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Asymptotic stability of linear non-autonomous difference equations with fixed delay
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Applied Mathematics and Computation 198 (2008) 787–798 www.elsevier.com/locate/amc
Asymptotic stability of linear non-autonomous difference equations with fixed delay q Y. Xu *, J.J. Zhao Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Abstract This paper studies the asymptotic stability of the linear non-autonomous difference equations with fixed delay. It is shown that the difference system is asymptotically stable if and only if the norm of the limit for the coefficient matrix is less than 1 under some conditions. Furthermore, the discretization of the pantograph equation by Runge–Kutta methods with variable stepsize is analyzed. Finally, some numerical experiments are made to demonstrate the main conclusions. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Difference equation; Delay differential equation; Stability; Numerical methods
1. Introduction Consider the difference equation: Dxn ¼ f ðn; xn ; xn1 ; . . . ; xnr Þ;
n 2 N;
ð1:1Þ
where Dxn ¼ xnþ1 xn and f : N Crþ1 ! C. It was studied as a class of generalized difference equation in Section 2 of [1]. The initial vector for this system requires the knowledge of initial data fxr ; xrþ1 ; . . . ; x0 g, which were called the initial string in [4]. The difference equation (1.1) can be thought of as a discrete analogue of the delay differential equation: x0 ðtÞ ¼ f ðtÞ; xðtÞ; xðaðtÞÞ; t > 0; xðtÞ ¼ x0 ðtÞ; t 2 inf faðsÞg; 0 ;
ð1:2Þ
sP0
q This paper is supported by the Natural Science Foundation of Heilongjiang Province (A200602), the project (HITC200710) supported by Science Research Foundation in Harbin Institute of Technology and the Project (HITQNJS.2006.053) supported by Development Program for Outstanding Young Teachers in Harbin Institute of Technology. * Corresponding author. E-mail address: [email protected] (Y. Xu).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.012
788
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
where f ; a and x0 are given functions with aðtÞ 6 t for all t P 0. In fact, this system can be found in a variety of scientific and engineering fields such as biology, economy, computer aided design, nonlinear dynamical systems and so on, which have a comprehensive list in [8]. Some papers dealt with the asymptotic stability of linear autonomous difference equation: xnþ1 ¼ P n xn ;
ð1:3Þ
where P n P is a constant matrix independent of n. The theory employed for the asymptotic stability of (1.3) is that the zero solution of this difference equation is asymptotically stable if and only if all the roots of its characteristic equation are inside the unit disk (see also [7]). Especially, the asymptotic stability has been studied of the zero solution for the difference equation, xnþ1 ¼ xn þ bxnk ;
n 2 N:
ð1:4Þ
A sufficient and necessary condition for its asymptotic stability was given in [12]. As far as we know, there are two special cases of difference equation (1.1) which are studied these years. One is the autonomous difference equation with unbounded delay (see also [9,10]). The other is the non-autonomous difference equation with bounded delay, i.e., the difference system (1.3) with the coefficient sequence fP n g depending on n. Iserles focused on the autonomous difference equation with unbounded delay: x2nþ1 ¼ kx2n þ 2lxn ; x2nþ2 ¼ kx2nþ1 þ lðxn þ xnþ1 Þ; where k; l 2 C. In [9,10], the asymptotic stability of this convenient model was analyzed by the Fourier transform. Yet, it is usually difficult to investigate the long time dynamical behavior of its solution due to the limited computer memory as shown in [13,15]. Yu [18] considered the following linear difference equation: xnþ1 ¼ xn þ bn xnkn ;
n 2 N;
ð1:5Þ
where fbn g is a sequence of non-negative real numbers, fk n g is a sequence of non-negative integers, and there exists a non-negative integer k such that k n 6 k for n 2 N. The sufficient conditions for the uniform stability and the global asymptotic stability are obtained. However, the method (see, for instance, the proof of Theorem 13 in [19]), used in the asymptotic stability analysis of the autonomous difference equation (1.3), does not apply for the above two kinds of difference equations. Therefore, the results about the stability of these two kinds of difference equations are very scarce. As it is pointed out in [16], although there are some theoretical study about the stability of delay differential equation (1.2) (see also [3,8,11,14]) and many numerical schemes to approximate this equation, the asymptotic behavior of these two types of systems (the continuous system and its discrete version) does not often coincide. Therefore, we pay attention to the criterion, on which the difference equation preserves the asymptotic stability (delay-independent type) of the delay differential equation. In this paper, we focus our attention on the asymptotic stability of a special linear non-autonomous difference equations with fixed delay. A sufficient and necessary condition such that the difference system is asymptotically stable is given. Applying this conclusion, the discretization of the pantograph equation is analyzed by using the numerical methods with variable stepsize. Using the above stable results of difference system, it is shown that the Runge–Kutta methods are asymptotically stable if and only if the modulus of stability function at infinity is less than 1, in which the proof is different from other papers (see also [2,17]). Finally, some numerical examples are presented to illuminate the main conclusions. 2. Asymptotic stability of difference equation In this paper, we consider the linear non-autonomous difference equation: y nþ1 ¼ Gnþ1 y n ; where the coefficient fGnþ1 g is a sequence of complex matrices, which depend on n.
ð2:1Þ
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For analyzing the property of this non-autonomous difference system, we recall the following definition. Definition 2.1. The difference equation (2.1) is called asymptotically stable, if the solution yn satisfies lim y n ¼ 0:
n!1
In order Ql to study the asymptotic stability of linear difference equation (2.1), we just need to estimate the norm of i¼0 G2N þi for some N 2 Z þ . Let ð2:2Þ
G ¼ lim Gn ; n!1
after simple calculations, we can obtain the norm of this matrix, which is independent of n. Here, for simplicity, we make the following assumptions. Assumption I kGk k ¼ kGkk ¼ nk ;
ð2:3Þ
with n = kGk. Assumption II kGnþN Gk 6 gn
for all n 2 Z þ ;
ð2:4Þ
where g is a constant and N is chosen such that gN < ln 2: 1g
ð2:5Þ
To facilitate the presentation of our results, for k 2 Z þ , we denote F k ¼ Gk G;
ð2:6Þ k
B2N ¼ F 2N ;
B2N þk ¼ GB2N þk1 þ F 2N þk ðG þ B2N þk1 Þ;
ð2:7Þ
then l Y
G2N þi ¼ Glþ1 þ B2N þl :
ð2:8Þ
i¼0
Lemma 2.1. Under conditions (2.3) and (2.4), for all l 2 Z þ , we have kB2N þl k 6
l Y
ðn þ gN þi Þ nlþ1 :
i¼0
Proof. From conditions (2.4) and (2.7), we can obtain kB2N k ¼ kG2N Gk 6 gN ; kB2N þ1 k ¼ kG B2N þ F 2N þ1 ðG þ B2N Þk 6 n gN þ gN þ1 ðn þ gN Þ ¼ ðn þ gN Þ ðn þ gN þ1 Þ n2 : Assume that kB2N þk1 k 6
k 1 Y ðn þ gN þi Þ nk ; i¼0
then under condition (2.3), we have k 1 Y ðn þ gN þi Þ nk kB2N þk k ¼ kGB2N þk1 þ F 2N þk ðG þ B2N þk1 Þk 6 n k
i¼0 k Y ¼ ðn þ gN þi Þ nkþ1 : i¼0
! þg
N þk
k 1 Y n þ ðn þ gN þi Þ nk k
i¼0
!
