Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 854517, 22 pages
doi:10.1155/2012/854517
Research Article
Nonlinear Stability and D-Convergence of
Additive Runge-Kutta Methods for
Multidelay-Integro-Differential Equations
Haiyan Yuan,1, 2 Jingjun Zhao,1 and Yang Xu1
1
2
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China
Correspondence should be addressed to Jingjun Zhao, hit zjj@hit.edu.cn
Received 30 December 2011; Accepted 19 February 2012
Academic Editor: Muhammad Aslam Noor
Copyright q 2012 Haiyan Yuan 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 is devoted to the stability and convergence analysis of the Additive Runge-Kutta
methods with the Lagrangian interpolation ARKLMs for the numerical solution of multidelayintegro-differential equations MDIDEs. GDN-stability and D-convergence are introduced and
proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASIstability give GDN-stability. A numerical example is given to illustrate the theoretical results.
1. Introduction
Delay differential equations arise in a variety of fields as biology, economy, control theory,
electrodynamics see, e.g., 1–5. When considering the applicability of numerical methods
for the solution of DDEs, it is necessary to analyze the stability of the numerical methods. In
the last three decades, many works had dealt with these problems see, e.g., 6. For the case
of nonlinear delay differential equations, this kind of methodology had been first introduced
by Torelli 7 and then developed by 8–12.
In this paper, we consider the following nonlinear multidelay-integro-differential equations
MDIDEs with m delays:
y t f
1
t, yt, yt − τ1 ,
f
g
1
t, s, ys ds
t−τ1
2
t
t, yt, yt − τ2 ,
t
g
t−τ2
2
t, s, ys ds
2
··· f
m
t, yt, yt − τm ,
t
Abstract and Applied Analysis
m
g
t, s, ys ds , t ∈ t0 , T ,
t−τ2
yt ϕt,
t ∈ t0 − τ, t0 ,
1.1
where τ1 ≤ τ2 ≤ · · · ≤ τm τ, f v : t0 , T × CN × CN × CN → CN , g v : t0 , T × CN × CN →
CN v 1, 2, . . . , m, and ϕ : t0 − τ, t0 → CN are continuous functions such that 1.1 has
a unique solution. Moreover, we assume that there exist some inner product ·, · and the
induced norm || · || such that
Re f v t, y1 , u1 , w1 − f v t, y2 , u2 , w2 , y1 − y2
2
≤ αv y1 − y2 βv u1 − u2 2 σv w1 − w2 2 , v 1, 2, . . . , m, t ≥ t0 ,
v t, y, u1 , w − f v t, y, u2 , w ≤ rv u1 − u2 ,
f
v
g t, s, w1 − g v t, s, w2 ≤ r
v w1 − w2 , t, s ∈ D
1.2
1.3
1.4
for all t ∈ t0 , T , for all y, y1 , y2 , u, u1 , u2 , w, w1 , w2 ∈ CN , −αv , βv , σv , rv , r
v are all
nonnegative constants. Throughout this paper, we assume that the problem 1.1 has unique
exact solution yt. Space discretization of some time-dependent delay partial differential
equations give rises to such delay differential equations containing additive terms with
different stiffness properties. In these situations, additive Runge-Kutta ARK methods are
used. Some recent works about ARK can refer to 13, 14. For the additive MDIDEs 1.1,
similar to the proof of Theorem 2.1 in 7, it is straightforward to prove that under the
conditions 1.2∼1.4, the analytic solutions satisfy
yt − zt ≤ max ϕt − ψt,
t0 −τ≤t≤t0
1.5
where zt is the solution of the perturbed problem to 1.1.
To demand the discrete numerical solutions to preserve the convergence properties of
the analytic solutions, Torelli 7 introduced a concept of RN-, GRN-stability for numerical
methods applied to dissipative nonlinear systems of DDEs such as 1.1 when g v t, s, ys 0, v 1, 2, . . . , m, which is the straightforward generalization of the well-known concept of
BN-stability of numerical methods with respect to dissipative systems of ODEs see also 9.
More recently, one has noticed a growing interesting the analysis of delay integro-differential
equations DIDEs. This type of equations have been investigated in various fields, such as
mathematical biology and control theory see 15–17. The theory of computational methods
for delay integro-differential equations DIDEs has been studied by many authors, and a
great deal of interesting results have been obtained see 18–22. Koto 23 dealt with the
linear stability of Runge-Kutta RK methods for systems of DIDEs; Huang and Vandewalle
24 gave sufficient and necessary stability conditions for exact and discrete solutions of
linear Scalar DIDEs. However, little attention has been paid to nonlinear multidelay-integrodifferential equations MDIDEs.
Abstract and Applied Analysis
3
So, the aim of this paper is the study of stability and convergence properties for
ARK methods when they are applied to nonlinear multidelay-integro-differential equations
MDIDEs with m delays.
2. The GDN-Stability of the Additive Runge-Kutta Methods
An additive Runge-Kutta method with the Lagrangian interpolation ARKLM of s stages
and m levels can be organized in the Butcher tableau:
1
C A1 A2 · · · Am
T
T
T
b1 b2 · · · bm
1
c1 a11 a12
1
1
c2 a21 a22
.
..
..
..
.
.
1
1
cs as1 as2
1
b1
v
1
b2
v
1
m
m1
· · · a1s
a11 a12
1
m
m
· · · a2s
a21 a22
.
.
..
..
. .. · · · ..
.
1
m
m
· · · ass
as1 as2
1
m
· · · bs
· · · b1
v
m
b2
m
· · · a1s
m
· · · a2s
..
..
.
.
m
· · · ass
2.1
m
· · · bs ,
v
where C c1 , c2 , . . . , cs T , bv b1 , b2 , . . . , bs , and Av aij si,j1 .
