Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 447-458
447
STABILITY ANALYSIS OF LINEAR MULTISTEP METHODS
FOR DELAY DIFFERENTIAL EQUATIONS
V.L. BAKKE and Z. JACKIEWICZ
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR 72701, USA
(Received August 26, 1985 and in revised form January 28, 1986)
ABSTRACT. Stability properties of linear multistep methods for delay differential
equations with respect to the test equation
ay(t) + by(t),
y’(t)
0 < %
i,
O,
t
are investigated. It is known that the solution of this equation is
bounded if and only if
lal
and we examine whether this property is inherited by
-b
multistep methods with Lagrange interpolation and by parametrized Adams methods.
KEY WORDS AND PHRASES:
Linear multistep method, delay differential equation, stabil-
ity analysis.
1980 AMS SUBJECT CLASSIFICATION CODE. 65L05.
i. INTRODUCTION
Consider the delay differential equation (DDE)
y’(t)
y(t)
0,
< t.
f(t,y(t),y(a(t))),
g(t),
t
t
t
o
J to
t
(1.1)
o
2
a(t)
R are continuous and t
R and a:[t 0, )
o
Such equations arise in a number of practical applications such as, for example,
where
f:[t0,)
R
control theory, electrodynamics, viscoelasticity, biomathematics, and medical sciences
(compare Hale [i]).
The convergence theory of numerical methods for DDEs is rather well developed and
resembles the convergence theory of corresponding numerical methods for ordinary differential equations (ODEs) (see Tavernini
[2-4], Jackiewicz [5-7], Zverkina [8]). In
this paper we wish to concentrate on stability analysis of some methods for (1.1).
To investigate the stability properties of numerical methods for (1.1), these
methods are usually applied, with a fixed positive step size
h,
to various test
equations with known region of stability. The simplest test equation is
y’(t)
y(t)
T
0,
q
qy(t-),
g(t),
-
0,
J
t
J
0,
(1.2)
real, and it is known (see Bellman and Cooke [9]) that the solution of
V.L. BAKKE and Z. JACKIEWICZ
448
this equation
tends to zero as
for all
t
The numerical method, with step size
T/m,
h
inherits this property is said to be
if and only if
q
where
DA0-stable
is a positive integer, which
m
and Cryer [I0] gives some necessary
and sufficient conditions for linear multistep method to be
duced a more general notion of
T/(m-u),
h
where
g
u
GDA0-stability
[0,i),
(-/2T,0).
q
DA0-stable.
which relaxes the condition
py(t) + qy(t-),
g(t),
y(t)
p
of
Cryer’s results
t > 0,
-T
O,
< t <
(1.3)
complex. He introduced the notion of Q- and GQ-stability related to
q
and
(1.2) (with
to
is complex and also considered a more general test equation
q
y’(t)
with
T/m
h
and investigated the properties of linear multistep
methods with respect to this concept. Barwell [ii] generalized some of
to the case where
He also intro-
complex) and P- and GP- stability related
q
Cryer’s DA 0- and
GDA0-stability
(1.3) which are analogues
to
and investigated the stability properties of some
simple multistep methods coupled with Lagrange interpolation. Still more general concepts of stability with respect to (1.2) were introduced by van der Houwen and
Sommeijer [12] and stability criteria were derived for a class of linear multistep
methods.
Stability analysis of numerical methods for DDEs is difficult since it is necessary
to consider difference equations of arbitrarily high order. To illustrate this point
suppose that the linear multistep method
k
Z
t
j yh
=0
i=0,1,...,
+
ih,
h
f( t
i+k-j
,Yh(t i+k-j )’ Yh (a
gh(s)’
Yh(S)
size
k
h Z
=0
i+k-j
t
)’
i+k-j
[to-,t0],
s
(1.4)
coupled with Lagrange interpolation of sufficiently high order, and step
z/m,
where
i=0,1,...;
m
is a positive integer, is applied to
is an approximate solution; and
Yh
(1.3).
Here,
t
t
i
is an approximation to
gh
o
g.
