asymptotical stability of numerical methods with ...

BIT Numerical Mathematics (2005) 0: 1-17 DOI: 10.1007/s10543-005-0022-3

° c Springer 2005

ASYMPTOTICAL STABILITY OF NUMERICAL METHODS WITH CONSTANT STEPSIZE FOR PANTOGRAPH EQUATIONS∗ M. Z. LIU1 , Z.W. YANG1 and G.D. HU2 † 1

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China. email: [email protected] 2 Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China. email: [email protected]

Abstract. In this paper, the asymptotical stability of the analytic solution and the numerical methods with the constant stepsize for the pantograph equations is investigated by using Razumikhin technique. Especially, the linear pantograph equations with constant coefficients and variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the θ-methods with the constant stepsize are obtained. As a result Z. Jackiewicz’s conjecture is partially proved. Finally, some experiments are given. AMS subject classification: 65L02, 65L05, 65L20 Key words: Pantograph equation, Razumikhin type theorem, asymptotical stability, numerical methods.

1

Introduction

In this paper we consider the pantograph equation ( x0 (t) = f (t, x(t), x(qt)), t > 0, (1.1) x(0) = x0 , where 0 < q < 1, x0 ∈ Rν and f : R+ × Rν × Rν → Rν is continuous. These systems can be found in modelling many phenomena like, for example, in the electrodynamics [10, 28] and in nonlinear dynamical systems [9] and [13]. In the recent years, numerous authors have considered ( x0 (t) = ax(t) + bx(qt), t > 0, (1.2) x(0) = x0 , where a, b, x0 are real constants and q ∈ (0, 1). We refer to [2, 7, 12, 13, 15, 16, 19, 21, 26] for an analytical study and to the papers [1, 3, 6, 14, 20, 22, 23, 24, ∗ Received † This

April 2004. Revised accepted July 2005. Communicated by Timo Eirola. paper is supported by the National Natural Science Foundation of China(10271036).

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M. Z. LIU ET AL.

25, 26, 27, 29, 31] for numerical investigations. The most difficult problem in the numerical computation over large intervals is the limited computer memory as shown in [26, 27]. There are two ways to avoid the storage problem. One way is transforming (1.2) to an equation with constant time lag and variable coefficients as [20, 25], ( x0 (t) = aet x(t) + bet x(t + log q), t > t0 , (1.3) x(t) = x0 (t), t0 + log q 6 t 6 t0 , where x0 (t) is a continuous function in [t0 + log q, t0 ] and applying the numerical methods with the constant stepsize. Another way is applying a numerical method to (1.2) with the variable stepsize. The resulting mesh is called a quasigeometric mesh [1, 11]. The other reason of applying a numerical method with the variable stepsize is that when using the numerical method with the constant stepsize to (1.2), the resulting difference equation is not of fixed order as shown in [4, 5, 14, 17]. In the recent years, there are only a few authors who have considered the asymptotical stability of the numerical methods with the constant stepsize for the pantograph equations [6, 13, 14, 22, 24]. The stepsize selection strategies of the two ways are the special cases of that in the paper [31] with hn = q −n/m (q −1/m − 1) and hn+1 = (1/m)q −[n/m]−1 (1 − q), where [n/m] is the integer part of n/m. Therefore the essential ideas of the two ways to avoid the storage problem are using numerical methods with the suitable variable stepsize. We have to note that in order to preserve the accuracy at a large time t, the storage problem remains problematic, since the number of data in the set S(t) = {xn : n ∈ Z+ such that tn ∈ [qt, t]}, which have to store in the computer, tends to infinity as t → ∞. The asymptotical stability conditions for the analytic solution and the numerical solution of the pantograph equation (1.1) are provided in the book [2]. It shows that for the variable coefficient pantograph equation (1.4)