The adoption of the method 2.1 for solving the problem 1.1 leads to
yn1 yn h
s
m v
n
n
vn
vn
,
bj f v tj , yj , y
j
, wj
v1 j1
n
yi
yn h
s
m 2.2
v
n
n
vn
vn
,
aij f v tj , yj , y
j
, wj
v1 j1
n
where tn t0 nh, tj
n
vn
tn cj h, yn , and yj , y
j
are approximations to the analytic
vn
solution ytn , ytn cj h, ytn cj h − τv of 1.1, respectively, and the argument y
j
determined by
vn
y
j
⎧ ⎪
⎨ϕ tn cj h − τv
r
n−mv Pv ⎪
LPv δv yj
⎩
is
tn cj h − τv ≤ 0
tn cj h − τv > 0,
2.3
Pv −d
with τv mv − δv h, δv ∈ 0, 1, integer mv ≥ r 1, r, d ≥ 0, and
LPv δv r δv − k
,
Pv − k
k−d
k/
Pv
Pv −d, −d 1, . . . , r.
2.4
4
Abstract and Applied Analysis
i
We assume mv ≥ r 1 is to guarantee that no unknown values yj with i ≥ n are used in
the interpolation procedure
vn
wj
is an approximation to w
n
tj
tn
j
:
n−mv tj
n
g v tj , s, ys ds,
2.5
which can be computed by a appropriate compound quadrature rule:
vn
wj
h
mv
n−q
n n−q
,
dq g v tj , tj
, yj
v 1, 2, . . . , m, j 1, 2, . . . , s.
2.6
q0
As for the quadrature rule 2.6, we usually adopt the compound trapezoidal rule, the
compound Simpsons rule or the compound Newton-Cotes rule, and so forth according to
the requirement of the convergence of the method see 19 and denote M max1≤v≤m {mv }
v
and η max1≤v≤m {ηv } with ηv satisfing m
q0 |dq | < ηv , v 1, 2, . . . , m.
n
In addition, we always put yj
ϕtn cj h, yn ϕtn whenever n ≤ 0.
In order to write 2.2, 2.3, 2.5, and 2.6 in a more compact way, we introduce some
G ⊗ IN
notations. The N × N identity matrix will be denoted by IN , e 1, 1, . . . , 1T ∈ RS , G
T
is the Kronecker product of matrix G and IN . For u u1 , u2 , . . . , us , v v1 , v2 , . . . , vs T ∈
CNS , we define the inner product and the induced norm in CNS as follows:
u, v s
ui , vi ,
s
u ui 2 .
i1
i1
⎡ vn ⎤
y
⎥
⎢ 1
⎢ vn ⎥
⎥
⎢y
2
⎥
⎢
⎢ .. ⎥,
⎢ . ⎥
⎦
⎣
vn
y
s
⎡ vn ⎤
w
⎥
⎢ 1
⎢ vn ⎥
⎥
⎢w2
⎥
⎢
⎢ .. ⎥,
⎢ . ⎥
⎦
⎣
vn
ws
2.7
Moreover, we also adopt that
yn
⎡ n ⎤
y
⎢ 1 ⎥
⎢ n ⎥
⎢y2 ⎥
⎥
⎢
⎢ .. ⎥,
⎢ . ⎥
⎦
⎣
n
ys
y
vn
wvn
⎡ n ⎤
t
⎢1 ⎥
⎢ n ⎥
⎢t2 ⎥
⎥
T n ⎢
⎢ .. ⎥,
⎢ . ⎥
⎣ ⎦
n
ts
⎤
vn
⎡
n
n
vn
f v t1 , y1 , y
1
, w1
⎢
⎥
⎢ v n n vn vn ⎥
⎢
⎥
t2 , y2 , y
2
, w2
⎢f
⎥
⎥.
f v T n , yn , y
vn , wvn ⎢
⎢
⎥
.
..
⎢
⎥
⎢
⎥
⎣
⎦
n
vn
vn
v n
ts , ys , y
s
, ws
f
2.8
Abstract and Applied Analysis
5
With the above notation, method 2.2,2.3, 2.5, and 2.6 can be written as
yn1 yn h
m
T
b
v f v Tn , yn , y
vn , wvn ,
v1
m
yn e
yn h
v f v Tn , yn , y
vn , wvn ,
A
v1
vn
y
j
vn
wj
⎧ ⎪
⎨e
ϕ tn cj h − τv ,
r
n−mv Pv ⎪
LPv δv yj
,
⎩
h
2.9
tn cj h − τv ≤ t0 ,
tn cj h − τv > t0 ,
Pv −d
mv
n−q
.
dq g v tn cj h, tn−q cj h, yj
q0
In 1997, Zhang and Zhou 25 introduced the extension of RN-stability to GDN-stability as
follows.
Definition 2.1. An ARKLM 2.1 for DDEs is called GDN-stable if, numerical approximations
yn and z n to the solution of 1.1 and its perturbed problem, respectively, satisfy
yn − zn ≤ C max ϕt − ψt,
t0 −τ≤t≺t0
n ≥ 0,
2.10
where constant C > 0 depends only on the method, the parameter αv , βv , σv , rv , r
v , and the
interval length T − t0 , ψt is the initial function to the perturbed problem of 1.1.
Definition 2.2. An ARKLM 2.1 is called strongly algebraically stable if matrices Mγμ are
nonnegative definite, where
T
T
Mγμ Bγ Aμ Aγ Bμ − bγ bμ ,
γ γ
γ
Bγ diag b1 , b2 , . . . , bs ,
2.11
for μ, γ 1, 2, . . . , m.