This leads to the difference equation
k
k
h Z B
j
j=0
Z a Yh(t
i+k-j
j
j=0
Yh(S)
i=0,1
of order
(pyh(t i+k-j
+ qYh (t i+k-j-m )’
gh(s),
m+k.
s e
[t0-,t0],
Now in order to establish some stability properties of
(1.4) we should be able to decide whether or not the solution
(or tends to zero as
t
(1.5)
Yh
of (1.5) is bounded
). This is, in general, a nontrivial task and satisfactory
results were obtained only for simplest numerical methods. Barwell [ii] established Pand GP-stability for first and second order backward differentiation methods and
Jackiewicz [13] determined stability regions for @-methods. Similar results with
respect to the test equation (1.2) were obtained by Cryer [I0] and Barwell [ii]. Cryer
proved that the @-methods are
DA0-stable
if and only if
Backward Euler method and the trapezoidal method are
@ e
[1/2,1]
GDA0-stable.
and that the
Barwell proved that
the Backward Euler method is Q-stable and conjectured that it is GQ-stable. For more
general methods some stability results were established by Wiederholt [14], Ai-Mutib
[15], and Oppelstrup [16]. Wiederholt determined numerically (via boundary locus
STABILITY ANALYSIS FOR DELAY DIFFERENTIAL EQUATIONS
(hp,hq)
method) the set of all
such that
449
(given by (1.5)) tends to zero as
Yh
t
for second order Milne predictor-corrector method and third order Adams predictor-corrector method for m=l,2, and 3. Similar results were obtained by Ai-Mutib for the
Runge-Kutta-Merson method, the trapezium rule and the fourth order implicit Runge-Kutta
method. Oppelstrup investigated stability properties of Runge-Kutta-Fehlberg method
combined with Hermite-Birkhoff interpolation with respect to the test equation (1.2).
More general test equations for functional differential equations were considered
by Bickart [17] and Brayton and Willoughby [18].
It is the purpose of this paper to present stability analysis of some linear multistep methods based on the test equation
ay(%t) + by(t),
y’(t)
y(0)
where
%
and
a, b,
0,
t
(1.6)
Y0’
0 < % < 1.
are real and
The importance of such equations in
practical applications is discussed, for example, by Fox et al [19]. It follows from
the results of Kato and McLeod [20] (see also Fox et al [19]) that all solutions of
lal
(1.6) are bounded if and only if
< -b
is inherited by the approximate solution
and we investigate whether this property
when (1.4) is applied to (1.6). The
Yh
problem is that the application of (1.4) to (1.6) leads to difference equations which
are not of fixed order. A a consequence, the approach of Barwell [II], Cryer [I0],
Jackiewicz [13] is not applicable (compare the discussion of this topic in Jackiewicz
[13]), and
a different method of attack is needed. Roughly speaking, the approach of
this paper consists of the following. Assuming that
M
by a constant
and
fly h
for conditions such that
constants
M
is true by
in the form
yM
and
Ay,
ll[to,ti
Yh(ti+l)
is bounded by
and
Yh(tj)
yM
for
J
for some
llyhll[ti,ti+l]
i
I,
are bounded
we are looking
are bounded by the same
respectively. Denoting the set of all
(hb,ha)
for which this
we obtain a lower bound on the stability region with respect to (1.6)
IAY"
This approach works reasonably well for simple multistep methods
such as, for example @-methods. We were also able to obtain some sufficient stability
conditions for linear multistep methods with Lagrange interpolation (Section 2) and
for low order parametrized Adams-Bashforth and Adams-Moulton methods (Section 3).
Unfortunately, this approach does not carry over to general linear multistep methods
with Hermite interpolation and high order parametrized Adams methods.
As a byproduct of our analysis we established that the stability region for @methods with respect to (1.3) is contained in the stability region with respect to
(1.6). This partially answers the conjecture posed in Jackiewicz [13].