x0 (t) = a(t)x(t) + b(t)x(qt),

t > 0,

where a(t), b(t) are real-valued continuous functions and q ∈ (0, 1), the sufficient conditions for the asymptotical stability are (i) there exists β > 0 such that a(t) 6 −β for all t > 0, (ii) there exists α ∈ (0, 1) such that

|b(t)| 6 α < 1 for all t > 0. |a(t)|

And the necessary and sufficient condition for the asymptotical stability of the 1 one-leg θ-method with a quasi-geometric mesh is < θ 6 1 under conditions (i) 2 and (ii). In [18], the author considers the linear θ-methods with the constant stepsize for the test equation of the pantograph equation (1.5)

x0 (t) = ax(λt) + bx(t),

t > 0,

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

3

where λ is an arbitrary constant satisfying the condition 0 < λ < 1. He conjectures that Sθ ∩ R = {(x, y) : |y| < −x and |y| < 2 + (1 − 2θ)x}, θ ∈ [0, 1], where R := {(x, y) : |y| < −x} and Sθ is the set of all points (hb, ha) for which the approximate solution xn → 0 as n → ∞. In the present paper we investigate the asymptotical stability of both the nonlinear pantograph equations (1.1) and the numerical methods applying to (1.4) by using Razumikhin techniques similar to those in [8, 12, 21]. As a result we obtain the stability conditions for (1.4), which turn out to be the same as (i) and (ii), and the numerical stability conditions for the linear θ-method and the one-leg θ-method with the constant stepsize, and give a positive answer to the conjecture in [18]. 2

Razumikhin type theorem of the analytic systems

In this section we assume that f (t, x, y) satisfies f (t, 0, 0) ≡ 0 for t ∈ R+ . And we also assume that (1.1) has on the interval [0, ∞) a unique continuous solution x(t) = x(t, x0 ) for any x0 ∈ Rν . Therefore x(t) ≡ 0 is the unique solution of (1.1) with x0 = 0, which is called the trivial solution of (1.1). Definition 2.1. [8, 12, 21] The trivial solution of (1.1) is said to be (i) stable, if for any given ² > 0, there exists a number η = η(²) > 0 such that if |x0 | 6 η, then |x(t, x0 )| 6 ² for all t > 0; (ii) asymptotically stable, if it is stable and there exists a η > 0 such that for any given γ > 0, there exists a number T = T (η, γ) > 0 such that if |x0 | 6 η, then |x(t, x0 )| 6 γ for all t > T ; (iii) globally asymptotically stable, if it is asymptotically stable and η = ∞; (iv) unstable, if stable fails to hold; where | · | is a norm in Rν . Definition 2.2. Given a continuous function V : R+ × Rν → R+ , the derivative of V along the solution of (1.1) is defined by (2.1)

1 V 0 (t, x(t)) = lim sup [V (t + h, x(t + h)) − V (t, x(t))], h h→0+

for (t, x) ∈ R+ × Rν . It is easy to see if V (t, x) has continuous partial derivatives with respect to t and x, then (2.1) can be represented by (2.2)

V 0 (t, x(t)) =

∂V (t, x(t)) ∂V (t, x(t)) + f (t, x(t), x(qt)). ∂t ∂x

Theorem 2.1. Suppose f : R+ ×Rν ×Rν → Rν is continuous, u, v : R+ → R+ are continuous, strictly increasing functions satisfying u(0) = v(0) = 0 and

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M. Z. LIU ET AL.

w : R+ → R+ is a continuous, nondecreasing function with w(0) = 0. Assume that there is a continuous function V : R+ × Rν → R+ such that u(|x|) 6 V (t, x) 6 v(|x|),

t ∈ R+ , x ∈ Rν .

The following statements are true: (i) The trivial solution of (1.1) is stable if (2.3)

V 0 (t, x(t)) 6 −w(|x(t)|), for V (qt, x(qt)) 6 V (t, x(t)).

