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Asymptotical stability of Runge-Kutta for a class of impulsive differential equations
- Gui-Lai Zhang^{1}Email author
- Received: 1 November 2015
- Accepted: 27 December 2015
- Published: 3 February 2016
Abstract
The aim of this paper is to study asymptotical stability of Runge-Kutta methods for a class of linear impulsive differential equations with piecewise continuous arguments. New results about the asymptotical stability region of Runge-Kutta methods for these equations are obtained by the theory of the Padé approximation. Finally, some numerical examples are given to illustrate the theoretical results.
Keywords
- asymptotical stability
- Runge-Kutta methods
- impulsive differential equations
- piecewise constant arguments
- stability region
1 Introduction
In the past two decades, the theory of impulsive differential equations has been developed rapidly [1–3]. Such equations consist of differential equations with impulse effects and emerge in modeling of real-world problems observed in engineering, physics, biology, etc. In addition to these, the theory of numerical methods for impulsive differential equations has also been studied extensively [4–7].
In 1984, Cooke and Wiener studied differential equations without impulses and they noted that such equations were comprehensively related to impulsive and difference equations [8]. Later, the case of discontinuous solutions of differential equations with piecewise continuous arguments was proposed as an open problem by Wiener [9]. On the other hand, it is well known that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, and optimal control models in economics do exhibit impulsive effects [3]. Recently, the existence and uniqueness, and oscillation of the exactly solutions of impulsive delay differential equations with piecewise constant arguments have been widely studied [10–13]. But to the best of our knowledge, up to now, there are few articles referring to numerical methods for impulsive delay differential equations with piecewise constant arguments.
The paper is organized as follows. In Section 2, we obtain the existence, uniqueness, and asymptotical stability of the exact solutions of (1.1). In Section 3, we study the asymptotical stability of the Runge-Kutta methods for (1.1). In Section 4, two special cases of Section 3 are studied, respectively: impulsive differential equations without piecewise constant argument and differential equations with piecewise constant arguments. In Section 5, some numerical examples are given to confirm the theoretical results.
2 Asymptotical stability of the exact solutions
Definition 2.1
- (1)
\(x(t)\) is continuous for \(t\in[0,+\infty)\) with the possible exception of the points \([t]\in[ 0, \infty)\),
- (2)
\(x(t)\) is right continuous and has left-hand limit at the points \([t]\in[ 0, \infty)\),
- (3)
\(x(t)\) is differentiable and satisfies \(x'(t)=px(t)+qx([t])\) for any \(t\in\mathbb{R}^{+}\) with the possible exception of the points \([t]\in[ 0, \infty)\) where one-sided derivatives exist,
- (4)
\(x(n)\) satisfies \(\Delta x(n)=rx(n^{-})\) for \(n\in\mathbb{Z}^{+}\).
Definition 2.2
By [15], p.179, Theorem 3, and [15], p.183, Theorem 9, we immediately obtain the following theorem.
Theorem 2.3
3 Runge-Kutta methods for (1.1)
Definition 3.1
- 1.
\(I-zA\) is invertible where \(z=hp\),
- 2.
for any given \(x_{k,l}\) (\(0\leq l\leq m\)) by relationship (3.1), such that \(\lim_{k\rightarrow\infty}X_{k}=0\) where \(X_{k}=(x_{k,0},x_{k,1},\ldots,x_{k,m})\).
Definition 3.2
The set of all pairs \((p,q,r)\) at the process (3.1) for equation (1.1) which is asymptotically stable is called stability region denoted by S.
Theorem 3.3
Proof
Lemma 3.4
Proof
Obviously, \(y_{1}=R(z)y_{0}\). Step by step, \(y_{m}=R(z)^{m} y_{0}\) is an approximation of \(y(1)\). Solving equation (3.5), we obtain \(y(1)=\mathrm{e}^{p}y_{0}\). Because the method is convergent, we have \(\lim_{h\rightarrow0, mh=1} y_{m}=y(1)\), which implies \(\lim_{m\rightarrow\infty,mh=1}R(z)^{m}=\mathrm{e}^{p}\). □
Lemma 3.5
It is easy to prove the following lemma. Therefore, the proof is omitted.