Let
n
1n
2n
mn
1n
2n
mn
yn , yj , y
j
, y
j
, . . . , y
j
, wj
, wj
, . . . , wj
n 1n 2n
mn
1n
2n
mn
zn , zj , z
j
, z
j
, . . . , z
j
,w
!j
,w
!j
,...,w
!j
s
j1
s
,
2.12
j1
be two sequences of approximations to problems 1.1 and its perturbed problem, respectively. From method 2.1 with the same step size h, and write
n
Ui
vn
Qi
n
n
n
vn y
vn − z
vn ,
yi − zi ,
U
U0 yn − zn ,
i
i
i
#
"
n
n
vn
vn
n n vn
vn
− f v ti , zi , z
i
,
h f v ti , yi , y
i
, wi
,w
!i
i 1, 2, . . . , s,
v 1, 2, . . . , m.
2.13
6
Abstract and Applied Analysis
Then 2.2 and 2.3 read
n1
U0
n
U0 s
m v
vn
bj Qj
2.14
,
v1 j1
n
Ui
n
U0 m s
v
vn
aij Qj
2.15
.
v1 j1
Our main results about GDN-stability are contained in the following theorem.
Theorem 2.3. Assume ARK method 2.2 is strongly algebraically stable, and then the corresponding
ARKLM 2.1 is GDN-stable, and satisfies
yn − zn ≤ C max ϕt − ψt2 ,
n ≥ 0,
t0 −τ≤t≤t0
2.16
where
$
m
C exp 6T − t0 ms βv L2v mv d 1
%
2
max ϕt − ψt ,
t0 −τ≤t≤t0
v1
& '
Lv max Lpν ,
−d≤pv ≤r
2.17
Proof. From 2.14 and 2.15 we get
n1 2
U0 (
n
U0
s
m v vn
n
bi Qi
, U0
s
m )
v vn
bi Qi
v1 i1
v1 i1
m s
2
n v
vn
n
bi Re Qi
, U0
U0 2
v1 i1
s
m u v
un
vn
bi bj Qi
, Qj
u,v1 i,j1
)
(
s
s
m m n 2
v
vn
n
v vn
U0 2
bi Re Qi
, Ui −
aij Qj
v1 i1
s
m u v
bi bj
2.18
v1 j1
vn
Qi
un
, Qj
u,v1 i,j1
s
m n 2
v
vn
n
U0 2
bi Re Qi
, Ui
−
v1 i1
s m u v
v u
u v
bi aij bj aij − bi bj
vn
Qi
un
, Qj
.
u,v1 i,j1
If the matrices Mγμ are nonnegative definite, then
s
m n1 2 n 2
v
vn
n
bj Re Qj
, Uj .
U0 ≤ U0 2
v1 j1
2.19
Abstract and Applied Analysis
7
Furthermore, by conditions 1.2∼1.4 and Schwartz inequality we have
vn
n
n
n
vn
vn
h Re f v tj , yj , y
j
, Uj
, wj
Re Qj
n n vn
vn
n
−f v tj , zj , z
j
, Uj
,w
!j
n 2
vn vn
vn 2
hσ
≤ hαv Uj hβv U
−
w
!
w
v
j
j
j
2.20
mv n 2
vn n−q 2
≤ hαv Uj hβv U
2h3 r
v2 ηv2 σv
Uj
j
q0
mv n 2
n−q 2
hαv Uj 2h3 r
v2 ηv2 σv
Uj
,
tn cj h − τv ≤ t0
q0
2
r
n−m
p
n 2
v
v
hαv Uj hβv Lpv δv Uj
pv −d
2h3 r
v2 ηv2 σv
mv n−q 2
Uj
,
2.21
tn cj h − τv > t0
q0
mv n 2
n−q 2
≤ hαv Uj 2h3 r
v2 ηv2 σv
Uj
,
tn cj h − τv ≤ t0
2.22
q0
2
r
n−m
p
n 2
v
v 2
≤ hαv Uj 2hβv Lv Uj
pv −d
m
v n−q 2
2h3 r
v2 ηv2 σv
Uj
, tn cj h − τv > t0 .
2.23
q0
For 2.23, we have
2
mv r
n−mv pv n−q 2
2h3 r
v2 ηv2 σv
U
2.23 ≤ 2hβv L2v U
j
j
pv −d
q0
2
mv
n−mv pv .
≤ 3hβv L2v U
j
pv −d
2.24
By the same way, we can also get
2.22 ≤
2
mv
n−mv pv 2
3hβv Lv Uj
.
P −d
v
2.25
8
Abstract and Applied Analysis
Substituting 2.25 and 2.24 into 2.19, yields
2
mv
s
m n−m
p
n1 2 n 2
v
v
v 2
3βv Lv bj Uj
U0 ≤ U0 2h
pv −d
v1 j1
2.26
s
m n−m p 2
n 2
v
≤ U0 6h
βv L2v bj mv d 1 max Uj v v −d≤pv ≤mv
v1 j1
m
n−m p 2
n 2
≤ U0 6hms
βv L2v mv d 1 max Uj v v $
≤ 1 6hms
m
2.27
j,pv ∈Ev
v1
%
βv L2v mv
*
+
n 2
n−mv pv 2
d 1 max U0 , max Uj
,
j,pv ∈Ev
v1
where Ev {j, Pv 1 ≤ j ≤ s, −d ≤ Pv ≤ r}.
Similar to 2.27, the inequalities:
%
$
*
m
+
n 2
n 2
n−m P 2
2
βv Lv mv d 1 max U0 , max Uj v v Ui ≤ 1 6hms
j,Pv ∈E
v1
2.28
follows for i 1, 2, . . . , s.
In the following, with the help of inequalities 2.27, 2.28, and induction we shall
prove the inequalities:
%n1
$
m
2
n 2
2
βv Lv mv d 1
max ϕt − ψt ,
Ui ≤ 1 6hms
t0 −τ≤t≤t0
v1
2.29
for n ≥ 0, i 1, 2, . . . , s.