790
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
Lemma 2.2. Under conditions (2.3)–(2.5), if n P 1, then for all l 2 Z þ , we have Y l gN G2N þi P nlþ1 2 e1g > 0: i¼0 Proof. According to Lemma 2.1, from (2.3) and (2.8), we can obtain Y l G2N þi ¼ kGlþ1 þ B2N þl k P kGlþ1 k kB2N þl k P nlþ1 i¼0 N lþ1 Þ 1g ð1g lþ1 1g n 2e Pn :
l Y
! ðn þ gN þi Þ nlþ1
i¼0
Under condition (2.5), we have gN
e1g < 2: Hence Y l gN G2N þi P nlþ1 2 e1g > 0: i¼0
Theorem 2.1. Under conditions (2.3)–(2.5), the linear non-autonomous difference equation (2.1) is asymptotically stable if and only if n < 1:
ð2:9Þ
Proof. Assume that condition (2.9) is satisfied, since n = kGk, then we have kGk < 1. In view of (2.2), for any fixed e with 1 kGk > e > 0, there exists N > 0 such that for n P N , kGn Gk < 1 kGk e; and ky nþ1 k 6 ðkGk þ kGn GkÞ ky n k < ð1 eÞky n k; which implies limn!1 y n ¼ 0. Conversely, assume that n P 1, it is easy to see from Lemma 2.2 and Theorem 4.10 in [6] that any subsequence of fy n g can not tend to zero. If n > 1, then according to Lemma 2.2, we have Y l lim G2N þi ¼ 1; l!1 i¼0 which implies that fy n g is unbounded. If n = 1, then according to Lemma 2.2, from (2.3) and (2.8), we can obtain Y l l Y ky 2N þl k ¼ G2N þi y 2N 1 6 kGlþ1 þ B2N þl k ky 2N 1 k 6 ð1 þ gN þi Þ ky 2N 1 k 6 2 ky 2N 1 k; i¼0 i¼0 which implies that fy n g is bounded. h Corollary 2.1. Under conditions (2.3)–(2.5), if n > 1, then any subsequence of solution fy n g for the linear nonautonomous difference (2.1) is unbounded.
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
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Corollary 2.2. Under conditions (2.3)–(2.5), if n ¼ 1, then any subsequence of solution fy n g for the linear nonautonomous difference (2.1) is bounded, but does not vanish. 3. Discretization of pantograph equation This section deals with the numerical discretization of linear pantograph equation: x0 ðtÞ ¼ kxðtÞ þ lxðqtÞ;
ð3:1Þ
t > 0;
where k; l 2 C, q 2 ð0; 1Þ with the initial condition such that xð0Þ ¼ x0 2 C. We are interested in studying if the delay-independent condition (see also [14]) jlj < jkj;
Rk < 0;
ð3:2Þ
under which the solution xðtÞ of pantograph equation (3.1) tends to zero (algebraically) as t approaches infinity, is preserved when a numerical scheme with variable stepsize is used to approximate its solutions. Here, the following grid points are proposed to perform this approximation, tn ¼ q1 tn1 for n = 1, 2, 3, . . ., which appears to be the special case of variable stepsize introduced in [17]. The advantage of this kind of stepsize is that it could be used to obtain the approximate of any grid point which we are interested. It is easy to see that for any n, we have qhn ¼ hn1
and
lim hn ¼ 1;
ð3:3Þ
n!1
and the stepsizes increase geometrically. Then we formulate the condition of numerical asymptotic stability. The condition will depend on parameter k and l, but turns out to be independent of q (see also Section 3 in [2]). Definition 3.1. The numerical method applied to Eq. (3.1) with condition (3.2) is called asymptotically stable, if the numerical solution xn satisfies lim xn ¼ 0
n!1
for any q 2 ð0; 1Þ and H ¼ ft0 ; t1 ; . . . ; tn ; . . .g. For the general pantograph equation (3.1), the Runge–Kutta method gives out the recurrence relation: 8 s P ðnþ1Þ > x ¼ x þ h bi ½kxi þ lxh ðtn þ ci hnþ1 Þ; > nþ1 n nþ1 < i¼1
s P > ðnþ1Þ ðnþ1Þ > ¼ xn þ hnþ1 aij ½kxj þ lxh ðtn þ cj hnþ1 Þ; : xi j¼1
T
T
where hnþ1 ¼ tnþ1 tn , matrix A ¼ ðaij Þss is non-singular, vectors b ¼ ðb1 ; b2 ; . . . ; bs Þ , c ¼ ðc1 ; c2 ; . . . ; cs Þ ; ðnþ1Þ and xi , xh ðtn þ ci hnþ1 Þ can be interpreted as an approximation to xðtn þ ci hnþ1 Þ, xðqðtn þ ci hnþ1 ÞÞ for i ¼ 1; . . . ; s, respectively. Using the results (3.3) of variable stepsize, the above difference system can be rewritten as xnþ1 ¼ rnþ1 xn þ anþ1 X ðnÞ ; X
ðnþ1Þ
¼ bnþ1 xn þ cnþ1 X
where X ðnÞ ¼
ðnÞ ðnÞ T ðx1 ; x2 ; . . . ; xðnÞ s Þ
ðnÞ
ð3:4Þ ;
and
ð3:5Þ