2. STABILITY ANALYSIS OF LAGRANGE INTERPOLATORY EXTENSIONS OF LINEAR MULTISTEP METHODS
A Lagrange interpolatory extension of the linear multistep method for ODEs with
coefficients
j, Bj,
j=0,1
k,
is the method defined by
k
j=0
Yh(t i+k-j
and
Yh(ti+k_l+rh)
k
h E
j=O
k
E U
j=0
F(t
i+k-j
(r)Yh(t i+k-j
(2.1)
(2.2)
V.L. BAKKE and Z. JACKIEWICZ
450
r
i=O,l,...,
g
[0,i],
where
F(t)
UO(1)
f(t,Yh(tv),Yh(a(t)))
0, j=l,2
1, Uj(1)
and (2.2) is Lagrange
k,
I,
interpolation formula, i.e.
k
E U.(r)
0,
(compare Tavernini [3], where a more general Hermite
# I; and
j=0 3
interpolatory extension is defined). This method can be written in the form
k
aoYh(ti+k_l+rh)
i=0,1
[0,i],
r
BjU0(r),
I a
j=l
k
h Z b (r)F(t
j
i+k-j
j=0
(r)Yh(t i+k-j
j
Uj(0)
(2 3)
ao %0; j(r) ajU0(r)-a0Uj(r), j=l,2,...,k; bj(r)=
the approximate solution obtained when the
where
Denote by
k.
j=0,1
+
UI(O)
Yh
method (2.3) is applied to (1.6). We propose the following definition.
DEFINITION. For given values of
to be absolutely stable, if
and
hb
x
Yh
(x,y)
bility is the set of all points
ha,
y
the method (2.3) is said
0 < % < I.
is bounded for
A region of absolute sta-
such that the method (2.3) is absolutely stable.
The method (2.3) is said to be A-stable if the region of absolute stability includes
{(x,y):
the stability region for (1.6), i.e. the set
IYl
-x}.
We have the following:
THEOREM i. Assume that
(x,y) satisfies the inequality
k
Z
=0
k
l-a
Z
YlYl
+ xB
+
r e
[0,I]}.
j =0
la 0 -xB01,
IBjl
(2.4)
k
y--sup{ Z
where
j=0
PROOF. It
< i
then
Assume that
F(t)
is
IUj(r) l:
l
ly(tj)l,, is
Here,
clear from (2.2) that if
llyll[0,ti
lYh(t.)
aYh(%t)
Then the method (2.3) is absolutely stable.
_<
M
+ bY h(t)
for
j
_<
s e
for
[0,ti]}.
Then it is easy to check, using (2.1) with
i.
lYh(ti+l)
that
sup{lYh(S) l:
llyII[0,ti
yM.
is bounded by
M
bounded by a constant
M
and
llyhll[0,ti+l
provided (2.4)
yM
holds, and the proof is complete.
To illustrate this theorem we apply it now to some simple methods for DDEs.
EXAMPLE i. Consider the @-methods with linear interpolation for DDEs (I.i). These
are the methods of the form
Yh(ti+l) Yh(ti) + rh[@F(ti+ I) + (l-@)F(ti)],
Yh(ti+rh) (l-r)Yh(ti) + rYh(ti+l),
Now
r e [0,i].
i=O,l
and the inequality (2.4) takes the form
y
+
(l-@)xl +
IYl
It is easy to verify that the solution of this inequality is
RI U 2
for
@
I
_<
R
for
0
@
and
where
i,
{(x’y):
IYl
-x
and
[Yl
2 + (l-2@)x}
and
2
{(x,y):
IY] -2 +
(In particular, the Backward Euler method (@
(2@-l)x}.
I) is A-stable). Comparing this with
Jackiewicz [13], p.391, we see that the stability region of @-methods with respect to
(1.3) is contained in the stability region of these methods with respect to (1.6), which
partially answers the conjecture posed in Jackiewicz [13].
