fractional part of t.Theorem 3.3.
Assume (2.3) holds, and there exists a function μn=μ(n) together with a constant λ such that α̂in and β̂in in (3.2) satisfy
(3.4)∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)[tn+ij-k]-1=μn∑i=0k(αi-hλβi)eλtn+i-k.
Then, the improved linear multistep methods (3.2) are of order p for (1.2).Proof.
It follows from Definition3.1 that
(3.5)Rn=∑i=0kα̂inx(tn+i-k)-h∑i=0kβ̂inax(tn+i-k)=∑i=0k(α̂in-haβ̂in)x(tn+i-k).
By Theorem 1.3, we have
(3.6)x(tn+i-k)={x0eatn+i-k(1+b)[tn+i-k],{tn+i-k}≠0,x0eatn+i-k(1+b)tn+i-k-1,{tn+i-k}=0.
Therefore,
(3.7)Rn=x0∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+x0∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)tn+ij-k-1=μnx0∑i=0k(αi-hλβi)eλtn+i-k=μnx0∑i=0k(αi-hλβi)eλh(n+i-k).
We can express the residual as a power series in h: collecting terms in Rn to obtain
(3.8)Rn=μnx0(r0+r1h+⋯+rqhq+⋯).
Then,
(3.9)rq=∑i=0kαiλq(n+i-k)qq!-∑i=0kλβiλq-1(n+i-k)q-1(q-1)!=∑i=0kαiλq(n+i-k)qq!-∑i=0kβiλq(n+i-k)q-1(q-1)!=∑i=0kαiλqq!∑j=0qq!j!(q-j)!ij(n-k)q-j-∑i=0kβiλq(q-1)!∑j=0q-1(q-1)!j!(q-1-j)!ij(n-k)q-1-j=∑i=0kαiλq((n-k)qq!+∑j=0q-1ij+1(n-k)q-j-1(j+1)!(q-1-j)!)-∑i=0kβiλq∑j=0q-1ij(n-k)q-j-1j!(q-1-j)!=λq(n-k)qq!∑i=0kαi+λq∑j=0q-1(n-k)q-j-1j!(q-1-j)!∑i=0kij(iαij+1-βi).
By (2.3),
(3.10)rq=0,q=0,1,…,p,rp+1≠0.
Therefore, Rn+k=O(hp+1). The proof is complete.


### 3.1. An Example

Denotel=m{n/m}, then we can define the coefficients of (3.2) as follows:(3.11)α̂in=(1+b)[(k-l-i)/m]+1αi,β̂in=(1+b)[(k-l-i)/m]+1βi.Theorem 3.4.
Assume that (2.3) holds. Then the improved linear multistep methods formed by (3.11) are of order p for (1.2).Proof.
We only need to verify that the condition in Theorem3.3 holds. Note that
(3.12)tn+i-k=n+i-km=[nm]+m{n/m}+i-km=[nm]+l+i-km.
Thus, {tn+i-k}=0 if and only if [(k-i-l)/m]=(k-i-l)/m, that is, i=i1,…,ic, 0≤ic<k. Therefore,
(3.13)[tn+i-k]+[k-i-lm]={[nm],if[k-i-lm]=k-i-lm,[nm]-1if[k-i-lm]≠k-i-lm.
Hence,
(3.14)∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)[tn+ij-k]-1=∑i=0,i≠i1,…,ick(αi-haβi)eatn+i-k(1+b)[tn+i-k]+[(k-i-l)/m]+1+∑j=1c(αij-haβij)eatn+ij-k(1+b)[tn+ij-k]+[(k-ij-l)/m]=(1+b)[n/m]∑i=0k(αi-haβi)eatn+i-k.
Thus, the conditions in Theorem 3.3 are satisfied with μn=(1+b)[n/m] and λ=a. Thus, the proof is complete.Remark 3.5.
Ifb=0 in (3.11), that is, the impulsive differential equations reduce to ODEs, we have α̂in=αi, β̂in=βi, 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 h=1/m can be chosen with arbitrary positive integer m without any restriction.Remark 3.7.
Ifm≥l≥k, then α̂in=αi, β̂in=βi. 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).