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Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798 1
rnþ1 ¼ 1 þ knþ1 bT ðI knþ1 AÞ e; l 1 1 anþ1 ¼ lnþ1 bT ðI knþ1 AÞ ¼ bT ðk1 nþ1 I AÞ ; k 1 bnþ1 ¼ ðI knþ1 AÞ e; 1
cnþ1 ¼ lnþ1 ðI knþ1 AÞ A; T
with e ¼ ð1; 1; . . . ; 1Þ 2 Rs . From Eq. (3.4), for l 2 Z þ , we have, xnþl ¼ r1 xnþl1 þ ðrnþl r1 Þ xnþl1 þ anþl X ðnþl1Þ ¼ rl1 xn þ ðrnþl r1 Þ xnþl1 þ r1 ðrnþl1 r1 Þ xnþl2 þ þ rl1 1 ðrnþ1 r1 Þ xn þ anþl ðnÞ X ðnþl1Þ þ r1 anþl1 X ðnþl2Þ þ þ rl1 ; 1 anþ1 X
ð3:6Þ
where r1 ¼ lim rnþ1 ¼ 1 bT A1 e: n!1
T T
Let X n ¼ ðxn ; ðX ðnÞ Þ Þ , then the difference equations (3.4) and (3.5) can be written as X nþ1 ¼ Gnþ1 X n ;
ð3:7Þ
where Gnþ1 ¼
rnþ1
anþ1
bnþ1
cnþ1
:
Define ð3:8Þ
G ¼ lim Gn ; n!1
then G¼
r1
a
0
c
;
ð3:9Þ
with l a ¼ lim anþ1 ¼ bT A1 ; n!1 k
l c ¼ lim cnþ1 ¼ I s : n!1 k
Here, we choose j lk j with j lk j < kbT Ajr11kj T
ð3:10Þ
, then
1 þ1
1
kGk1 ¼ j1 b A ej ¼ jr1 j;
ð3:11Þ
through some calculations, for k 2 Z þ , we have k
kGk k1 ¼ jr1 j ;
ð3:12Þ
which means that the difference equation (3.7) satisfies Assumption I (2.3) with n ¼ jr1 j. Here, N is chosen such that 1 kk1 N A k1 ¼ krN k1 < 1;
ðkbT A1 k1 þ 1Þ qN < ln 2; 1q N
q where rN ¼ kð1qÞ A1 .
krN k1 s < 1; 1 krN k1
ð3:13Þ ð3:14Þ ð3:15Þ
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
793
Lemma 3.1. Under condition (3.2), for all n 2 Z þ , we have kGnþN Gk1 6 qn : Proof. From (3.13), for n 2 Z þ , we have 1 1 1 1 1 ðI k1 kðI knþN AÞ1 k1 ¼ kk1 nþN A nþN A Þ k1 6 kknþN A k1
1 1
1 kk1 N A k1
¼
krN k1 qn : 1 krN k1
Thus, from (3.2) and (3.14), we can obtain kGnþN Gk1 ¼ maxfjrnþN r1 j þ kbnþN k1 ; kanþN ak1 þ kcnþN ck1 g l ¼ max jbT A1 ðI knþN AÞ1 ej þ kðI knþN AÞ1 ek1 ; bT A1 ðI knþN AÞ1 k 1 l 1 þ ðI knþN AÞ k 1 krN k1 krN k1 T 1 T 1 n l n 6 max ðkb A k1 þ 1Þ q s; ðkb A k1 þ 1Þ q 6 qn : k 1 krN k1 1 krN k1
This conclusion proves that the difference equation (3.7) satisfies Assumption II (2.4) with g ¼ q. Although the following result has been already obtained in our next paper (see also Theorem 3.1 in [17]), here we include with another new proof in order to point out how the coefficients are involved in the asymptotic analysis. Theorem 3.1. Assume the matrix A is regular, then Runge–Kutta method ðA; b; cÞ, applied to Eq. (3.1) with condition (3.2), is asymptotically stable if and only if jr1 j < 1;
ð3:16Þ T
1
where r1 ¼ 1 b A e is the limit of stability function of Runge–Kutta method. Proof. Assume that jr1 j P 1, if the method is asymptotically stable, then lim xn ¼ 0:
ð3:17Þ
n!1
Case I. If jr1 j > 1, from (3.10), for any given e1 with 0 < e1 < jr12j1, there exists an integer N 0 such that, for all n P N 0 , we have jrn r1 j < e1 ;
kan ak1 < e1 :
ð3:18Þ
In view of (3.4) and (3.17), we know that lim anþ1 X ðnÞ ¼ 0:
ð3:19Þ
n!1
From (3.17) and (3.19), there exists n0 and k 0 such that, for all n P N 0 , jxn0 j P jxn j;
kak0 þ1 X ðk0 Þ k1 P kanþ1 X ðnÞ k1 :
Here, we can choose
j lk j
ð3:20Þ
so small that
e1 jxn0 j P kak0 þ1 X ðk0 Þ k1 :
ð3:21Þ þ
From (3.6), according to (3.18) and (3.21), for all l 2 Z , we arrive at jxn0 þl j P jrl1 xn0 j ½ðjrn0 þl r1 j jxn0 þl1 j þ þ jr1 j l1
þ jr1 j kan0 þl1 X ðn0 þl2Þ k1 þ þ jr1 j
l1
jrn0 þ1 r1 j jxn0 jÞ þ ðkan0 þl X ðn0 þl1Þ k1 l
kan0 þ1 X ðn0 Þ k1 Þ P jr1 j jxn0 j
½ð1 þ jr1 j þ þ jr1 jl1 Þe1 jxn0 j þ ð1 þ jr1 j þ þ jr1 jl1 Þe1 jxn0 j ¼
jr1 jl ðjr1 j 1 2e1 Þ þ 2e1 jxn0 j > 0: jr1 j 1
794
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
This implies that lim jxn0 þl j ¼ 1;
ð3:22Þ
l!1
which contradicts (3.17). Case II. If jr1 j ¼ 1, then in view of Corollary 2.2, we know fX n g is bounded, but does not vanish. Since fX n g is bounded, then there exist X and subsequence fX ðnk Þ g such that lim X ðnk Þ ¼ X :
k!1
From (3.5), we arrive at X ðnk þ1Þ ¼ bnk þ1 xnk þ cnk þ1 X ðnk Þ ; then, for l 2 Z þ , we have
l lim X ðnk þ1Þ ¼ X ; k!1 k
l 2 lim X ðnk þ2Þ ¼ X ; k!1 k l l ðnk þlÞ lim X ¼ ð Þ X ; k!1 k
ð3:23Þ
Here, we write X ðn1 Þ X ðn1 þ1Þ X ðn1 þlÞ
X ðn2 Þ X ðn2 þ1Þ X
ðn2 þlÞ
X ðnk Þ X ðnk þ1Þ
... ...