(ii) The trivial solution of (1.1) is asymptotically stable if w(s) > 0 for s > 0 and there is a continuous nondecreasing function p(s) > s for s > 0 such that (2.4)

V 0 (t, x(t)) 6 −w(|x(t)|), for V (qt, x(qt)) < p(V (t, x(t))).

Proof. (i) For any given ² > 0, we choose η = η(²) > 0 such that v(2η) 6 u(²). Let |x0 | 6 η, x(t) = x(t, x0 ) be the solution of (1.1). Then from the continuity of the solution x(t), there is a T0 > 0 such that for 0 6 t 6 T0 , |x(t)| 6 2η, which implies V (t, x(t)) 6 v(2η) for 0 6 t 6 T0 . Suppose there exists a t∗ > T0 such that V (t∗ , x(t∗ )) > v(2η). Let t = max{t : V (s, x(s)) 6 v(2η), s ∈ [0, t]}. Then V (t, x(t)) = v(2η) > V (qt, x(qt)) and V 0 (t, x(t)) > 0, which contradicts (2.3) and we have u(|x(t)|) 6 V (t, x(t)) 6 v(2η) 6 u(²) for all t > 0, which implies |x(t)| 6 ² for all t > 0. (ii) Let H > 0 and η0 > 0 satisfy v(2η0 ) = u(H). In view of (i), |x0 | 6 η0 implies that for all t > 0, |x(t)| 6 H and V (t, x(t)) 6 v(2η0 ) = u(H). For any given 0 < γ < H, we will prove that there is a T = T (γ, η0 ) > 0 such that |x0 | 6 η0 implies |x(t)| 6 γ for t > T . This is true if we can show that |x0 | 6 η0 implies that V (t, x(t)) 6 u(γ) for all t > T . Let d > 0 be such that p(s) − s > d for u(γ) 6 s 6 v(2η0 ), N = N (γ) be the smallest nonnegative integer such that v(2η0 ) 6 u(γ) + N d, and σ = inf{w(s) : v(2η0 ) v(2η0 ) v −1 (u(γ)) 6 s 6 H}. We denote ti = , si = and 2i σ(1 − q)q σ(1 − q)q 2i+1 Ii = [si , ti+1 ]. We will prove that for i 6 N (2.5)

V (t, x(t)) 6 u(γ) + (N − i)d, for t > ti .

Clearly, (2.5) holds for i = 0. Suppose (2.5) holds for 0 6 i < N , then we claim that there is a t∗ ∈ Ii such that (2.6)

V (t∗ , x(t∗ )) 6 u(γ) + (N − i − 1)d.

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

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Otherwise we have for all t ∈ Ii qt > ti , V (t, x(t)) > u(γ) + (N − i − 1)d, hence p(V (t, x(t))) > V (t, x(t)) + d > u(γ) + (N − i)d > V (qt, x(qt)). Thus

V 0 (t, x(t)) 6 −w(|x(t)|) 6 −σ,

and consequently V (t, x(t)) 6 V (si , x(si )) − σ(t − si ) 6 v(2η0 ) − σ(t − si ). Hence

· u(|x(ti+1 )|) 6 V (ti+1 , x(ti+1 )) 6 v(2η0 ) 1 −

1

¸

q 2i+2

< 0,

which is a contradiction to the fact that u(s) > 0 for s > 0. Therefore (2.6) holds, which together with (2.4) implies V (t, x(t)) 6 u(γ) + (N − i − 1)d for t > ti+1 . Let T =

v(2η0 ) , we can conclude by induction σ(1 − q)q 2N u(|x(t)|) 6 V (t, x(t)) 6 u(γ), for t > T.