STABILITY ANALYSIS FOR DELAY DIFFERENTIAL EQUATIONS
451
EXAMPLE 2. Consider the trapezoidal method with quadratic interpolation
Yh(ti+rh)
i=0,1,
+
(r+l)(2-r)Yh(ti)
r e [0,i].
r
(r-l)Yh(ti+l)
This method can be written in the form
Yh(ti+l)
r
h
+
Yh(ti)
[F(ti+l)+F(ti)],
(l-r2)yh(ti)
+
(r+l)Yh(ti+ I)
Yh(ti+rh)
r e [0,I] (see Tavernini [3]).
i=0,1
r
+ (r+l)h[F(ti+l
)+F(t )]
i
+
Now
+
r
(r-l)Yh(ti_l),
5
y
and the inequality (2.4) reads
5
1/2xl + IYl
The solution of this inequality is the set
R
IYl
{(x,y):
-x
IYl
and
J
8
}.
For comparison, for the trapezoidal rule with linear interpolation (this corresponds
to @
in Example i) the solution of (2.4) is
R*
and
C
IYl
{(x,y):
J
IYl
and
-x
J 2},
This suggest that the trapezoidal rule with linear interpolation may have
better stability properties than trapezoidal rule with quadratic interpolation.
EXAMPLE 3. Consider the method
Yh(ti_l+rh)
i--0,1
r
[0,I]
(l+r)hF(ti),
or, equivalently
Yh(ti)
Yh(ti-2)
+
l+r
Yh(ti_l+rh)
i=0,1
+
Yh(ti_2)
--Yh(ti)
2hF(ti)’
+
Yh(ti_2),
This method is not of type (2.3), but the same approach can be
r e [0,i].
used to obtain a sufficient condition for absolute stability. It is clear that y
and inequality (2.4) takes the form
11
+ 21y
2x
{(x,y):
A2
{(x,y):
A U A 2,
(x,y)
Therefore, this method is absolutely stable if
A
I.
where
IYl _< -x},
IYl _< x i}.
In particular, it is A-stable.
EXAMPLE 4. Consider the backward differentiation methods:
k
Y.
k,
j=0
hF(t
Yh (t i+k-j
i+k
k
Yh(ti+k_l+rh)
i=0,1
r e
[0,I],
6,
k=l,2
Y. U
j=0
k ’J
(r)Yh(t i+k-j
where
k
k,0
k,j
i=
k
U
k
j(r)
H
r+m-I
m-j
k=1,2
j=l,2
k,
j=l,2
k
452
V.L. BAKKE and Z. JACKIEWICZ
(the case k=l corresponds to @--I in Example i). Now ineqltality (2.4) reads
k
lk,jl + ykly
Z
Yk
k
:= sup{ Z
j=O
la k j I:
[0,i]},
r
]k
_<
0-
and its solution is
U
Ak Ak,
Ak, 2,
where
k
Ak,
(x,y)
x
AE, 2
{(x,y):
x
+
klyl
_<
ykly
_<
j=ljl_l( l-(j)
k
k
(note that
r. -(1-())
0
2).
k
for
I( l+(j))}
J =ljZ
k
Unfortunately, this approach cannot be used
to determine the stability of these methods in the neighborhood of the origin for
3. STABILITY ALYSIS OF PRAMETRIZED AD/S METHODS
We will consider the (explicit) Adams-Bashforth and the (implicit) Adams-Moulton
formulas. The Adams-Bashforth methods are given by
k
Yh(ti+k_l+rh)
i=0,1
r e
(0,I],
Yh(ti+k_l)
+ h
I
j=0
bk,j(r)F(t i+k-l-j ),
(3.I)
where
k
r
s__.) ds.
f
bk, j(r)
(3.2)
0 m0 m-3
mj
The method (3.1) can also be written in the form
k
Yh(ti+k_l+rh)
i=0,1
r e (0,i],
The coefficients
Co,
where
V
h
Y.c.(r)VJF(t i+k-i
(3 3)
j=O
is the backward difference operator of order
k,
j=0,1
+
Yh(ti+k_l)
are independent of
k,
j.
and are given by
.r
(-l)Jf (-S)ds.
m
c.(r)
0
Using the generating function
G(t,r) :=
Z c (r)t
m= 0
l-(l-t)-r
m
ln(l-t)
m
we also have
m
c0(r)
Expressing
VJF(ti+k_ I)
ZO1.j,. Cm_
r,
(r)
j=
in
-r
(-l)m+l( m+l
).