Lemma 3.6
Assume \(p\neq0\), \(p+q\neq0\), and \(f(x)=|x+\frac{q}{p} (x-1)|\). Then \(f(x)\) is decreasing for \(x<\frac{q}{p+q}\). On the other hand, \(f(x)\) is increasing for \(x>\frac{q}{p+q}\).
Theorem 3.7
- 1.
when \(h\leq\min\{h_{1},-\frac{\varsigma}{p}\}\), \(H_{1}\subseteq S\) if and only if j is even,
- 2.
when \(h\leq h_{2}\), \(H_{2}\subseteq S\) if and only if j is odd,
- 3.
when \(h\leq\min\{h_{1},\frac{\eta}{p}\}\), \(H_{3}\subseteq S\) if and only if k is odd,
- 4.
when \(h\leq h_{2}\), \(H_{4}\subseteq S\) if and only if k is even,
The high order Runge-Kutta methods for ( 1.1 )
Gauss-Legendre | Radau IA, IIA | Lobatto IIIA, IIIB | Lobatto IIIC | |
---|---|---|---|---|
(j,k) | (v,v) | (v − 1,v) | (v − 1,v − 1) | (v − 2,v) |
\(H_{1}\subseteq S\) | v is even | v is odd | v is odd | v is even |
\(H_{2}\subseteq S\) | v is odd | v is even | v is even | v is odd |
\(H_{3}\subseteq S\) | v is odd | v is odd | v is even | v is odd |
\(H_{4}\subseteq S\) | v is even | v is even | v is odd | v is even |
Proof
For brevity, we only prove case 1 of the theorem; the others, which can be proved similarly, are omitted. By Lemma 3.4, we see that \(\lim_{m\rightarrow\infty,mh=1}R(z)^{m}=\mathrm{e}^{p}\). \(\mathrm{e}^{p}\leq \frac{q}{p+q}\) implies \(R(z)^{m}\leq\frac{q}{p+q}\) for \(h\leq h_{1}\). Assume \(h\leq\min\{h_{1},-\frac{\varsigma}{p}\}\).
⟹) \(H_{1}\subseteq S\) implies \(|R(z)^{m}+\frac{q}{p} (R(z)^{m}-1)|\leq|\mathrm{e}^{p}+\frac{q}{p} (\mathrm{e}^{p}-1)|\). Because \(\mathrm{e}^{p}\leq\frac{q}{p+q}\), \(R(z)^{m}\leq\frac{q}{p+q}\), by Lemma 3.6, we obtain \(R(z)^{m}\leq\mathrm{e}^{p}\), which implies \(R(z)\geq\mathrm{e}^{z}\). By Lemma 3.5, we see that j is even. □
4 Special cases
In this section, two special cases are studied: the first special case \(q=0\), where equation (1.1) is changed as linear impulsive ordinary differential equations; second special case \(r=0\), where equation (1.1) is changed as linear differential equations with piecewise continuous argument without impulsive perturbations.
4.1 Linear impulsive ordinary differential equations
Theorem 2.3 is changed to the following result.
Theorem 4.1
From Theorem 3.7, we immediately obtain the following results.
Theorem 4.2
- 1.
\(H_{5}\subseteq S_{1}\) for an arbitrary consistent Runge-Kutta method,
- 2.
when \(h\leq h_{3}\), \(H_{7}\subseteq S_{1}\) if and only if j is odd,
- 3.
when \(h\leq h_{3}\), \(H_{8}\subseteq S_{1}\) if and only if k is even,
The high order Runge-Kutta methods for ( 4.1 )
Gauss-Legendre | Radau IA, IIA | Lobatto IIIA, IIIB | Lobatto IIIC | |
---|---|---|---|---|
(j,k) | (v,v) | (v − 1,v) | (v − 1,v − 1) | (v − 2,v) |
\(H_{7}\subseteq S_{1}\) | v is odd | v is even | v is even | v is odd |
\(H_{8}\subseteq S_{1}\) | v is even | v is even | v is odd | v is even |
The results obtained in this subsection are consistent with the results Ran et al. in [4].
4.2 Linear differential equations with piecewise continuous argument
Theorem 2.3 is changed to the following result.
Theorem 4.3
Theorem 4.4
- 1.