X ðnk þlÞ
...
ð3:24Þ
Thus, from (3.24), there exists an integer p0 such that kX ðnp0 Þ X k1 < 1; and there exists integers pi with pi > pi1 , for i = 1, 2, . . ., such that ðn þiÞ l i 1 X pi X < 2i : k 1 Then, kX
ðnpi þiÞ
l i 1 k1 < X þ 2i : k 1
Hence, under condition (3.2), there exists subsequence fX ðnpl þlÞ g such that lim X ðnpl þlÞ ¼ 0;
l!1
ð3:25Þ
where p0 < p1 < < pl < . From (3.17) and (3.25), we have lim X npl þl ¼ 0;
l!1
which contradicts Corollary 2.2. Hence, if the method is asymptotically stable, then jr1 j < 1. Conversely, the conclusion can be easily derived from the result of Theorem 2.1. h
ð3:26Þ
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
795
Remark 3.1. The conjecture (L-stability be a necessary condition for the asymptotic stability of numerical methods) which is formulated in [2] has been settled by Theorem 3.1. 4. Numerical examples To illuminate the stability properties of the non-autonomous difference system (2.1), we consider the following linear pantograph equation: 1 0 ð4:1Þ x ðtÞ ¼ 3xðtÞ þ x t ; t > 0; 4 with the initial condition xð0Þ ¼ 1. Its discretization obtained by Runge–Kutta methods can be written as X nþ1 ¼ Gnþ1 X n ;
ð4:2Þ
where 1
Gnþ1 ¼
1 3hnþ1 bT ðI þ 3hnþ1 AÞ e 1
ðI þ 3hnþ1 AÞ e
hnþ1 bT ðI þ 3hnþ1 AÞ
1
1
hnþ1 ðI þ 3hnþ1 AÞ A
! :
Under variable stepsize (3.3), we can obtain limn!1 kGn k ¼ j1 bT A1 ej. Quoting the condensed representation (Butch-array) of Runge–Kutta methods (see also [5]). Example 4.1. The two-stage Radau IA method reads
ð4:3Þ
Example 4.2. The two-stage Radau IIA method reads
ð4:4Þ
Example 4.3. The two-stage Lobatto IIIC method reads
ð4:5Þ
Example 4.4. The two-stage Gauss-Legendre method reads
ð4:6Þ
796
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
Example 4.5. The three-stage Gauss-Legendre method reads
ð4:7Þ
Figs. 1–3 demonstrate the difference Eq. (4.2) is asymptotically stable, in which the Butch-arrays are given by three kinds of two-stage Runge–Kutta methods (4.3)–(4.5). However, Figs 4 and 5 show that the difference equation (4.2) is not asymptotically stable if the Butcharrays are chosen as Gauss-Legendre methods 4.6 and 4.7. In fact, it can be easily seen that the norm of 7
x 10 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
7
8
9
4
8
4
x 10
Fig. 1. Radau IA method.
7
x 10 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
Fig. 2. Radau IIA method.
7
x 10
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798 6
x 10 2 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
7
8
9
4
x 10
Fig. 3. Lobatto IIIc method. 7
x 10 0 –2 –4 –6 –8 –10
1
2
3
4
5
6
7
8
9
10
5
x 10 Fig. 4. Two-stage Gauss-Legendre method. 12
x 10 5 4 3 2 1 0 –1 –2 –3 –4
1
2
3
4
5
6
7
8
9
10
Fig. 5. Three-stage Gauss-Legendre method.
4
x 10
797
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Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
the limit for the coefficient matrix in these five examples is 12, 14, 0, 1, 1, respectively. Therefore, these five figures coincide with Theorem 2.1. References [1] R.P. Agarwal, P.J.Y. Wong, Advanced topics in difference equations, Mathematics and its Applications, vol. 404, Kluwer, Dordrecht, 1997. [2] A. Bellen, N. Guglielmi, L. Torelli, Asymptotic stability properties of h-methods for the pantograph equation, Appl. Numer. Math. 24 (1997) 279–293. [3] M.D. Buhmann, A. Iserles, Stability of the discretized pantograph differential equation, Math. Comput. 60 (1993) 575–589. [4] K.L. Cooke, A.F. Ivanov, On the discretization of a delay differential equation, J. Differ. Equat. Appl. 6 (2000) 105–119. [5] K. Dekker, J.G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984. [6] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. [7] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [8] A. Iserles, On the generalized pantograph functional–differential equation, Eur. J. Appl. Math. 4 (1993) 1–38. [9] A. Iserles, The asymptotic behaviour of certain difference equations with proportional delay, Comp. Math. Appl. 28 (1994) 141–152. [10] A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math. 24 (1997) 295–308. [11] Y. Kuang, A. Feldstein, Monotonic and oscillatory solutions of a linear neutral delay equation with infinite lag, SIAM J. Math. Anal. 21 (1990) 1633–1641. [12] S.A. Kuruklis, The asymptotic stability of xnþ1 axn þ bxnk ¼ 0, J. Math. Anal. Appl. 188 (1994) 719–731. [13] Y.K. Liu, On the h-method for delay differential equations with infinite lag, J. Comput. Appl. Math. 71 (1996) 177–190. [14] Y.K. Liu, Asymptotic behaviour of functional–differential equations with proportional time delays, Eur. J. Appl. Math. 7 (1996) 11–30. [15] Y.K. Liu, Numerical investigation of the pantograph equation, Appl. Numer. Math. 24 (1997) 309–317. [16] E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659. [17] Y. Xu, M.Z. Liu, H-stability of Runge–Kutta methods with general variable stepsize for pantograph equation, Appl. Math. Comp. 148 (2004) 881–892. [18] J.S. Yu, Asymptotic stability for a linear difference equation with variable delay, Comp. Math. Appl. 36 (1998) 203–210. [19] M. Zennaro, P-stability properties of Runge–Kutta methods for delay differential equations, Numer. Math. 49 (1986) 305–318.