In fact, it is clear from 2.27, 2.28, and mv ≥ r 1 such that
%
$
m
2
0 2
2
βv Lv mv d 1 max ϕt − ψt ,
Ui ≤ 1 6hms
v1
t0 −τ≤t≤t0
i 0, 1, 2, . . . , s.
2.30
Suppose for n ≤ k k ≥ 0 that
%n1
$
m
2
n 2
2
βv Lv mv d 1
max ϕt − ψt ,
Ui ≤ 1 6hms
v1
t0 −τ≤t≤t0
i 0, 1, 2, . . . , s.
2.31
Abstract and Applied Analysis
9
Then from 2.27 and 2.28, mv ≥ r 1 and 1 6hms
that
m
v1
βv L2v mv d 1 > 1, we conclude
%k2
$
m
2
k1 2
2
βv Lv mv d 1
max ϕt − ψt ,
Ui
≤ 1 6hms
t0 −τ≤t≤t0
v1
i 0, 1, 2, . . . , s.
2.32
This completes the proof of inequalities 2.29. In view of 2.29, we get for n ≥ 0 that
%n1
$
m
2
n 2
2
βv Lv mv d 1
max ϕt − ψt
U0 ≤ 1 6hms
$
t0 −τ≤t≤t0
v1
≤ exp n 16hms
m
%
βv L2v mv
d 1
v1
$
≤ exp 6T − t0 ms
m
βv L2v mv
t0 −τ≤t≤t0
%
d 1
v1
2
max ϕt − ψt
2.33
2
max ϕt − ψt .
t0 −τ≤t≤t0
As a result, we know that method 2.1 is GDN-stable.
3. D-Convergence
In order to study the convergence of numerical methods for MDIDEs, we have to mention
the concept of the convergence for stiff ODEs.
In 1981, Frank et al. 26 introduced the important concept of B-convergence for
numerical methods applied to nonlinear stiff initial value problems of ordinary differential
equations. Later, there have been rapid developments in the study of B-convergence, and a
significant number of important results have already been found for Runge-Kutta methods.
In fact, B-convergence result is nothing but a realistic global error estimate based on onesided Lipschitz constant 27. In this section, we start discussing the convergence of ARKLM
2.1 for MDIDEs 1.1 with conditions 1.2–1.4. The approach to the derivation of these
estimates is similar to that used in 25. We assume the analytic solution yt of 1.1 is smooth
enough, and its derivatives used later are bounded by
i
,
D yt ≤ M
i,
t ∈ t0 − τ, T ,
3.1
where
yi t,
D yt yi t0 jh − 0 ,
i
t ∈ t0 j − 1 h, t0 jh ,
t t0 jh.
3.2
10
Abstract and Applied Analysis
If we introduce some notations
Y n
⎤
⎡
ytn c1 h
⎢ytn c2 h⎥
⎥
⎢
⎢
⎥,
..
⎦
⎣
.
Y
vn
⎤
⎡
ytn c1 h − τv ⎢ytn c2 h − τv ⎥
⎥
⎢
⎢
⎥,
..
⎦
⎣
.
ytn cs h
w
vn
⎤
⎡
wtn c1 h − τv ⎢wtn c2 h − τv ⎥
⎥
⎢
⎢
⎥.
..
⎦
⎣
.
ytn cs h − τv 3.3
wtn cs h − τv With the above notations, the local errors in 2.9 can be defined as
ytn1 ytn h
T
b
v f v Tn , Y n , Y
vn , w
vn Qn ,
3.4
v f v T n , Y n , Y
vn , w
vn rn ,
A
3.5
m
v1
Y n e
ytn h
m
v1
vn vn
vn T
, Y2
, . . . , Y
s
,
Y
vn Y
1
3.6
with
vn
Y
j
⎧ ⎪
⎨ϕ tn cj h − τv ,
r
n−mv Pv vn
⎪
LPv δv yj
ρj
,
⎩
tn cj h − τv ≤ t0 ,
tn cj h − τv > t0 ,
3.7
Pv −d
vn
wj
h
mv
n−q
vn
Rj
dq g v tn cj h, tn−q cj h, yj
.
3.8
q0
˘
vn w
vn
If we take y̆n ytn , y̆n Y n , y̆vn Y
vn , and w
Then we can get the perturbed scheme of 2.9,
m
T
vn
˘
vn Qn ,
y̆n1 y̆n h b
v f v T n , y̆n , y
˘
,w
3.9
v1
m
˘
vn rn ,
v f v T n , y̆n , y
˘ vn , w
y̆n e
y̆n h A
3.10
v1
vn
y
˘ j
⎧ ⎪
⎨e
ϕ tn cj h − τv ,
r
n−mv Pv ⎪
LPv δv y̆j
ρj vn ,
⎩
tn cj h − τv ≤ 0,
tn cj h − τv > 0,
3.11
Pv −d
vn
wj
h
mv
q0
n−q
vn
Rj
dq g v tn cj h, tn−q cj h, yj
.
3.12
Abstract and Applied Analysis
11
nT
With perturbations, Qn ∈ CN , rn r1
vn T
Rs
,
n
ρ1
T
n
, ρ2
T
n
, . . . , ρs
T
nT
, r2
nT T
, . . . , rs
vn
, Rvn R1
vn
, R2
,...,
ρn ∈ CNS , according to Taylor formula and the formula
in 28, pages 205–212, Qn , rn and ρn can be determined respectively, as follows:
Qn l1
n
ri
P
l1
n
ρi
⎞
⎛
s
m hl ⎝ 1 v
n
−
b cl−1 ⎠Dl ytn R0 ,
l − 1! l v1 j1 j j
3.13
⎞
⎛
s
m hl ⎝ 1 l v
n
c −
a cl−1 ⎠Dl ytn Ri ,
l − 1! l i v1 j1 ij j
3.14
P
r
m hq1 n
δv − Pv Dq1 y ξi ,
q 1 ! v1 Pv −d
n
ξi
∈ tn−mv −d ci h, tn−mv r ci h,
3.15
n
n
n
2i hi1 , i 0, 1, 2, . . . , s, h ∈ 0, h0 , h0 depends
where q d r, Ri , and ξi satisfy ||Ri || ≤ M
2
,i in
only on the method, and Mi i 0, 1, 2, . . . , s depends only on the method and some M
3.2.