\(H_{9}\subseteq S_{2}\) for arbitrary consistent Runge-Kutta method,
- 2.
when \(h\leq\min\{h_{1},-\frac{\varsigma}{p}\}\), \(H_{11}\subseteq S_{2}\) if and only if j is even,
- 3.when \(h\leq h_{2}\), \(H_{12}\subseteq S_{2}\) if and only if k is even. (See Table 3.)Table 3
The high order Runge-Kutta methods for ( 4.4 )
Gauss-Legendre
Radau IA, IIA
Lobatto IIIA, IIIB
Lobatto IIIC
(j,k)
(v,v)
(v − 1,v)
(v − 1,v − 1)
(v − 2,v)
\(H_{11}\subseteq S_{2}\)
v is even
v is odd
v is odd
v is even
\(H_{12}\subseteq S_{2}\)
v is even
v is even
v is odd
v is even
The results obtained in this subsection are consistent with the results of Liu et al. in [14].
5 Numerical experiments
Obviously, we have \(|(1+r)(\mathrm{e}^{p}+\frac{q}{p} (\mathrm{e}^{p}-1))|<1\). Hence the exact solution of (5.1) is asymptotically stable.
Obviously, we have \(|(1+r)(\mathrm{e}^{p}+\frac{q}{p} (\mathrm{e}^{p}-1))| =2(3-\mathrm{e})<1\). Hence the exact solution of (5.2) is asymptotically stable.
The errors of the Runge-Kutta methods for ( 5.1 )
m | The implicit Euler | 2-Lobatto IIIA | 2-Lobatto IIIC | |||
---|---|---|---|---|---|---|
AE | RE | AE | RE | AE | RE | |
10 | 0.2731 | 2.1864 | 0.0023 | 0.0188 | 0.0049 | 0.0393 |
20 | 0.0956 | 0.7651 | 5.8209 | 0.0047 | 0.0012 | 0.0096 |
40 | 0.0406 | 0.3251 | 1.4524e−004 | 0.0012 | 2.9546e−004 | 0.0024 |
80 | 0.0188 | 0.1504 | 3.6293e−005 | 2.9057e−004 | 7.3237e−005 | 5.8636e−004 |
160 | 0.0090 | 0.0724 | 9.0721e−006 | 7.2634e−005 | 1.8227e−005 | 1.4594e−004 |
320 | 0.0044 | 0.0355 | 2.2680e−006 | 1.8158e−005 | 4.5464e−006 | 3.6400e−005 |
Ratio | 2.2977 | 2.2977 | 4.0083 | 4.0083 | 4.0435 | 4.0435 |
The errors of the Runge-Kutta methods for ( 5.2 )
m | The implicit Euler | 2-Lobatto IIIA | 2-Lobatto IIIC | |||
---|---|---|---|---|---|---|
AE | RE | AE | RE | AE | RE | |
100 | 0.0013 | 0.3930 | 2.5918e−006 | 8.0380e−004 | 5.2281e−006 | 0.0016 |
200 | 7.0145e−004 | 0.2175 | 6.4811e−007 | 2.0100e−004 | 1.3014e−006 | 4.0362e−004 |
400 | 3.6926e−004 | 0.1145 | 1.6204e−007 | 5.0254e−005 | 3.2470e−007 | 1.0070e−004 |
800 | 1.8947e−004 | 0.0588 | 4.0510e−008 | 1.2564e−005 | 8.1098e−008 | 2.5151e−005 |
1,600 | 9.5971e−005 | 0.0298 | 1.0128e−008 | 3.1409e−006 | 2.0265e−008 | 6.2848e−006 |
3,200 | 4.8298e−005 | 0.0150 | 2.5319e−009 | 7.8522e−007 | 5.0649e−009 | 1.5708e−006 |
Ratio | 1.9326 | 1.9227 | 3.9997 | 3.9997 | 4.0064 | 4.0064 |
Declarations
Acknowledgements
The author would like to thank the referees for their helpful comments and suggestions. This work is supported by the Research Fund for Northeastern University at Qinhuangdao XNB201415, the NSF of Hebei Province A2015501130 and Research project of higher school science and technology in Hebei province ZD2015211.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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