Applied Mathematics and Computation 198 (2008) 787–798 www.elsevier.com/locate/amc
Asymptotic stability of linear non-autonomous difference equations with fixed delay q Y. Xu *, J.J. Zhao Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Abstract This paper studies the asymptotic stability of the linear non-autonomous difference equations with fixed delay. It is shown that the difference system is asymptotically stable if and only if the norm of the limit for the coefficient matrix is less than 1 under some conditions. Furthermore, the discretization of the pantograph equation by Runge–Kutta methods with variable stepsize is analyzed. Finally, some numerical experiments are made to demonstrate the main conclusions. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Difference equation; Delay differential equation; Stability; Numerical methods
1. Introduction Consider the difference equation: Dxn ¼ f ðn; xn ; xn1 ; . . . ; xnr Þ;
n 2 N;
ð1:1Þ
where Dxn ¼ xnþ1 xn and f : N Crþ1 ! C. It was studied as a class of generalized difference equation in Section 2 of [1]. The initial vector for this system requires the knowledge of initial data fxr ; xrþ1 ; . . . ; x0 g, which were called the initial string in [4]. The difference equation (1.1) can be thought of as a discrete analogue of the delay differential equation: x0 ðtÞ ¼ f ðtÞ; xðtÞ; xðaðtÞÞ; t > 0; xðtÞ ¼ x0 ðtÞ; t 2 inf faðsÞg; 0 ;
ð1:2Þ
sP0
q This paper is supported by the Natural Science Foundation of Heilongjiang Province (A200602), the project (HITC200710) supported by Science Research Foundation in Harbin Institute of Technology and the Project (HITQNJS.2006.053) supported by Development Program for Outstanding Young Teachers in Harbin Institute of Technology. * Corresponding author. E-mail address: [email protected] (Y. Xu).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.012
788
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
where f ; a and x0 are given functions with aðtÞ 6 t for all t P 0. In fact, this system can be found in a variety of scientific and engineering fields such as biology, economy, computer aided design, nonlinear dynamical systems and so on, which have a comprehensive list in [8]. Some papers dealt with the asymptotic stability of linear autonomous difference equation: xnþ1 ¼ P n xn ;
ð1:3Þ
where P n P is a constant matrix independent of n. The theory employed for the asymptotic stability of (1.3) is that the zero solution of this difference equation is asymptotically stable if and only if all the roots of its characteristic equation are inside the unit disk (see also [7]). Especially, the asymptotic stability has been studied of the zero solution for the difference equation, xnþ1 ¼ xn þ bxnk ;
n 2 N:
ð1:4Þ
A sufficient and necessary condition for its asymptotic stability was given in [12]. As far as we know, there are two special cases of difference equation (1.1) which are studied these years. One is the autonomous difference equation with unbounded delay (see also [9,10]). The other is the non-autonomous difference equation with bounded delay, i.e., the difference system (1.3) with the coefficient sequence fP n g depending on n. Iserles focused on the autonomous difference equation with unbounded delay: x2nþ1 ¼ kx2n þ 2lxn ; x2nþ2 ¼ kx2nþ1 þ lðxn þ xnþ1 Þ; where k; l 2 C. In [9,10], the asymptotic stability of this convenient model was analyzed by the Fourier transform. Yet, it is usually difficult to investigate the long time dynamical behavior of its solution due to the limited computer memory as shown in [13,15]. Yu [18] considered the following linear difference equation: xnþ1 ¼ xn þ bn xnkn ;
n 2 N;
ð1:5Þ
where fbn g is a sequence of non-negative real numbers, fk n g is a sequence of non-negative integers, and there exists a non-negative integer k such that k n 6 k for n 2 N. The sufficient conditions for the uniform stability and the global asymptotic stability are obtained. However, the method (see, for instance, the proof of Theorem 13 in [19]), used in the asymptotic stability analysis of the autonomous difference equation (1.3), does not apply for the above two kinds of difference equations. Therefore, the results about the stability of these two kinds of difference equations are very scarce. As it is pointed out in [16], although there are some theoretical study about the stability of delay differential equation (1.2) (see also [3,8,11,14]) and many numerical schemes to approximate this equation, the asymptotic behavior of these two types of systems (the continuous system and its discrete version) does not often coincide. Therefore, we pay attention to the criterion, on which the difference equation preserves the asymptotic stability (delay-independent type) of the delay differential equation. In this paper, we focus our attention on the asymptotic stability of a special linear non-autonomous difference equations with fixed delay. A sufficient and necessary condition such that the difference system is asymptotically stable is given. Applying this conclusion, the discretization of the pantograph equation is analyzed by using the numerical methods with variable stepsize. Using the above stable results of difference system, it is shown that the Runge–Kutta methods are asymptotically stable if and only if the modulus of stability function at infinity is less than 1, in which the proof is different from other papers (see also [2,17]). Finally, some numerical examples are presented to illuminate the main conclusions. 2. Asymptotic stability of difference equation In this paper, we consider the linear non-autonomous difference equation: y nþ1 ¼ Gnþ1 y n ; where the coefficient fGnþ1 g is a sequence of complex matrices, which depend on n.
ð2:1Þ
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
789
For analyzing the property of this non-autonomous difference system, we recall the following definition. Definition 2.1. The difference equation (2.1) is called asymptotically stable, if the solution yn satisfies lim y n ¼ 0:
n!1
In order Ql to study the asymptotic stability of linear difference equation (2.1), we just need to estimate the norm of i¼0 G2N þi for some N 2 Z þ . Let ð2:2Þ
G ¼ lim Gn ; n!1
after simple calculations, we can obtain the norm of this matrix, which is independent of n. Here, for simplicity, we make the following assumptions. Assumption I kGk k ¼ kGkk ¼ nk ;
ð2:3Þ
with n = kGk. Assumption II kGnþN Gk 6 gn
for all n 2 Z þ ;
ð2:4Þ
where g is a constant and N is chosen such that gN < ln 2: 1g
ð2:5Þ
To facilitate the presentation of our results, for k 2 Z þ , we denote F k ¼ Gk G;
ð2:6Þ k
B2N ¼ F 2N ;
B2N þk ¼ GB2N þk1 þ F 2N þk ðG þ B2N þk1 Þ;
ð2:7Þ
then l Y
G2N þi ¼ Glþ1 þ B2N þl :
ð2:8Þ
i¼0
Lemma 2.1. Under conditions (2.3) and (2.4), for all l 2 Z þ , we have kB2N þl k 6
l Y
ðn þ gN þi Þ nlþ1 :
i¼0
Proof. From conditions (2.4) and (2.7), we can obtain kB2N k ¼ kG2N Gk 6 gN ; kB2N þ1 k ¼ kG B2N þ F 2N þ1 ðG þ B2N Þk 6 n gN þ gN þ1 ðn þ gN Þ ¼ ðn þ gN Þ ðn þ gN þ1 Þ n2 : Assume that kB2N þk1 k 6
k 1 Y ðn þ gN þi Þ nk ; i¼0
then under condition (2.3), we have k 1 Y ðn þ gN þi Þ nk kB2N þk k ¼ kGB2N þk1 þ F 2N þk ðG þ B2N þk1 Þk 6 n k
i¼0 k Y ¼ ðn þ gN þi Þ nkþ1 : i¼0
! þg
N þk
k 1 Y n þ ðn þ gN þi Þ nk k
i¼0
!