Combining 2.2, 2.3, 2.5, and 2.6 with 3.9, 3.10, 3.11, and 3.12 yields the
n1
y̆n1 − yn1 :
following recursion scheme for the ε0
n1
ε0
n
ε0 h
m
T
vn
, wvn − f v T n , y̆n , y
vn , wvn
b
v f v T n , y̆n , y
˘
v1
n
εn e
ε0 h
vn
v
˘
− wvn Qn,
gn εn H vn w
m
v f v T n , y̆n , y
˘ vn , wvn − f v T n , y̆n , y
vn , wvn
A
v1
vn
v
˘
gn εn H vn w
− wvn rn ,
3.16
n1
where ε0
vn
gi
n T
y̆n1 − yn1 , εn ε1
1
0
vn
Hi
n T
, ε2
n T T
, . . . , εs
y̆n − yn ,
vn vn n
n
n
˘
i
dθ,
f2 tn ci h, yi θ y̆i − yi , y
˘ i
,w
1
0
f4 tn vn
n
˘
vn
ci h, yi , y
˘ i
,w
i
θ
˘
vn
w
i
−
i 1, 2, . . . , s,
vn
wi
3.17
dθ,
12
Abstract and Applied Analysis
here, fi x1 , x2 , x3 , x4 is the Jacobian matrix ∂fx1 , x2 , x3 , x4 /∂xi i 1, 2, 3, 4.
mv
vn
"
#
˘
H vn w
− wvn hH vn
dq g v tn , tn−q , y̆n−q − g v tn , tn−q , yn−q
q1
Rvn
"
#
hH vn d0 g v tn , tn , y̆n − g v tn , tn , yn
hH
vn
hH vn
mv
"
#
dq g v tn , tn−q , y̆n−q − g v tn , tn−q , yn−q
q1
Rvn
1
v
vn
hH
d0 g3 tn , tn , yn θ y̆n − yn dθ · εn .
hH
vn
0
Assume that I
s − h
3.18, we can get
n1
ε0
⎧
⎨
m
⎩
v1
IN h
m
v1
3.18
v g vn hH vn d0 g vn is regular, from 3.16 and 3.17,
A
3
$
%
m
−1
vn
v
vn
vn
bvT I
s − h
g
hH
d0 g 3
A
v1
⎫
⎬
vn
n
ε
×
e g vn hH vn d0 g3
⎭ 0
h
m
$
%
m
−1
vn
v
vn
vn
bvT g vn hH vn d0 g vn I
s − h
g
hH
d0 g 3
A
3
v1
×
rn h2
m
v1
v H vn Rvn
A
v1
h
m
b
v1
$
× h
·
$
vT
m
v
A
I
s − h
m
v
A
g
vn
hH
v1
g
vn
hH
vn
vn
d0 g 3
%−1
%
vn
vn
d0 g 3
⎧
⎨
v1
⎩
vn
f v T n , y̆n , y
˘
, wvn − f v T n , y̆n , y
vn , wvn
⎫
"
#⎬
hH vn
dq g v tn , tn−q , y̆n−q − g v tn , tn−q , yn−q
⎭
q1
mv
Qn h2
m
T
b
v H vn Rvn .
v1
Now, we introduce the concept of D-convergence from 25.
3.19
Abstract and Applied Analysis
13
n
Definition 3.1. An ARKLM 2.1 with yn ytn n ≤ 0, yi ytn ci h n < 0 and
vn
y
i
ytn ci h − τv n < 0 is called D-convergence of order p if this method, when
applied to any given DIDEs 1.1 subject to 1.2–1.4; produce an approximation sequence
yn and the global error satisfies a bound of the form:
ytn − yn ≤ Ctn hP ,
h ∈ 0, h0 ,
3.20
where the maximum stepsize h0 depends on characteristic parameter αv , βv , σ v , r v , r v and
,i in 3.2, delay τv , characteristic
the method, the function Ct depends only on some M
parameters αv , βv , σv , rv , r
v , v 1, 2, . . . , m, and the method.
Definition 3.2. The ARKLM 2.2, 2.3, 2.5, and 2.6 is said to be DA-stable if the
ν
−
matrix Is − m
ν1 A ξ is regular for ξ ∈ C : {ξ ∈ C | Re ξ ≤ 0}, and |Ri ξ| ≤ 1 for all ξ ∈
−
C , i 0, 1, . . . , s.
Where
Ri ε1 1 m
v
Ai ε1
v1
v
A0 bv ,
v
Ai
Is −
m
−1
A
ν1
v
v
v
ai1 , ai2 , . . . , ais
ν
T
,
ξ
e,
3.21
i 0, 1, . . . , s.
Definition 3.3. The ARKLM 2.2, 2.3, 2.5, and 2.6 is said to be ASI-stable if the matrix
M ν −1
ν
−
−
Is − M
ν1 A ξ is regular for ξ ∈ C , and Is −
ν1 A ξ is uniformly bounded for ξ ∈ C .
Definition 3.4. The ARKLM 2.2, 2.3, 2.5, and 2.6 is said to be DAS-stable if the
m
νT
ν
−
ν −1
ξIs − M
i 0, 1, . . . , s
matrixIs − M
ν1 A ξ is regular for ξ ∈ C , and
ν1 A ξ
ν1 Ai
is uniformly bounded for ξ ∈ C− .