The proof is completed. Remark 2.1. If u(s) → ∞, as s → ∞, then one can determine H such that u(H) = v(2η) for any given η > 0. The proof of Theorem 2.1 show that the asymptotical stability of x = 0 for any given initial value x0 , i.e., the trivial solution of (1.1) is globally asymptotically stable. Now we consider the equation ( x0 (t) = a(t)x(t) + b(t)x(qt), t > 0, (2.7) x(0) = x0 , where x0 ∈ R, a(t) and b(t) are continuous real functions and 0 < q < 1. Theorem 2.2. Suppose 0 6 α < 1 and β > 0 such that (2.8)

a(t) 6 −β < 0, |b(t)| 6 −αa(t).

Then the trivial solution of (2.7) is globally asymptotically stable.

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1 1 Proof. In fact, let α < µ < 1 and define V (t, x) = x2 , u(s) = v(s) = s2 , 2 2 β 1 w(s) = (µ − α)s2 and p(s) = 2 s > s. Then from V (qt, x(qt)) < p(V (t, x(t))), µ µ we can obtain 1 |x(qt)| < |x(t)|, µ which together with µa(t) + |b(t)| 6 µa(t) − αa(t) = (µ − α)a(t) 6 −β(µ − α) implies V 0 (t, x(t)) = x(t)x0 (t) = a(t)x2 (t) + b(t)x(t)x(qt) |b(t)| 2 )x (t) 6 −w(|x(t)|). µ From Theorem 2.1 and Remark 2.1 we can conclude the solution of (2.7) is globally asymptotically stable. Remark 2.2. By the similar arguments as the proof of Theorem 2.2, we can obtain the globally asymptotical stability of the following equations when (2.8) holds, ( x0 (t) = a(t)xp (t) + b(t)xp (qt), t > 0, (2.9) x(0) = x0 , < (a(t) +

where x0 ∈ R, a(t) and b(t) are continuous real functions, 0 < q < 1 and p is an odd. Corollary 2.3. Suppose |b| < −a, then the trivial solution of (1.2) is globally asymptotically stable. Remark 2.3. Because of the limitation of Theorem 2.1, we can only get the sufficient condition for the global asymptotical stability. In fact, the trivial solution of (1.2) is globally asymptotically stable if and only if |b| < −a ( see [2, 13, 16, 19, 26]). 3

Razumikhin type theorem of the discrete systems

In this section we will consider the discrete systems with the form ( x(n + 1) = f (n, x(n), x(qc1 (n)), · · · , x(qcr (n))), n > 0, (3.1) x(0) = x0 , where f : Z+ × Rν × Rν × · · · × Rν → Rν is continuous, x0 ∈ Rν , 0 < q < 1, qn 6 qci (n) 6 n + 1, i = 1, 2, · · · , r and x(m + δ) = (1 − δ)x(m) + δx(m + 1), where 0 < δ < 1 and m is an integer. We assume that Jn = {[qn], [qn] + 1, · · · , n, n + 1}, f (n, 0, 0, · · · , 0) ≡ 0 and (3.1) has a unique solution for x0 ∈ Rν denoted by x(n) = x(n, x0 ). The solution x(n) ≡ 0 is the trivial solution of (3.1). Similar to Definition 2.1, we can define the stability, asymptotical stability of (3.1), respectively. The following lemma is useful to prove Theorem 3.2. Lemma 3.1. Assume that α > β > 0 and 0 < q < 1. Then

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

7

(i) [α − β] 6 [α] − [β]. (ii) [q([α] + 1)] > [qα]. Proof. (i) Since [α] 6 α for any α > 0, we have [α − β] + [β] 6 α − β + β = α, which implies [α − β] + [β] 6 [α], i.e., [α − β] 6 [α] − [β]. (ii) The statement is true from q[α] + q > qα − q + q = qα. Theorem 3.2. Suppose f : Z+ ×Rν ×Rν ×· · ·×Rν → Rν and u, v : R+ → R+ are continuous, strictly increasing functions satisfying u(0) = v(0) = 0 and w : R+ → R+ is continuous, nondecreasing function with w(0) = 0 and w(s) > 0 for s > 0. Assume that there is a continuous function V : R+ × Rν → R+ such that u(|x|) 6 V (t, x) 6 v(|x|), t ∈ R+ , x ∈ Rν . Then the trivial solution of (3.1) is asymptotically stable if there is a continuous nondecreasing function p(s) > s for s > 0 such that

