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Oscillation of Runge-Kutta methods for advanced impulsive differential equations with piecewise constant arguments
- Gui-Lai Zhang^{1}Email author
https://doi.org/10.1186/s13662-016-1067-0
© The Author(s) 2017
- Received: 20 October 2016
- Accepted: 22 December 2016
- Published: 27 January 2017
Abstract
The purpose of this paper is to study oscillation of Runge-Kutta methods for linear advanced impulsive differential equations with piecewise constant arguments. We obtain conditions of oscillation and nonoscillation for Runge-Kutta methods. Moreover, we prove that the oscillation of the exact solution is preserved by the θ-methods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution. We give some numerical examples to confirm the theoretical results.
Keywords
- impulsive differential equations
- piecewise constant arguments
- Runge-Kutta methods
- oscillation
- Padé approximation
1 Introduction
The rest of the paper is organized as follows. In Section 2, the results about oscillation of the exact solutions of (1.1) in [6] are introduced. In Section 3, the conditions of oscillation and nonoscillation for Runge-Kutta methods are obtained. In Section 4, the conditions of oscillation and nonoscillation for θ-methods are obtained. Moreover, it is proved that the oscillation of the exact solution is preserved by the θ-methods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution with the order of accuracy 1 (\(\theta\neq\frac{1}{2}\)) and 2 (\(\theta=\frac{1}{2}\)). In the last section, two simple numerical examples are given to confirm the theoretical results.
2 Preliminaries
Definition 2.1
- (1)
\(x:[0,\infty)\rightarrow\mathbb{R}\) 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 \(x'(t)+ax(t)+bx([t])+cx([t+1])=0\) 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 \(\Delta x(n)=d x(n)\) for \(n\in\mathbb{N}\) and \(x(0)=x_{0}\).
Theorem 2.2
Definition 2.3
See [6]
The solution \(x(t)\) of (1.1) is said to be oscillatory if there exist two real-valued sequences \((t_{n})_{n\geq0}, (t'_{n})_{n\geq 0} \subseteq[0,\infty)\) such that \(t_{n}\rightarrow\infty\), \(t'_{n}\rightarrow\infty\) as \(n\rightarrow\infty\) and \(x(t_{n})\leq0\leq x(t'_{n})\) for \(n\geq N\) with N sufficiently large. Otherwise, the solution is called nonoscillatory.
3 Runge-Kutta methods
3.1 Oscillation of Runge-Kutta methods
Definition 3.1
A nontrivial solution \(x_{n}\) of (3.1)-(3.2) is said to be oscillatory if there exists a sequence \(n_{k}\) such that \(n_{k}\rightarrow \infty\) as \(k\rightarrow\infty\) and \(x_{n_{k}}x_{n_{k+1}}\leq0\); otherwise, it is called nonoscillatory. We say that the Runge-Kutta method (3.1)-(3.2) for (1.1) is oscillatory if all the nontrivial solutions of (3.1)-(3.2) are oscillatory; we say that the Runge-Kutta method (3.1)-(3.2) for (1.1) is non-oscillatory if all the nontrivial solutions of (3.1)-(3.2) are nonoscillatory.
Definition 3.2
We say that the Runge-Kutta method preserves oscillations of (1.1) if (1.1) oscillates and there is \(h_{0}\) such that (3.1)-(3.2) oscillates for \(h< h_{0}\).
Theorem 3.3
Proof
Lemma 3.4
Lemma 3.5
- (i)
\(R(z)<\mathrm{e}^{z}\) for all \(z>0\) if and only if k is even,
- (ii)
\(R(z)>\mathrm{e}^{z}\) for \(0< z<\eta\) if and only if k is odd;
- (i)
\(R(z)> \mathrm{e}^{z}\) for all \(z<0\) if and only if j is even,
- (ii)
\(R(z)<\mathrm{e}^{z}\) for \(\varsigma< z<0\) if and only if j is odd,
Theorem 3.6
- (1)
\(a<0\), and k is odd for \(h=\frac{1}{m}<\min\{\frac{\eta }{-a},\frac{\delta_{2}}{-a}\}\);
- (2)
\(a>0\), and j is odd for \(h=\frac{1}{m}<\min\{\frac{\zeta }{-a},\frac{\delta_{1}}{-a}\}\).