Lemma 3.5. Suppose the ARKLM 2.2, 2.3, 2.5, and 2.6 is DA- DAS- and ASI-stable, then
there exist positive constants h0 , γ 1 , γ 2 , γ 3 , which depend only on the method and the parameter
αv , βv , σ v , r v , r v such that
M
ν A ξ ≤ γ1 ,
Is −
ν1
−1 m
m
T
IN ≤ 1 γ2 h,
v ξ I
s −
v ξ
e
A
A
i
v1
v1
−1 m
m vT
NS
A
ξ I
s −
v ξ
v
A
≤ γ3 v, v ∈ C ,
i
v1
v1
h ∈ 0, h0 ,
3.22
i 0, 1, 2 . . . , s.
Proof. This Lemma can be proved in the similar way as that of in 29, Lemmas 3.5–3.7.
14
Abstract and Applied Analysis
Theorem 3.6. Suppose the ARKLM 2.2, 2.3, 2.5, and 2.6 is DA- DAS- and ASI-stable,
then there exist positive constants h0 , γ 3 , γ 4 , γ 5 , which depend only on the method and the
parameters αv , βv , σ v , r v , r
v , such that for h ∈ o, h0 ,
6
⎧
⎪
n−mv pv n−q n−1 n1 ⎪
⎪
hγ
max
max
max
max
1
hγ
ε
,
ε
,
ε
ρ
⎪
4
5
0
i
i
i
⎪
1≤i≤s
i,pv ∈E
i,q∈Eq
⎪
⎪
⎪
⎪
n−1 ⎨ ⎪
i 0,
Q
γ3 γ
n−1 ,
n 6
εi ≤
⎪
⎪
n−mv pv n−q n n1 ⎪
⎪
hγ
max
max
1
hγ
max
max
,
ε
,
ε
ρ
ε
4
5
⎪
0
i
i
i ⎪
1≤i≤s
i,pv ∈E
i,q∈Eq
⎪
⎪
⎪
⎪
⎩ n
i 1, 2, . . . , s,
Q
γ3 γ
n ,
3.23
n
n
n
n
where ε0 y̆n − yn , εi y̆i − yi , E {i, pv | 1 ≤ i ≤ s, −d ≤ pv ≤ γ}, Eq {i, q | 1 ≤
n Qn h2 m b
vT H vn Rvn , r
n rn h2 m A
v H vn Rvn .
i ≤ s, 1 ≤ q ≤ m}, Q
v1
v1
Proof. Using 3.19 and Lemma 3.5, for h ∈ 0, h0 , we obtain that
n1
ε0
n ≤ 1 γ2 h ε0 γ3 rn Q
n
⎧
m v ⎨ v n n vn vn hγ3 T , y̆ , y
˘
− f v T n , y̆n , y
vn , wvn
,w
A ⎩f
v1
⎫
mv
"
#⎬
hH vn
dq g v tn , tn−q , y̆n−q − g v tn , tn−q , yn−q
⎭
q1
⎧
m vT ⎨ v n n vn vn h
T , y̆ , y
˘
− f v T n , y̆n , y
vn , wvn
,w
b ⎩f
v1
⎫
mv
"
#⎬
hH vn ·
dq g v tn , tn−q , y̆n−q − g v tn , tn−q , yn−q
⎭
q1
3.24
n ≤ 1 γ2 h ε0 γ3 rn Q
n
3.25a
⎧ s s
m ⎨
"
#
av f v tn , yn , y
˘ vn , wvn − f v tn , yn , y
vn , wvn
hγ3
j
ij
j
j
j
j
j
j
j
⎩ i1 j1
v1
2 ⎫1/2
mV
"
#
⎬
n−q
n−q
vn
hHj
− g v tn , tn−q , yj
dq g v tn , tn−q , y̆j
⎭
q1
3.25b
Abstract and Applied Analysis
m s
vn
v
n
n
vn
n
n
vn
vn
− f v tj , yj , y
j
, wj
, wj
h f v tj , yj , y
˘ j
bj
v1 j1
⎫
mV
"
#⎬
n−q
n−q
vn
.
− g v tn , tn−q , yj
dq g v tn , tn−q , y̆j
hHj
⎭
q1
15
3.25c
For
⎧ s s
m ⎨
"
#
av f v tn , yn , y
˘ vn , wvn − f v tn , yn , y
vn , wvn
hγ3
j
ij
j
j
j
j
j
j
j
⎩ i1 j1
v1
2 ⎫1/2
mV
"
#
⎬
n−q
n−q
vn
hHj
− g v tn , tn−q , yj
dq g v tn , tn−q , y̆j
⎭ 3.25b.