790
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
Lemma 2.2. Under conditions (2.3)–(2.5), if n P 1, then for all l 2 Z þ , we have Y l gN G2N þi P nlþ1 2 e1g > 0: i¼0 Proof. According to Lemma 2.1, from (2.3) and (2.8), we can obtain Y l G2N þi ¼ kGlþ1 þ B2N þl k P kGlþ1 k kB2N þl k P nlþ1 i¼0 N lþ1 Þ 1g ð1g lþ1 1g n 2e Pn :
l Y
! ðn þ gN þi Þ nlþ1
i¼0
Under condition (2.5), we have gN
e1g < 2: Hence Y l gN G2N þi P nlþ1 2 e1g > 0: i¼0
Theorem 2.1. Under conditions (2.3)–(2.5), the linear non-autonomous difference equation (2.1) is asymptotically stable if and only if n < 1:
ð2:9Þ
Proof. Assume that condition (2.9) is satisfied, since n = kGk, then we have kGk < 1. In view of (2.2), for any fixed e with 1 kGk > e > 0, there exists N > 0 such that for n P N , kGn Gk < 1 kGk e; and ky nþ1 k 6 ðkGk þ kGn GkÞ ky n k < ð1 eÞky n k; which implies limn!1 y n ¼ 0. Conversely, assume that n P 1, it is easy to see from Lemma 2.2 and Theorem 4.10 in [6] that any subsequence of fy n g can not tend to zero. If n > 1, then according to Lemma 2.2, we have Y l lim G2N þi ¼ 1; l!1 i¼0 which implies that fy n g is unbounded. If n = 1, then according to Lemma 2.2, from (2.3) and (2.8), we can obtain Y l l Y ky 2N þl k ¼ G2N þi y 2N 1 6 kGlþ1 þ B2N þl k ky 2N 1 k 6 ð1 þ gN þi Þ ky 2N 1 k 6 2 ky 2N 1 k; i¼0 i¼0 which implies that fy n g is bounded. h Corollary 2.1. Under conditions (2.3)–(2.5), if n > 1, then any subsequence of solution fy n g for the linear nonautonomous difference (2.1) is unbounded.
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
791
Corollary 2.2. Under conditions (2.3)–(2.5), if n ¼ 1, then any subsequence of solution fy n g for the linear nonautonomous difference (2.1) is bounded, but does not vanish. 3. Discretization of pantograph equation This section deals with the numerical discretization of linear pantograph equation: x0 ðtÞ ¼ kxðtÞ þ lxðqtÞ;
ð3:1Þ
t > 0;
where k; l 2 C, q 2 ð0; 1Þ with the initial condition such that xð0Þ ¼ x0 2 C. We are interested in studying if the delay-independent condition (see also [14]) jlj < jkj;
Rk < 0;
ð3:2Þ
under which the solution xðtÞ of pantograph equation (3.1) tends to zero (algebraically) as t approaches infinity, is preserved when a numerical scheme with variable stepsize is used to approximate its solutions. Here, the following grid points are proposed to perform this approximation, tn ¼ q1 tn1 for n = 1, 2, 3, . . ., which appears to be the special case of variable stepsize introduced in [17]. The advantage of this kind of stepsize is that it could be used to obtain the approximate of any grid point which we are interested. It is easy to see that for any n, we have qhn ¼ hn1
and
lim hn ¼ 1;
ð3:3Þ
n!1
and the stepsizes increase geometrically. Then we formulate the condition of numerical asymptotic stability. The condition will depend on parameter k and l, but turns out to be independent of q (see also Section 3 in [2]). Definition 3.1. The numerical method applied to Eq. (3.1) with condition (3.2) is called asymptotically stable, if the numerical solution xn satisfies lim xn ¼ 0
n!1
for any q 2 ð0; 1Þ and H ¼ ft0 ; t1 ; . . . ; tn ; . . .g. For the general pantograph equation (3.1), the Runge–Kutta method gives out the recurrence relation: 8 s P ðnþ1Þ > x ¼ x þ h bi ½kxi þ lxh ðtn þ ci hnþ1 Þ; > nþ1 n nþ1 < i¼1
s P > ðnþ1Þ ðnþ1Þ > ¼ xn þ hnþ1 aij ½kxj þ lxh ðtn þ cj hnþ1 Þ; : xi j¼1
T
T
where hnþ1 ¼ tnþ1 tn , matrix A ¼ ðaij Þss is non-singular, vectors b ¼ ðb1 ; b2 ; . . . ; bs Þ , c ¼ ðc1 ; c2 ; . . . ; cs Þ ; ðnþ1Þ and xi , xh ðtn þ ci hnþ1 Þ can be interpreted as an approximation to xðtn þ ci hnþ1 Þ, xðqðtn þ ci hnþ1 ÞÞ for i ¼ 1; . . . ; s, respectively. Using the results (3.3) of variable stepsize, the above difference system can be rewritten as xnþ1 ¼ rnþ1 xn þ anþ1 X ðnÞ ; X
ðnþ1Þ
¼ bnþ1 xn þ cnþ1 X
where X ðnÞ ¼
ðnÞ ðnÞ T ðx1 ; x2 ; . . . ; xðnÞ s Þ
ðnÞ
ð3:4Þ ;
and
ð3:5Þ
792