(3.2)

∆V (n) = ∆V (n, x(·)) = V (n + 1, x(n + 1)) − V (n, x(n)) = V (n + 1, f (n, x(n), x(qc1 (n)), · · · , x(qcr (n))) − V (n, x(n)) 6 −w(|x(n)|), for V (m, x(m)) < p(V (n + 1, x(n + 1))), m ∈ Jn .

Proof. (i) For any given ² > 0, we choose η such that v(η) 6 u(²). Let |x0 | 6 η and x(n) = x(n, x0 ) be the solution of (3.1). Then V (0, x(0)) 6 v(η). Suppose there exists an n∗ > 1 such that V (n∗ , x(n∗ )) > v(η) then there is an n 6 n∗ satisfying V (m, x(m)) 6 v(η) for 0 6 m 6 n, V (n + 1, x(n + 1)) > v(η), therefore ∆V (n) > 0. On the other hand, V (m, x(m)) 6 v(η) < V (n + 1, x(n + 1)) < p(V (n + 1), x(n + 1)) for m ∈ Jn , which implies by (3.2) that ∆V (n) 6 −w(|x(n+1)|) 6 0. This is a contradiction. Therefore u(|x(n)|) 6 V (n, x(n)) 6 v(η) 6 u(²) for all n, which implies |x(n)| 6 ² for all n. (ii) Let H > 0 and η0 > 0 satisfy v(η0 ) = u(H). By (i) we have |x0 | 6 η0 implies that |x(n)| 6 H and V (n, x(n)) 6 v(η0 ) for all n ∈ Z+ . For any given 0 < γ < H, we will prove that there is an N ∗ = N ∗ (γ, η0 ) ∈ Z+ such that |x(n)| 6 γ for n > N ∗ if |x0 | 6 η0 . This is true if we can show that |x0 | 6 η0 implies that V (n, x(n)) 6 u(γ) for n > N ∗ .

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M. Z. LIU ET AL.

Let d =

inf

u(²)6s6v(η0 )

(p(s) − s), N = N (γ) be the smallest integer such that

v(η0 ) 6 u(γ) + N d, σ = inf{w(s) : v −1 (u(γ)) 6 s 6 · H} and K0 be ¸ an inv(η0 ) v(η0 ) K0 teger such that q < . We denote ni = , mi = v(η0 ) + 2σ σ(1 − q)q 2i+K0 · ¸ v(η0 ) and Ii = {mi + 1, · · · , ni+1 − 1}. We will prove that for σ(1 − q)q 2i+1+K0 i6N (3.3)

V (n, x(n)) 6 u(γ) + (N − i)d, for n > ni .

Clearly, (3.3) holds for i = 0. Suppose (3.3) holds for 0 6 i < N , then we claim that there is an n∗ ∈ Ii such that (3.4)

V (n∗ , x(n∗ )) 6 u(γ) + (N − i − 1)d.

Otherwise, for all n ∈ Ii , we have (3.5)

V (n, x(n)) > u(γ) + (N − i − 1)d.

It is known from Lemma 3.1 that if n ∈ Ii , m ∈ Jn , then m > ni , and by (3.3) (3.6)

V (m, x(m)) 6 u(γ) + (N − i)d.