F(ti+k_l_ v)
(3.3) in terms of
(3.1) we can easily find the following relationship between
bk,j(r)
k
Z
(-I)
(3.4)
and comparing (3.3) and
b
k
and
(m) Cm(r)"
c.:
(3.5)
m=j
Consider now the Adams-Moulton formulas for DDEs. These methods read
k
Yh(ti+k_l+rh)
i=0,1
r e (0,i],
+
Yh(ti+k_l)
hj=0E bk,j(r)F(ti+k_j),
(3.6)
where
r
bk,j(r
k
f( H
0
s+m-l)dsm-j"
(3.7)
STABILITY ANALYSIS FOR DELAY DIFFERENTIAL EQUATIONS
453
Another representation of (3.6) is
k
Yh(ti+k_l+rh)
i=O,l
(0,1],
r e
hj=07,, c.(r)j JF(ti+k )’
+
Yh(ti+k_l)
where the coefficients
j=O,1
cj,
(3.8)
k.
are given by
.-l+r
(-I) 3 f (-s
-I
c.(r)
j)ds.
3
Using the generating function
,
,
(l-t)-(l-t)
in(l-t)
E c (r) t m
m
m=O
G (t,r)
l-r
we also have
,
m
CO(r)
E
j=0
r,
j--
*
(-l)m-l(m+l-rI)
Cm_j(r)
(compare Tavernini [3]). The relationship between
,
b
b
c.
is
,
k
()Cm(r).
m=j
E
(-i)
j(r)=
k
and
k
(3.9)
(3.10)
Now we list some properties of the coefficients of the Adams-Bashforth and Adams-Noulton
methods which will be useful later on. To be consistent with Henrici [21] we will use
the following notation:
:=
8k,
PI.
P2.
m
0,
m
YO
I,
m
,
m
Z
P3.
i=O
P4.
bk,j(1),
8k
O,
<
_>
m
*
:=
8k,
* :=
bk,j(1),
yj
cj(1).
i.
O.
m
> 0
cj(1),
0.
Ym’
Yi
0
_<
:=
j
0
k
k
E
j=l
P5.
Uk
P6.
8k, 0
0,
PT.
gk,1
gk,0
P8
Wk
Bk’O
,
IBk, J
0
,
8k,1
0,
0,
k
2;
k=O
k
1.
O,
k
u
0
k
k
3
1.
k
X
k,0
k*
:=
j=l
]Bk,jl
O* + B,I j=Z21Bk,j
P9.
u
PIO.
llbk,jll[O,l]
I;
w
k
<
O,
2.
k
k
O,
Bk,
Ibk,j(1)l;
k=1,2,3,4,5;
,
llbk,jlI[O,ll
u
Ibk,j(1)
O,
k
O,
k
_>
k
6.
j=O,l
k.
The properties P1-P3 are well known (compare Henrici [21]), where the first few values
of
Ym
and
,
Ym
are also listed, or Hall
[22]). P4 follows from (3.5) for
and PI. It is easy to check, using (3.5) for
k
E
Hence,
k
u
I,
k
E
E
m=O m=j
Ym
m=O
UK
u
I,
,
+
E
m=l
E
,
25
u
0
3
2
establishes P5. P6 follows from (3.10) for
r=l
k
E
m=1 j=l
ym
and
r=l,
()m-j
Uk+
j=0,1,
we obtain
,
k,1
,
Bk,O
k
X (l+i)y
i=2
r=l,
j=O,
that
k
k
()Ym-3
u
r=l,
,
i
+
k
E
m=l
(2-2m)y m
3 which
for k
k
and P1-P3. Using (3.10) for
u
V.L. BAKKE and Z. JACKIEWICZ
454
which, together with P2, establishes P7. In view of (3.10) for r=l we have
k
E
w*
k
*
k
*
E
mYm +
Ym + m=l
m=O
k
k
k
Z
Z
m=2 j=2
k
* +
*
E
E
j=2 m=j
)Ym
k
() Ym*
Z (l+m+2m-m -I)
m=2
k
Z 2my
m=2
Ym*
k
E
m=2
* +
2YI
YO
(l+m)Ym*
m*
which proves P8. Similarly,
u
k
*
+ E
k
2
2m)y *
m
m.