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798 1
rnþ1 ¼ 1 þ knþ1 bT ðI knþ1 AÞ e; l 1 1 anþ1 ¼ lnþ1 bT ðI knþ1 AÞ ¼ bT ðk1 nþ1 I AÞ ; k 1 bnþ1 ¼ ðI knþ1 AÞ e; 1
cnþ1 ¼ lnþ1 ðI knþ1 AÞ A; T
with e ¼ ð1; 1; . . . ; 1Þ 2 Rs . From Eq. (3.4), for l 2 Z þ , we have, xnþl ¼ r1 xnþl1 þ ðrnþl r1 Þ xnþl1 þ anþl X ðnþl1Þ ¼ rl1 xn þ ðrnþl r1 Þ xnþl1 þ r1 ðrnþl1 r1 Þ xnþl2 þ þ rl1 1 ðrnþ1 r1 Þ xn þ anþl ðnÞ X ðnþl1Þ þ r1 anþl1 X ðnþl2Þ þ þ rl1 ; 1 anþ1 X
ð3:6Þ
where r1 ¼ lim rnþ1 ¼ 1 bT A1 e: n!1
T T
Let X n ¼ ðxn ; ðX ðnÞ Þ Þ , then the difference equations (3.4) and (3.5) can be written as X nþ1 ¼ Gnþ1 X n ;
ð3:7Þ
where Gnþ1 ¼
rnþ1
anþ1
bnþ1
cnþ1
:
Define ð3:8Þ
G ¼ lim Gn ; n!1
then G¼
r1
a
0
c
;
ð3:9Þ
with l a ¼ lim anþ1 ¼ bT A1 ; n!1 k
l c ¼ lim cnþ1 ¼ I s : n!1 k
Here, we choose j lk j with j lk j < kbT Ajr11kj T
ð3:10Þ
, then
1 þ1
1
kGk1 ¼ j1 b A ej ¼ jr1 j;
ð3:11Þ
through some calculations, for k 2 Z þ , we have k
kGk k1 ¼ jr1 j ;
ð3:12Þ
which means that the difference equation (3.7) satisfies Assumption I (2.3) with n ¼ jr1 j. Here, N is chosen such that 1 kk1 N A k1 ¼ krN k1 < 1;
ðkbT A1 k1 þ 1Þ qN < ln 2; 1q N
q where rN ¼ kð1qÞ A1 .
krN k1 s < 1; 1 krN k1
ð3:13Þ ð3:14Þ ð3:15Þ
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
793
Lemma 3.1. Under condition (3.2), for all n 2 Z þ , we have kGnþN Gk1 6 qn : Proof. From (3.13), for n 2 Z þ , we have 1 1 1 1 1 ðI k1 kðI knþN AÞ1 k1 ¼ kk1 nþN A nþN A Þ k1 6 kknþN A k1
1 1
1 kk1 N A k1
¼
krN k1 qn : 1 krN k1
Thus, from (3.2) and (3.14), we can obtain kGnþN Gk1 ¼ maxfjrnþN r1 j þ kbnþN k1 ; kanþN ak1 þ kcnþN ck1 g l ¼ max jbT A1 ðI knþN AÞ1 ej þ kðI knþN AÞ1 ek1 ; bT A1 ðI knþN AÞ1 k 1 l 1 þ ðI knþN AÞ k 1 krN k1 krN k1 T 1 T 1 n l n 6 max ðkb A k1 þ 1Þ q s; ðkb A k1 þ 1Þ q 6 qn : k 1 krN k1 1 krN k1
This conclusion proves that the difference equation (3.7) satisfies Assumption II (2.4) with g ¼ q. Although the following result has been already obtained in our next paper (see also Theorem 3.1 in [17]), here we include with another new proof in order to point out how the coefficients are involved in the asymptotic analysis. Theorem 3.1. Assume the matrix A is regular, then Runge–Kutta method ðA; b; cÞ, applied to Eq. (3.1) with condition (3.2), is asymptotically stable if and only if jr1 j < 1;
ð3:16Þ T
1
where r1 ¼ 1 b A e is the limit of stability function of Runge–Kutta method. Proof. Assume that jr1 j P 1, if the method is asymptotically stable, then lim xn ¼ 0:
ð3:17Þ
n!1
Case I. If jr1 j > 1, from (3.10), for any given e1 with 0 < e1 < jr12j1, there exists an integer N 0 such that, for all n P N 0 , we have jrn r1 j < e1 ;
kan ak1 < e1 :
ð3:18Þ
In view of (3.4) and (3.17), we know that lim anþ1 X ðnÞ ¼ 0:
ð3:19Þ
n!1
From (3.17) and (3.19), there exists n0 and k 0 such that, for all n P N 0 , jxn0 j P jxn j;
kak0 þ1 X ðk0 Þ k1 P kanþ1 X ðnÞ k1 :
Here, we can choose
j lk j
ð3:20Þ
so small that
e1 jxn0 j P kak0 þ1 X ðk0 Þ k1 :
ð3:21Þ þ
From (3.6), according to (3.18) and (3.21), for all l 2 Z , we arrive at jxn0 þl j P jrl1 xn0 j ½ðjrn0 þl r1 j jxn0 þl1 j þ þ jr1 j l1
þ jr1 j kan0 þl1 X ðn0 þl2Þ k1 þ þ jr1 j
l1
jrn0 þ1 r1 j jxn0 jÞ þ ðkan0 þl X ðn0 þl1Þ k1 l
kan0 þ1 X ðn0 Þ k1 Þ P jr1 j jxn0 j
½ð1 þ jr1 j þ þ jr1 jl1 Þe1 jxn0 j þ ð1 þ jr1 j þ þ jr1 jl1 Þe1 jxn0 j ¼
jr1 jl ðjr1 j 1 2e1 Þ þ 2e1 jxn0 j > 0: jr1 j 1
794
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
This implies that lim jxn0 þl j ¼ 1;
ð3:22Þ
l!1
which contradicts (3.17). Case II. If jr1 j ¼ 1, then in view of Corollary 2.2, we know fX n g is bounded, but does not vanish. Since fX n g is bounded, then there exist X and subsequence fX ðnk Þ g such that lim X ðnk Þ ¼ X :
k!1
From (3.5), we arrive at X ðnk þ1Þ ¼ bnk þ1 xnk þ cnk þ1 X ðnk Þ ; then, for l 2 Z þ , we have
l lim X ðnk þ1Þ ¼ X ; k!1 k
l 2 lim X ðnk þ2Þ ¼ X ; k!1 k l l ðnk þlÞ lim X ¼ ð Þ X ; k!1 k
ð3:23Þ
Here, we write X ðn1 Þ X ðn1 þ1Þ X ðn1 þlÞ
X ðn2 Þ X ðn2 þ1Þ X
ðn2 þlÞ
X ðnk Þ X ðnk þ1Þ
... ...