In view of (3.5) and (3.6) p(V (n, x(n))) > V (n, x(n)) + d > u(γ) + (N − i)d > V (m, x(m)), From (3.2) we have for all mi + 1 6 n 6 ni+1 − 1, ∆V (n) 6 −w(|x(n)|) 6 −σ, hence V (n, x(n)) 6 V (mi + 1, x(mi + 1)) − σ(n − mi − 1) 6 v(η0 ) − σ(n − mi − 1), which by Lemma 3.1 implies that V (ni+1 , x(ni+1 )) < 0. This is a contradiction to the fact that u(s) > 0 for s > 0, therefore (3.4) holds, which together with (3.2) implies V (n, x(n)) 6 u(γ) + (N − i − 1)d for n > ni+1 . · ¸ v(η0 ) Let N ∗ = . Then we can conclude by induction for n > N ∗ , σ(1 − q)q 2N +K0 u(|x(n)|) 6 V (n, x(n)) 6 u(γ). Consequently |x(n)| 6 γ for all n > N ∗ . Remark 3.1. If u(s) → ∞, as s → ∞, then similarly, we can draw the conclusion that the trivial solution of (3.1) is globally asymptotically stable.

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

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In the rest of this section we investigate the following scalar system  r X   x(n + 1) = a(n)x(n) + bi (n)x(qci (n)), n > 0, (3.7) i=1   x(0) = x0 , where x0 ∈ R, 0 < q < 1, a(n), bi (n), i = 1, 2, · · · , r, are real functions and qn 6 qci (n) 6 n + 1, i = 1, 2, · · · , r. Corollary 3.3. Suppose there is an 0 6 η < 1 such that for all n > 0, (3.8)

|a(n)| +

r X

|bi (n)| 6 η,

i=1

then the trivial solution of the system (3.7) is globally asymptotically stable. Proof. Let η < µ < 1 and define V (t, x) = |x|, u(s) = v(s) = s, p(s) =

1 s, µ

and

µ−η s. µ Then from V (m, x(m)) < p(V (n + 1, x(n + 1))) for m ∈ Jn , we can obtain w(s) =

|x(m)| < in particular |x(qci (n))|
0. Similarly the one-leg θ-method (4.2) is called asymptotically stable if the above holds for application of (4.2) to (2.7). Theorem 4.1. Assume that there is A > 0 such that a(t) > −A, for all t > 0 and (2.8) holds. If 1+α 6 θ 6 1. 2

(4.7)

Then (i) the one-leg θ-method is asymptotically stable; (ii) the linear θ-method is asymptotically stable, if, in addition, a(t) is nonincreasing. Proof. (i) Let an = a(tn+θ ), bn = b(tn+θ ) and ¯ ¯ ¯ han ¯¯ |bn |h In = ¯¯1 + + . 1 − θhan ¯ 1 − θhan Then it is easy to see that if 1 + (1 − θ)han > 0, then from (2.8) (4.8)

In 6 1 +

(1 − α)han (1 − α)hβ < 1. 61− 1 − θhan 1 + θhβ

Otherwise 1 + (1 − θ)han < 0, then from (2.8) and (4.7), we have (4.9)

In − 1 6

−2 + (2θ − 1 − α)han −2 − (2θ − 1 − α)hA 6 < 0. 1 − θhan 1 + θhA

From Corollary 3.3 we can conclude that for any given x0 and any h > 0, the solution of (4.6) tends to zero as n tends to infinity. (ii) Let an = a(tn ), bn = b(tn ) and ¯ ¯ ¯ 1 + (1 − θ)han ¯ ¯ ¯ + θ|bn+1 |h + (1 − θ)|bn |h . In = ¯ 1 − θhan+1 ¯ 1 − θhan+1 1 − θhan+1 Therefore, if 1 + (1 − θ)han > 0, then from an+1 6 an , 1 + (1 − θ)han − θαhan+1 − (1 − θ)αhan − θhan+1 + θhan+1 1 − θhan+1 (1 − θ)(1 − α)han + θ(1 − α)han+1 =1+ 1 − θhan+1 (1 − α)han (1 − α)hβ 61+ 61− < 1. 1 − θhan 1 + θhβ

In 6

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M. Z. LIU ET AL.