## 3.1. An Example

Denotel=m{n/m}, then we can define the coefficients of (3.2) as follows:(3.11)α̂in=(1+b)[(k-l-i)/m]+1αi,β̂in=(1+b)[(k-l-i)/m]+1βi.Theorem 3.4.
Assume that (2.3) holds. Then the improved linear multistep methods formed by (3.11) are of order p for (1.2).Proof.
We only need to verify that the condition in Theorem3.3 holds. Note that
(3.12)tn+i-k=n+i-km=[nm]+m{n/m}+i-km=[nm]+l+i-km.
Thus, {tn+i-k}=0 if and only if [(k-i-l)/m]=(k-i-l)/m, that is, i=i1,…,ic, 0≤ic<k. Therefore,
(3.13)[tn+i-k]+[k-i-lm]={[nm],if[k-i-lm]=k-i-lm,[nm]-1if[k-i-lm]≠k-i-lm.
Hence,
(3.14)∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)[tn+ij-k]-1=∑i=0,i≠i1,…,ick(αi-haβi)eatn+i-k(1+b)[tn+i-k]+[(k-i-l)/m]+1+∑j=1c(αij-haβij)eatn+ij-k(1+b)[tn+ij-k]+[(k-ij-l)/m]=(1+b)[n/m]∑i=0k(αi-haβi)eatn+i-k.
Thus, the conditions in Theorem 3.3 are satisfied with μn=(1+b)[n/m] and λ=a. Thus, the proof is complete.Remark 3.5.
Ifb=0 in (3.11), that is, the impulsive differential equations reduce to ODEs, we have α̂in=αi, β̂in=βi, 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 h=1/m can be chosen with arbitrary positive integer m without any restriction.Remark 3.7.
Ifm≥l≥k, then α̂in=αi, β̂in=βi. 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 solutionx(t)≡0 of (1.2) is asymptotically stable if and only if |(1+b)ea|<1.The corresponding property of the numerical solution is described as follows.Definition 4.2.
The numerical solutionxn is called asymptotically stable for (1.2) with |(1+b)ea|<1 if limn→∞xn=0 for arbitrary stepsize h=1/m.Theorem 4.3.
Assume there exist constantsϵ, C>0 and a consequence of functions γi=γi(n,i), i=0,…,k such that |1/γi|<C for arbitrary n and 0≤i≤k, and the following equality holds
(4.1)α̂in-haβ̂in=(αi-hϵβi)γi,i=0,1,…,k.
Then, limn→∞xn=0 if the corresponding linear multistep methods (2.1) are A-stable and ℜϵ<0.Proof.
Denote(4.2)yn+i-k=γixn+i-k.
Then by (4.1), the process (3.2) becomes
(4.3)∑i=0k(αi-hϵβi)yn+i-k=0.
Therefore, yn can be viewed as the numerical solution of the equation y′(t)=ϵy(t) calculated by linear multistep methods (2.1).
On the other hand, we know thatℜϵ<0 and the methods are A-stable. Therefore, limn→∞yn=0. The conclusion is obvious in view of that
(4.4)|xn|<C|yn|.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 ℜa<0 hold.Proof.
It is obvious that for (3.11):
(4.5)α̂in-haβ̂in=(αi-haβi)γi,i=0,1,…,k,
where γi=(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/γi| 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)x′(t)=2x,t>0,t≠d,Δx=(e-2-1.27)x,t=d,x(0+)=2.
Theorem 4.1 implies that limt→∞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.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:(4.7)α̂in={(1+b){tn+i-k}αi,{tn+i-k}≠0,(1+b)αi,{tn+i-k}=0,β̂in={(a+ln⁡(1+b))a(1+b){tn+i-k}βi,{tn+i-k}≠0,(a+ln⁡(1+b))a(1+b)βi,{tn+i-k}=0,
where we define β̂in=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)∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)[tn+ij-k]-1=∑i=0k(αi-h(a+ln⁡(1+b))βi)eatn+i-k(1+b)tn+i-k=∑i=0k(αi-h(a+ln⁡(1+b))βi)e(a+ln⁡(1+b))tn+i-k.
Thus, the conditions in Theorem 3.3 are satisfied with μn=1, and λ=a+ln(1+b). The proof is compete.Theorem 4.7.
Assume thatℜ(a+ln(1+b))<0. Then, limn→∞xn=0 if the corresponding linear multistep methods (2.1) are A-stable, where xn is obtained by the improved linear multistep method (4.7).Proof.
Define(4.9)γi={(1+b){tn+i-k},{tn+i-k}≠0,(1+b),{tn+i-k}=0.
Then (4.1) is satisfied, and |1/γi|≤max{1,|1/(1+b)|}=C. 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(4.10)y(t)={(1+b){t}x(t),t≥0,t≠d,(1+b)x(d),t=d.
Then, it is easy to see that y(t) is continuous for t∈[0,+∞).Lemma 4.8.
Assume that (4.10) holds. Then x(t) is the solution of (1.2) if and only if y(t) is the solution of
(4.11)y′(t)=(a+ln⁡(1+b))y(t),y(0)=x0.Proof.
Necessity. In view of Theorem1.3 and (4.10), we obtain that in the case t≠d:
(4.12)y(t)=(1+b){t}eat(1+b)[t]x0=eat(1+b)tx0=e(a+ln⁡(1+b))tx0.
and y(k)=(1+b)x(k)=e(a+ln(1+b))kx0, 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 yn as follows:(4.13)∑i=0kαiyn+i-k=h(a+ln⁡(1+b))∑i=0kβiyn+i-k,xn={(1+b)-{tn}yn,if{tn}≠0,(1+b)-1yn,if{tn}=0.
It is obvious that the methods (4.7) and (4.13) are the same.