q1
3.26
Then
⎧
⎧
s
s 7
m ⎨
72 ⎨
7 v 7 v n n ˘ vn vn
tj , yj , y
j
2
, wj
3.25b ≤ hγ3
7aij 7 f
⎩ i1 j1
⎩
v1
n
n
vn
vn 2
, wj
−f v tj , yj , y
j
mV
"
n−q
vn
v
hH
g
t
d
,
t
,
y̆
q
n
n−q
j
j
q1
−g v
2 ⎫⎫1/2
⎬⎬
#
n−q
tn , tn−q , yj
⎭⎭
⎧ ⎧
m
s ⎨ s 7
s 7
m ⎨
7 2
7
7 v 72 2 ˘ vn
7 v 72 2 vn 2 v
vn ≤ hγ3
− y
j
2
dq
2
7aij 7 rv y
j
2
7aij 7 h Hj
⎩ i1 ⎩ j1
v1
j1
q1
⎫⎫1/2
2 ⎬⎬
n−q
n−q − g v tn , tn−q , yj
·g v tn , tn−q , y̆j
⎭⎭
s
s 7
m 7 2
7 v 72 2 ˘ vn
vn 2
≤ hγ3
− y
j
7aij 7 rv y
j
v1
i1
j1
m
s 7
s m 7
n−q
n−q 2
7 v 72 vn 2 v 2 v
tn , tn−q , y̆j
− g v tn , tn−q , yj
2h2 γ3
dq · g
7aij 7 Hj
v1
i1 j1
q1
16
Abstract and Applied Analysis
≤ 2hγ3
s s 7
m 7 7 v 7 ˘ vn
vn − y
j
7aij 7rv y
j
v1 i1 j1
2h2 γ3
mv
s s m 7
77
7 n−q
n−q 7 v 77 vn 7 v
tn , tn−q , y̆j
− g v tn , tn−q , yj
dq 7aij 77Hj
7 · g
v1 i1 j1 q1
≤ 2hγ3
s 7
m 7 7 v 7 ˘ vn
vn − y
j
7aij 7rv y
j
v1 i,j1
2h2 γ3
mv
dq
q1
s 7
m 77
7 n−q 7 v 77 vn 7 n−q
− yj
7aij 77Hj
7
rv y̆j
.
v1 i,j1
3.27
For
⎧
m s
⎨
v
n ˘ vn
vn
n
vn
vn
v n
v n
f
h t
−
f
t
b
,
y
,
y
,
w
,
y
,
y
,
w
j
j ⎩
j
j
j
j
j
j
j
v1 j1
vn
hHj
mV
"
n−q
dq g v tn , tn−q , y̆j
q1
−g
v
n−q
tn , tn−q , yj
3.28
6
# 3.25c,
then
3.25c ≤ h
s 7
m 7 7 ν 7 ˘ vn
vn − y
j
7bj 7γv y
j
v1 j1
⎫
⎧
mv
s 7
m ⎬
77
7⎨
n−q
n−q 7 v 77 vn 7
h2
− g v tn , tn−q , yj
dq g v tn , tn−q , y̆j
7bj 77Hj
7
⎭
⎩
v1 j1
q1
⎧
⎫
mv
s 7
s 7
m m 7 vn
77
7⎨
⎬
n−q 7 ν 7 ˘
7 v 77 vn 7
n−q
vn ≤h
− y
j
dq r
v y̆j
− yj
7bj 7γv y
j
h2
7bj 77Hj
7
.
⎩
⎭
v1 j1
v1 j1
q1
3.29
Combine 3.27, 3.29, and 3.25a, we have
n1
ε0
n ≤ 1 γ2 h ε0 γ3 rn Q
n
⎞
⎛
m
s 7
s 7
7 7 vn
7
7
7
7
v
v
vn h
γv ⎝2γ3
− y
j
7aij 7 7bj 7⎠y
˘ j
v1
h2
m
v1
i,j1
r
v
mv
q1
dq
s
j1
2γ3
j1
6
s 7
7 7
7 7
7 7
7 7 v 7 7 vn 7 7 v 7 7 vn 7 n−q 7aij 7 · 7Hj
7 7bj 7 · 7Hj
7 εj
.
i1
3.30
Abstract and Applied Analysis
17
Moreover, it follows from 2.5 and 3.11 that
vn
˘
vn − y
j
y
j
≤ sup
r 7
7
vn n−mv Pv 7Lp δv 7 max ε
ρ
.
v
j
j
−d≤Pr ≤r
δv ∈0,1 Pv −d
3.31
Substituting 3.31 in 3.30, we get
6
n1 n n−mv Pv n−q 0
0 2
ε0 ≤ 1 γ4 h γ6 h max ε0 , max εj
, max εj
Q
n
0
γ3 rn hγ5
vn max ρj
,
j,v∈Em
j,pv ∈E
j.q∈Eq
h ∈ 0, h0 ,
3.32
where
0
γ4
γ2 0
γ5
0
γ6
E
&
r 7
7
7Lp δv 7,
sup
v
δv∈0,1 Pv −d
m
γ
v
v1
mv s
0
γ5
⎞
s 7
s 7
7 7
7
7
7
7
v
v
rv ⎝2γ3
7aij 7 7bj 7⎠,
⎛
v1
i,j1
j1
6
s 7
7 7
7
7 v
7 v
vn 7
vn 7
2γ3 7aij · Hj
7 7bj · Hj
7 ,
dq
m
q1 j1
3.33
i1
'
& '
j, Pv | 1 ≤ j ≤ s, −d ≤ Pv ≤ γ ,
Eq j, q | 1 ≤ j ≤ s, 1 ≤ q ≤ mv ,
& '
Em j, v | 1 ≤ j ≤ s, 1 ≤ v ≤ m .
By Lemma 3.5, similar to 3.32, we can obtain the inequalities:
6
n n n−mv Pv n−q i
2 i
εi ≤ 1 hγ4 h γ6 max ε0 , max εj
, max εj
j,Pv ∈E
i
hγ5
vn max ρj
Qn γ3 rn ,
j,v∈Em
j,q∈Eq
3.34
i 1, 2, . . . , s, h ∈ 0, h0 ,
where
i
γ4
r 7
7
i
7Lp δv 7,
γ2 γ5 sup
v
δv∈0,1 Pv −d
i
γ6
m
γ
v
v1
mv s
q1 j1
i
dq 2γ3
i
γ5 m
⎞
s 7
s 7
7 7
7 v 7
7 v 7
rv ⎝2γ3
7aij 7 7aij 7⎠,
⎛
v1
i,j1
j1
7 7
7
7 v
7 v
vn 7
vn 7
7aij · Hj
7 7aij · Hj
7 .
s 7
3.35
i1
i
i
Setting γ4 max0≤i≤s γ4 , γ5 max0≤i≤s γ5 , γ6 max0≤i≤s γ6 .