m=2
* > 0 for k=1,2,3,4,5, u * < 0
6
k
proves P9. PIO follows easily from (3.2) and (3.7).
It is clear that
u
*
and
< u
Uk+
k
k
for
!
6
which
We are now in a position to formulate our stability theorems.
A
Denote by
THEOREM 2.
k
I1
+
+
Ik,ox
(x,y)
the set of all
Y
IIk,jllx
)-i.
j=1
+
-
and
k
Z
j=O
and put
Y>UIA Y.
A
IB k
l(Ixl
(x,y)
Then if
g
YlYl)
+
A
(hb,ha)
k
y E Ig
k ’J
j=0
<
such that
lYl
(3.11)
(3 12)
the Adams-Bashforth method (3.1) is absolutely
stable.
PROOF. The method (3.1) applied to (1.6) takes the form
Yh(ti+k-I +rh)
i=O,l
r
Let
(0,i].
g
ly(tj)l
that
Yh(t i+k-i
_<
M
for
M
+
k
h I
llyhll[0,tk_1
llyhll[O,ti+k_l
< yM,
and
i+k-1,
y %(t
)I
h i+k
Then, using (3.13) for r--I it follows that
M.
Assume
where
_>
g
A,
as in P5 and put
v
i.
(3.11) is satisfied
if
llYll[O,ti+k
Similarly, using (3.13) and el0 we can see that
(x,y)
M
(3.13)
bYh(ti+k-l-j)
be a constant such that
j=O,l
satisfied. Therefore, if
+
j=obk,j (r)(a Yh (%ti+k-l-j)
<_
the approximate solution
yM
if (3.12) is
Yh
given by (3.13) is
bounded and the theorem follows.
k
Define
A
Y
u
k
k
Z
j=0
:=
Bk
’J
I.
It is easy to see, using P4
that
can be written in the form
A
where
B
{(x,y):
x
0
UkX
+
YvklY
Vk(-X
< 0
YlYl)
+
2}
and
C
{(x,y):
It is clear in view of P5 that
Vk(-X-
+
x
O,
By
is empty for
YlYl)
k
Y
i}.
therefore this theorem applies
3,
only for k=0,1, and 2. It follows after some computations that the boundary of
k=0,1, and 2 is given by
+(2+VkX)/(3v k),
Y
+UkX/(Vk(2Bk,oX-l)),
2/v
k
x
-I/Bk
_<-i/Bk, O,
0
x
_<
O.
A
for
STABILITY ANALYSIS FOR DELAY DIFFERENTIAL EQUATIONS
These boundaries for
are plotted in Fig. I.
0
y
6/11
-1
-2
Fig. i.
455
Stability regions for Adams-Bashforth methods for k=0,1,2.
The coordinates of the points
Qi(xi,Yi),
i=0,I,2,
are given in Table
Xk
Yk
0
-I
2
-2/3
-12/23
I/3
i/4
2/253
below.
Table
For k=O we already obtained a better bound on stability region in Example
for @=0.
Our next theorem deals with the Adams-Moulton methods (3.6).
TttEOREM 3
Denote by
I1
+
tk,lXl
(x,y)
the set of all
k
+
Y.
j=2
I1 k ’J lxl
+ Y
Y..
j=0
I1 k ’J lYJ
and
k
r.
j=0
and put
A*
Y
>UlA Y
Then if
(hb,ha)
such that
k
lk,jl(Ix
+
(x,y)
A*
wlYl)
<
-
Ik
,
(3.14)
(3.15)
the Adams-Moulton method (3 6) is absolutely
stable.