## 4.1. Another Improved Linear Multistep Methods

In this section, we give another improved linear multistep methods. We define the coefficients as follows:(4.7)α̂in={(1+b){tn+i-k}αi,{tn+i-k}≠0,(1+b)αi,{tn+i-k}=0,β̂in={(a+ln⁡(1+b))a(1+b){tn+i-k}βi,{tn+i-k}≠0,(a+ln⁡(1+b))a(1+b)βi,{tn+i-k}=0,
where we define β̂in=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)∑i=0,i≠i1,…,ick(α̂in-haβ̂in)eatn+i-k(1+b)[tn+i-k]+∑j=1c(α̂ijn-haβ̂ijn)eatn+ij-k(1+b)[tn+ij-k]-1=∑i=0k(αi-h(a+ln⁡(1+b))βi)eatn+i-k(1+b)tn+i-k=∑i=0k(αi-h(a+ln⁡(1+b))βi)e(a+ln⁡(1+b))tn+i-k.
Thus, the conditions in Theorem 3.3 are satisfied with μn=1, and λ=a+ln(1+b). The proof is compete.Theorem 4.7.
Assume thatℜ(a+ln(1+b))<0. Then, limn→∞xn=0 if the corresponding linear multistep methods (2.1) are A-stable, where xn is obtained by the improved linear multistep method (4.7).Proof.
Define(4.9)γi={(1+b){tn+i-k},{tn+i-k}≠0,(1+b),{tn+i-k}=0.
Then (4.1) is satisfied, and |1/γi|≤max{1,|1/(1+b)|}=C. 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(4.10)y(t)={(1+b){t}x(t),t≥0,t≠d,(1+b)x(d),t=d.
Then, it is easy to see that y(t) is continuous for t∈[0,+∞).Lemma 4.8.
Assume that (4.10) holds. Then x(t) is the solution of (1.2) if and only if y(t) is the solution of
(4.11)y′(t)=(a+ln⁡(1+b))y(t),y(0)=x0.Proof.
Necessity. In view of Theorem1.3 and (4.10), we obtain that in the case t≠d:
(4.12)y(t)=(1+b){t}eat(1+b)[t]x0=eat(1+b)tx0=e(a+ln⁡(1+b))tx0.
and y(k)=(1+b)x(k)=e(a+ln(1+b))kx0, 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 yn as follows:(4.13)∑i=0kαiyn+i-k=h(a+ln⁡(1+b))∑i=0kβiyn+i-k,xn={(1+b)-{tn}yn,if{tn}≠0,(1+b)-1yn,if{tn}=0.
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:(5.1)x0,0=x0,xwm,l=-α0-haβ0α2-haβ2xwm,l-2-α1-haβ1α2-haβ2xwm,l-1,l=2,…,m,w=0,1,…,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,1=-(1+b)α0-haβ0α2-haβ2xwm,m-1-α1-haβ1α2-haβ2x(w+1)m,0,
where we assume that x0,1 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.Table 4
Equation (5.1) in case of mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.964613490354896e+0028.950836048980485e-001204.971904772277776e+0011.197198445982186e-001401.248351989479124e+0011.553305410469574e-002803.126205023072544e+0002.002751471856357e-0031607.821288588893367e-0012.607915957975049e-0043201.955988918562071e-0013.491638393382512e-005Ratio3.998636451705015e+0007.469032196798133e+000Table 5
Equation (5.1) in case of 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)103.675982365982200e + 0029.516633893408510e − 003209.552174783206283e + 0012.325825408788229e − 003402.442645505222754e + 0015.749450957563962e − 004806.181500463695556e + 0001.429301543146577e − 0041601.555180585986818e + 0003.563220614588580e − 0053203.900496418718831e − 0018.895535536623811e − 006Ratio3.987135018310406e + 0004.005627991612696e + 000Table 6
Equation (4.13) in case of mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.533815775376497e + 0034.656258034474448e − 006204.273673528511026e + 0021.126844488719137e − 006401.116157821124107e + 0022.771258855727155e − 007802.844156657094209e + 0016.871261282181962e − 0081607.173499151609576e + 0001.710731889481565e − 0083201.800995925495954e + 0004.267991404738325e − 009Ratio3.983073503974838e + 0004.008283352169618e + 000Table 7
Equation (4.13) in case of 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)103.034120865054534e + 0039.093526915804340e − 006208.173504073904187e + 0022.227110126540310e − 006402.159842038101851e + 0025.509781502155420e − 007805.579637010861916e + 0011.370190298999319e − 0071601.419913379106947e + 0013.416407701184454e − 0083203.582733454190020e + 0008.529607353757740e − 009Ratio3.963212438944776e + 0004.005351664492912e + 0 00