Combining 3.32 with 3.34, we immediately obtain the conclusion of this theorem.
18
Abstract and Applied Analysis
Now, we turn to study the convergence of ARKLM 2.1 for 1.1. It is always assumed
that the analytic solution yt of 1.1 is smooth enough on each internal of the form t0 j −
1h, t0 jh j is a positive integer as 3.2 defined.
Theorem 3.7. Assume ARKLM 2.1 with stage order p is DA-, DAS- and ASI-stable, then the
ARKLM 2.1 is D-convergent of order min{p, q 1, s 1}, where q d r.
Proof. By Theorem 3.6, we have for h ∈ 0, h0 6
⎧
⎪
n−1−m
p
n−q
n−1
⎪
v
v
⎪
, max εj
⎪ 1 hγ4 h2 γ6 max ε0 , max εi
⎪
⎪
i,pv ∈E
j,q∈Eq ⎪
⎪
⎪
⎨ T hp1 T hq2 T hs2 , i 0
1
2
n 6
-3
εi ≤
⎪
n−m
p
n−q
⎪
n
v
v
2
⎪
1 hγ4 h γ6 max ε0 , max εi
, max εj
⎪
⎪
⎪
i,pv ∈E
j,q∈Eq
⎪
⎪
⎪
⎩ T hp1 T hq2 T hs2 , i 1, 2, . . . , s,
1
2
3.36
3
where
s
2
22 ,
T1 M0 γ3 M
i
r
γs
|δv − Pv |Mq1 ,
q 1 ! pv −d
i1
m
m
vT
vR
v vR
g vs ξ.
T3 b H
A H
v1
T2 3.37
v1
It follows from an induction to 3.36 for n that
n εi ≤
⎧ n
j ⎪
⎪
1 2hγ4 T1 hp1 T2 hq2 T3 hs2 , i 0,
⎪
⎪
⎨ j0
n1
⎪
j ⎪
⎪
1 2hγ4 T1 hp1 T2 hq2 T3 hs2 , i 1, 2, . . . , s.
⎪
⎩
j0
Hence, for h ∈ 0, h0 , we arrive at
n
n j ytn − yn 1 2hγ4 T1 hp1 T2 hq2 T3 hs2
ε0 ≤
j0
n1
− 1
1 2hγ4
T1 hp1 T2 hq2 T3 hs2
2hγ4
3.38
Abstract and Applied Analysis
19
8
6
4
2
0
−2
−4
−6
−8
−10
0
5
10
15
20
25
30
35
Figure 1: Values yn with h 0.1.
8
6
4
2
0
−2
−4
−6
0
5
10
15
20
25
30
35
Figure 2: Values zn with h 0.1.
8
9
exp 2n 1hγ4 − 1 T1 hp1 T2 hq2 T3 hs2
≤
2hγ4
8
9
exp 2T − t0 γ4 exp 2h0 γ4 − 1 ≤
T1 hp T2 hq1 T3 hs1 .
2γ4
3.39
Therefore, the ARKLM 2.1 is D-Convergent of order min{p, q 1, s 1}, q r d.
20
Abstract and Applied Analysis
Table 1: Comparison between the numerical solutions for different h and the exact solution.
t
Numerical solution
for h 0.2
Numerical solution
for h 0.1
Numerical solution
for h 0.05
Exact solution y(t)
2.87569414
2.751727517
2.448183694
1.212021722
1.165089633
1.0957815
0.898712327
0.801604459
0.626189996
0.818730753
0.670320046
0.548811636
0.2
0.4
0.6
4. Some Examples
Consider the following initial value problem of multidelay-integro-differential equations:
t
81 yt92
8
9
ysds
y t −10 99i
8
92 500 yt − 1 − yt − 2 − 500
t−1
1 1 yt
8
92
t
1 exp−t
4
500
ysds − exp−t 10 − 99i ×
8
92 , 0 ≤ t ≤ 3,
t−2
1 1 exp−t
4
yt exp−t,
4.1
−2 ≤ t ≤ 0,
and its perturbed problem:
z t −10 99i
4
500
t
1 zt2
1 1 zt2
500zt − 1 − zt − 2 − 500
zsds
t−1
zsds − exp−t 104 − 99i ×
t−2
zt exp−t 0.01,
t
8
92
1 exp−t
8
92 ,
1 1 exp−t
0 ≤ t ≤ 3,
4.2
−2 ≤ t ≤ 0.
It an be easily verified that a1 a2 −9.5 × 103 , β1 β2 250, σ1 σ2 250, and r
1 r
2 1,
with analytic solution yt exp−t, where the inner product is standard inner product. We
apply the two-stages and two-order additive R-K method:
1 1
1
1
2
1
2
0 1 0
1
0 1
2
1
0 1
2
4.3
Abstract and Applied Analysis
21
1 to the problem 4.1 and its perturbed problem 4.2. Since the order of the method
is 2, we adopt the compound trapezoidal rule for computing the integer part.
According to the result of Theorems 2.3 and 3.7, the corresponding method for
DIDEs is GDN-stable and D-convergent. We denote the numerical solution of
problem 4.1 and 4.2 yn and zn , where yn and zn are approximations to ytn and
ztn , respectively. The values yn and zn with h 0.1 are listed in Figure 1, where
the abscissa and ordinate denote variable n and yn , resp. and Figure 2, where the
abscissa and ordinate denote variable n and zn , resp.. It is shown in Table 1 that
the numerical solutions are toward to the exact solutions as h → 0.
It is obvious that the corresponding method for MDIDEs is GDN-stable and Dconvergent.
Acknowledgments
This work was supported by the National Natural Science Foundation of China 11101109
and the Natural Science Foundationof Hei-long-jiang Province of China A201107.
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