Otherwise, let 1 + (1 − θ)han < 0. Then from (4.7) and (4.9), we can obtain −1 − (1 − θ)(1 + α)han − θαan+1 h 1 − θhan+1 −2 + (2θ − 1 − α)han+1 61+ 1 − θhan+1 −2 − (2θ − 1 − α)hA 61+ < 1. 1 + θhA

In = (4.10)

From Corollary 3.3 we can conclude the solution of (4.4) tends to zero as n tends to infinity. Remark 4.1. From the proof of Theorem 4.1 we can see that if a(t) is unbounded and (2.8) holds, then the statements in Theorem 4.1 are also true for 1+α < θ 6 1. 2 From Remark 3.2 we can obtain the asymptotical stability of the numerical solutions of the θ-methods for (1.2). 1 |b| 1 6 θ 6 1, then the Corollary 4.2. Suppose a < 0, |b| < −a and − 2 2 a linear θ-method and the one-leg θ-method for (1.2) are asymptotically stable for all q ∈ (0, 1). 1 Remark 4.2. When q = with the integer L > 2, the authors in [6, 22, 24] L proved that the necessary and sufficient condition of the asymptotical stability of 1 the linear θ-methods with the constant stepsize is 6 θ 6 1, which is stronger 2 than that in Corollary 4.2. 4.3

The numerical stability region

In this subsection we will investigate the asymptotical stability region of the linear θ-method and the one-leg θ-method, (4.1) and (4.2), when applying to (1.2). Applying (4.1) to (1.2), denoting x(l) = xl , l = 0, 1, 2, · · · , we have the recurrence relation x(n + 1) = (4.11)

1 + (1 − θ)ha x(n) 1 − θha θbhx(q(n + 1)) + (1 − θ)bhx(qn) + , n = 0, 1, 2, · · · . 1 − θha

In the case of (4.2), we have (4.12) x(n + 1) =

1 + (1 − θ)ha bh x(n) + x(q(n + θ)), n = 0, 1, 2, · · · . 1 − θha 1 − θha

Definition 4.2. Method (4.1) is called asymptotically stable at (¯ a, ¯b) if and only if any application of (4.1) to (1.2) generates numerical solution xn such

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

13

that xn → 0 as n → ∞, whenever a, b, q and h satisfy 0 < q < 1, h > 0, a ¯ = ah and ¯b = bh. Definition 4.3. The asymptotical stability region of method (4.1) is the subset of R2 consisting of all pairs (¯ a, ¯b) at which method (4.1) is asymptotically stable, denoted by Sθ . Similarly we can define the asymptotical stability region of the one-leg θmethod (4.2), denoted by S¯θ . From Theorem 4.1 and Remark 3.2 we can prove the following theorems. Theorem 4.3. Assume that θ ∈ [0, 1] and a ¯, ¯b satisfy |¯b| < −¯ a, ¯ |b| < 2 + (1 − 2θ)¯ a.

(4.13) (4.14)

Then the linear θ-method and the one-leg θ-method for (1.2) are asymptotically stable at (¯ a, ¯b). Proof. From the proof of Theorem 4.1 and Remark 3.2 it is obvious that I < 1 as 1 + (1 − θ)ha < 0 implies that the solutions of (4.11) and (4.12) is asymptotically stable, where I=

−1 − (1 − θ)ha |b|h + . 1 − θha 1 − θha

It is easy to see from (4.14) that I
0, then I=

1 + (1 − θ)ha + h|b| 1 + (1 − θ)ha − ha < = 1. 1 − θha 1 − θha

Otherwise 1 + (1 − θ)ha < 0, then in view of |b|h(1 − θ) < 1, we have

1 6 θ 6 1, a < −|b| and 2

−2 + (2θ − 1)ha + h|b| < −2 + (2θ − 1)h(−|b|) + h|b| = 2(|b|h(1 − θ) − 1) < 0.