### 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 [0,500] in Figure 2. We can see from the figure that the method can preserve the stability of the analytic solution.Figure 2
The numerical solution obtained by (4.13) in case of the 2-step BDF method to (4.6).


## 5.1. Convergence

The improved 2-step linear multistep methods (3.11) takes the form:(5.1)x0,0=x0,xwm,l=-α0-haβ0α2-haβ2xwm,l-2-α1-haβ1α2-haβ2xwm,l-1,l=2,…,m,w=0,1,…,x(w+1)m,0=(1+b)xwm,m,x(w+1)m,1=-(1+b)α0-haβ0α2-haβ2xwm,m-1-α1-haβ1α2-haβ2x(w+1)m,0,
where we assume that x0,1 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.Table 4
Equation (5.1) in case of mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.964613490354896e+0028.950836048980485e-001204.971904772277776e+0011.197198445982186e-001401.248351989479124e+0011.553305410469574e-002803.126205023072544e+0002.002751471856357e-0031607.821288588893367e-0012.607915957975049e-0043201.955988918562071e-0013.491638393382512e-005Ratio3.998636451705015e+0007.469032196798133e+000Table 5
Equation (5.1) in case of 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)103.675982365982200e + 0029.516633893408510e − 003209.552174783206283e + 0012.325825408788229e − 003402.442645505222754e + 0015.749450957563962e − 004806.181500463695556e + 0001.429301543146577e − 0041601.555180585986818e + 0003.563220614588580e − 0053203.900496418718831e − 0018.895535536623811e − 006Ratio3.987135018310406e + 0004.005627991612696e + 000Table 6
Equation (4.13) in case of mid-point rule for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)101.533815775376497e + 0034.656258034474448e − 006204.273673528511026e + 0021.126844488719137e − 006401.116157821124107e + 0022.771258855727155e − 007802.844156657094209e + 0016.871261282181962e − 0081607.173499151609576e + 0001.710731889481565e − 0083201.800995925495954e + 0004.267991404738325e − 009Ratio3.983073503974838e + 0004.008283352169618e + 000Table 7
Equation (4.13) in case of 2-step BDF methods for (2.7) and (2.8).
mAbsolute errors for (2.7)Absolute errors for (2.8)103.034120865054534e + 0039.093526915804340e − 006208.173504073904187e + 0022.227110126540310e − 006402.159842038101851e + 0025.509781502155420e − 007805.579637010861916e + 0011.370190298999319e − 0071601.419913379106947e + 0013.416407701184454e − 0083203.582733454190020e + 0008.529607353757740e − 009Ratio3.963212438944776e + 0004.005351664492912e + 0 00


## 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 [0,500] in Figure 2. We can see from the figure that the method can preserve the stability of the analytic solution.Figure 2
The numerical solution obtained by (4.13) in case of the 2-step BDF method to (4.6).



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*Source: 652928-2012-05-13.xml*