PROOF. The proof is similar to that of Theorem 2. The method (3.6) applied to
(1.6) takes the form
Yh(ti+k-1 +rh)
i=O,l
(0,i].
r
(l-hb
k
+
Yh (ti+k-l)
For
0)Yh(ti+k)
r=l
k
h I
j=0
(r)(ay h(At
+
i+k-j
bYh (t i+k-j )),
(3.16)
this can be written in the form
(l+hb *
k
+ hb
bk,
T.
j=2
k
l)Yh(ti+k_l)
Bk, Yh ti+k-J )"
+ ha
k
Y.
j=0
*
k ’J Yh (xt i+k-j
(3.17)
V.L. BAKKE and Z. JACKIEWICZ
456
Denote by
i
the smallest integer such that
0
lly
constant such that
i+k-1,
i
using (3.17) that
10+k
Assume that
N.
li[0,i+k_lJ
lYh(ti+k) M
llyhll[0,ti+k !
and
i 0,
ll[o,ti0+k
flY
< yM,
< tl
t_
i0+k
ly_(t.)
n
y > I.
where
M
and let
< N
for
be a
]=0,1
Then it is easy to check
if (3.14) is satisfied. Similarly, using (3.16) and
PI0 it follows that
approximate solution
yM
if (3.15) holds. Therefore, if
(x,y)
A*
the
defined by (3.16) is bounded, which is our claim.
Yh
For k=0 (this corresponds to the backward Euler method) condition (3.14) and
(3.15) take the form
71Yl _<
Ixl + YlYl <
+
Y- I,
and it can be verified that the boundary of the region
is given by
+x/(2x-l),
x
<_
0,
_+(x-2)/(2x-l),
x
_>
2.
Y
We obtained a better bound on stability region in Example
To get some idea
for @=I.
we rewrite (3.14) and (3.15) in a different
k
form. Define
and
as in P8 and P9 and put
It can be checked
j=0
P6-P8
that
by considering special cases and using
looks like for
how the region
k >
*
*
w
v
u
A* B* C*,
where
B*
{(x,y):
and
C
,
UkX +
x < 0
{(x,y):
x
YvklY
Vk(-X
0
< 0
WkX
YvklY
+
+ y[y[)
2}
i}.
In view of P9 the set
5.
is empty for k
6, therefore, Theorem 3 is applicable only
V
It follows after some computations that the boundary of the region
A*
for
k
for
k=i,2,3,4, and 5 is given by
y
,
, ,
,
Z(2-Wk)/((Vk(3-(Wk+Vk)X)),
, , , ,
ZUkX/(Vk((Uk+Vk)X-l)),
2/w
,
k
, -I/Sk, I,
x
-I/Sk,
x
0.
Using the table of coefficients of Adams-Moulton methods given in Lapidus and Seifeld
[23] we can compute
plotted in Fig. 2.
,
,
k, v k, and w k and these boundaries, for k=1,2,3,4,5, are
The coordinates of the points
k=1,2,3,4,5, are given
u
, ,
Qk(Xk,Yk),
in Table 2 below
,
k
x
2
3
4
5
-3/2
-24/19
-360/323
-1440/1427
k
-2
Yk
2/5
36/119
88/425
4562/39232
303840/8845621
Table 2
As mentioned above, we were not able to apply the approach used in this paper to the
general linear multistep methods with Hermite interpolation or high order Adams methods.
STABILITY ANALYSIS FOR DELAY DIFFERENTIAL EQUATIONS
457
Y
X
3
6
Fig. 2.
90/49
38/45
Stability regions for Adams-Moulton methods for k=1,2,3,4,5.
REMARK. It was pointed out to us by Professor M. Zennaro that the main theorem in
the paper by Jackiewicz [13] gives only sufficient, not necessary and sufficient condition for absolute stability of O-methods for delay differential equations.
ACKNOWLEDGEMENT. This research was performed under Arkansas EPSCOR grant PRM-8011447.
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