14

M. Z. LIU ET AL.

Therefore I=

−1 − (1 − θ)ha + h|b| −2 + (2θ − 1)ha + h|b| =1+ < 1. 1 − θha 1 − θha

1 . Then the numerical solution is unstable when b = 0. 2 Remark 4.3. Let θ ∈ [0, 1] and R = {(¯ a, ¯b) : |¯b| < −¯ a}. Theorem 4.3 shows that S¯θ ∩R ⊇ {(¯ a, ¯b) : |¯b| < −¯ a and |¯b| < 2+(1−2θ)¯ a} and Sθ ∩R ⊇ {(¯ a, ¯b) : |¯b| < −¯ a and |¯b| < 2 + (1 − 2θ)¯ a}, which proves one part of Jackiewicz’s conjecture in [18]. Necessity. Let θ
−m, tn = 0, n = −(m + 1), hni

n . m m −N In the following experiments T = q . We denote the error at t = T by ERROR, the number of data in the set {xn : n ∈ Z+ such that tn ∈ [0, T ]} by N D, the number of data in the set {xn : n ∈ Z+ such that tn ∈ [qT, T ]} by N Dq and the running times by TIME. 1 In Table 5.1 a = −1E − 4, b = 5E − 5, θ = 1, m = 1E3, T ≈ 1000 and h = . m Table 5.1 shows that N D, N Dq and TIME for (4.1) are much larger than those in (5.1). While the error of (4.1) is much smaller than that of (5.1). In Table 5.2 a = −1E − 4, b = 5E − 5, θ = 1, h = 1E − 2, T ≈ 1000 and (1 − q)T h= . Table 5.2 shows that N D and TIME for (4.1) are much smaller m than those in (5.1). While the errors for (4.1) and (5.1) are approximatively the same. 1 In Table 5.3 a = −2E − 1, b = 1E − 1, q = 0.1, h = = 1E − 2. The analytic m solution and the numerical solutions at t = 1E + 1, 1E + 2, 1E + 3, 1E + 4 are listed in the table. We can see from the table that method (4.1) is better than (5.1) if m is fixed and t is sufficiently large. and hn+1 = tn+1 − tn , where

is the integer part

STABILITY OF NUMERICAL METHODS FOR PANTOGRAPH EQUATIONS

Table 5.1: The comparison with h = q = 0.1, T = 1E3 (4.1) (5.1) 1E6 5E3 9E5 1E3 2.1025E − 9 1.7591E − 6 1.2420 < 0.01

ND N Dq ERROR TIME

ND N Dq ERROR TIME

1 m

q = 0.5, T = 1024 (4.1) (5.1) 1.024E6 1.2E4 5.12E5 1E3 1.7122E − 9 5.9935E − 7 1.2520 < 0.01

Table 5.2: The comparison with h = q = 0.1, T = 1E3 (4.1) (5.1) 1E5 4.5E5 9E4 9E4 2.1562E − 8 1.9546E − 8 0.11 0.24

15

(1 − q)T m

q = 0.5, T = 1024 (4.1) (5.1) 1.024E5 6E5 5.12E4 5.12E3 1.7684E − 8 1.1712E − 8 0.12 0.36

Table 5.3: The numerical solutions of the two methods t x(t) θ = 0.3 xn xn xn θ=1 xn

of of of of

(4.1) (5.1) (4.1) (5.1)

1E + 1 5.4123E − 1 5.4117E − 1 5.4080E − 1 5.4138E − 1 5.4231E − 1

1E + 2 2.7532E − 1 2.7529E − 1 2.7508E − 1 2.7540E − 1 2.7592E − 1

1E + 3 1.3790E − 1 1.3788E − 1 1.3778E − 1 1.3794E − 1 1.3820E − 1

1E + 4 6.8961E − 2 6.8953E − 2 8.5566E + 9 6.8981E − 2 6.9111E − 2

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