X ðnk þlÞ
...
ð3:24Þ
Thus, from (3.24), there exists an integer p0 such that kX ðnp0 Þ X k1 < 1; and there exists integers pi with pi > pi1 , for i = 1, 2, . . ., such that ðn þiÞ l i 1 X pi X < 2i : k 1 Then, kX
ðnpi þiÞ
l i 1 k1 < X þ 2i : k 1
Hence, under condition (3.2), there exists subsequence fX ðnpl þlÞ g such that lim X ðnpl þlÞ ¼ 0;
l!1
ð3:25Þ
where p0 < p1 < < pl < . From (3.17) and (3.25), we have lim X npl þl ¼ 0;
l!1
which contradicts Corollary 2.2. Hence, if the method is asymptotically stable, then jr1 j < 1. Conversely, the conclusion can be easily derived from the result of Theorem 2.1. h
ð3:26Þ
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
795
Remark 3.1. The conjecture (L-stability be a necessary condition for the asymptotic stability of numerical methods) which is formulated in [2] has been settled by Theorem 3.1. 4. Numerical examples To illuminate the stability properties of the non-autonomous difference system (2.1), we consider the following linear pantograph equation: 1 0 ð4:1Þ x ðtÞ ¼ 3xðtÞ þ x t ; t > 0; 4 with the initial condition xð0Þ ¼ 1. Its discretization obtained by Runge–Kutta methods can be written as X nþ1 ¼ Gnþ1 X n ;
ð4:2Þ
where 1
Gnþ1 ¼
1 3hnþ1 bT ðI þ 3hnþ1 AÞ e 1
ðI þ 3hnþ1 AÞ e
hnþ1 bT ðI þ 3hnþ1 AÞ
1
1
hnþ1 ðI þ 3hnþ1 AÞ A
! :
Under variable stepsize (3.3), we can obtain limn!1 kGn k ¼ j1 bT A1 ej. Quoting the condensed representation (Butch-array) of Runge–Kutta methods (see also [5]). Example 4.1. The two-stage Radau IA method reads
ð4:3Þ
Example 4.2. The two-stage Radau IIA method reads
ð4:4Þ
Example 4.3. The two-stage Lobatto IIIC method reads
ð4:5Þ
Example 4.4. The two-stage Gauss-Legendre method reads
ð4:6Þ
796
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
Example 4.5. The three-stage Gauss-Legendre method reads
ð4:7Þ
Figs. 1–3 demonstrate the difference Eq. (4.2) is asymptotically stable, in which the Butch-arrays are given by three kinds of two-stage Runge–Kutta methods (4.3)–(4.5). However, Figs 4 and 5 show that the difference equation (4.2) is not asymptotically stable if the Butcharrays are chosen as Gauss-Legendre methods 4.6 and 4.7. In fact, it can be easily seen that the norm of 7
x 10 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
7
8
9
4
8
4
x 10
Fig. 1. Radau IA method.
7
x 10 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
Fig. 2. Radau IIA method.
7
x 10
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798 6
x 10 2 1.5 1 0.5 0 –0.5 –1 –1.5
1
2
3
4
5
6
7
8
9
4
x 10
Fig. 3. Lobatto IIIc method. 7
x 10 0 –2 –4 –6 –8 –10
1
2
3
4
5
6
7
8
9
10
5
x 10 Fig. 4. Two-stage Gauss-Legendre method. 12
x 10 5 4 3 2 1 0 –1 –2 –3 –4
1
2
3
4
5
6
7
8
9
10
Fig. 5. Three-stage Gauss-Legendre method.
4
x 10
797
798
Y. Xu, J.J. Zhao / Applied Mathematics and Computation 198 (2008) 787–798
the limit for the coefficient matrix in these five examples is 12, 14, 0, 1, 1, respectively. Therefore, these five figures coincide with Theorem 2.1. References [1] R.P. Agarwal, P.J.Y. Wong, Advanced topics in difference equations, Mathematics and its Applications, vol. 404, Kluwer, Dordrecht, 1997. [2] A. Bellen, N. Guglielmi, L. Torelli, Asymptotic stability properties of h-methods for the pantograph equation, Appl. Numer. Math. 24 (1997) 279–293. [3] M.D. Buhmann, A. Iserles, Stability of the discretized pantograph differential equation, Math. Comput. 60 (1993) 575–589. [4] K.L. Cooke, A.F. Ivanov, On the discretization of a delay differential equation, J. Differ. Equat. Appl. 6 (2000) 105–119. [5] K. Dekker, J.G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984. [6] S.N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. [7] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [8] A. Iserles, On the generalized pantograph functional–differential equation, Eur. J. Appl. Math. 4 (1993) 1–38. [9] A. Iserles, The asymptotic behaviour of certain difference equations with proportional delay, Comp. Math. Appl. 28 (1994) 141–152. [10] A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math. 24 (1997) 295–308. [11] Y. Kuang, A. Feldstein, Monotonic and oscillatory solutions of a linear neutral delay equation with infinite lag, SIAM J. Math. Anal. 21 (1990) 1633–1641. [12] S.A. Kuruklis, The asymptotic stability of xnþ1 axn þ bxnk ¼ 0, J. Math. Anal. Appl. 188 (1994) 719–731. [13] Y.K. Liu, On the h-method for delay differential equations with infinite lag, J. Comput. Appl. Math. 71 (1996) 177–190. [14] Y.K. Liu, Asymptotic behaviour of functional–differential equations with proportional time delays, Eur. J. Appl. Math. 7 (1996) 11–30. [15] Y.K. Liu, Numerical investigation of the pantograph equation, Appl. Numer. Math. 24 (1997) 309–317. [16] E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659. [17] Y. Xu, M.Z. Liu, H-stability of Runge–Kutta methods with general variable stepsize for pantograph equation, Appl. Math. Comp. 148 (2004) 881–892. [18] J.S. Yu, Asymptotic stability for a linear difference equation with variable delay, Comp. Math. Appl. 36 (1998) 203–210. [19] M. Zennaro, P-stability properties of Runge–Kutta methods for delay differential equations, Numer. Math. 49 (1986) 305–318.