Proof
3.2 Nonoscillation of Runge-Kutta methods
Lemma 3.7
- (1)
\(a<0\), and k is even for \(h=\frac{1}{m}<-\frac{\delta_{2}}{a}\);
- (2)
\(a>0\), and j is even for \(h=\frac{1}{m}<-\frac{\delta_{1}}{a}\).
Proof
Lemma 3.8
- (1)
\(f(x)\) is decreasing if \(a<0\);
- (2)
\(f(x)\) is increasing if \(a>0\).
Proof
- (1)
\(f(x)\) is decreasing if \(a<0\),
- (2)
\(f(x)\) is increasing if \(a>0\).
Theorem 3.9
- (1)
\(a<0\), and k is even for \(h=\frac{1}{m}<-\frac{\delta_{2}}{a}\);
- (2)
\(a>0\), and j is even for \(h=\frac{1}{m}<-\frac{\delta_{1}}{a}\).
Proof
4 Piecewise linear interpolation of θ-methods
4.1 Oscillation of θ-methods
Lemma 4.1
- (i)for \(a>0\),$$\begin{aligned}& \biggl(1+\frac{z}{1-z\theta}\biggr)^{m}\geq\mathrm{e}^{-a} \quad \textit{if and only if}\quad \varphi(-1)\leq\theta\leq1, \\& \biggl(1+\frac{z}{1-z\theta}\biggr)^{m}\leq\mathrm{e}^{-a} \quad \textit{if and only if}\quad 0\leq\theta\leq\frac{1}{2}; \end{aligned}$$
- (ii)for \(a<0\),where \(\varphi(x)=\frac{1}{x} -\frac{1}{\mathrm{e}^{x}-1}\).$$\begin{aligned}& \biggl(1+\frac{z}{1-z\theta}\biggr)^{m}\geq\mathrm{e}^{-a} \quad \textit{if and only if}\quad \frac{1}{2} \leq\theta\leq1, \\& \biggl(1+\frac{z}{1-z\theta}\biggr)^{m}\leq\mathrm{e}^{-a} \quad \textit{if and only if}\quad 0\leq\theta\leq\varphi(1), \end{aligned}$$
Applying Lemma 4.1, we can obtain the following two results.
Theorem 4.2
- (1)
\(\frac{1}{2} \leq\theta\leq1\) and \(a<0\) for \(h=\frac{1}{m}\), \(m>-a\);
- (2)
\(0 \leq\theta\leq\frac{1}{2}\) and \(a>0\) for \(h=\frac{1}{m}\), \(m>a\).
Theorem 4.3
- (1)
\(0 \leq\theta\leq\varphi(1)\) and \(a<0\) for \(h=\frac{1}{m}\), \(m>-a\);
- (2)
\(\varphi(-1) \leq\theta\leq1\) and \(a>0\) for \(h=\frac{1}{m}\), \(m>a\).
4.2 Piecewise linear interpolation of θ-methods
Theorem 4.4
- (1)
\(x(k)x(k+1)<0\);
- (2)
\(x(t)\) has at most one zero at \([k,k+1]\).
Proof
Theorem 4.5
- (1)
\(\bar{x}(t)-x(t)=O(h)\) (\(\theta\neq\frac{1}{2}\)), \(\bar {x}(t)-x(t)=O(h^{2})\) (\(\theta=\frac{1}{2}\));
- (2)
\(\bar{x}(t)\) has at most one zero in \([k,k+1]\) for any integer k.
Proof
In the following, we will prove that \(M\neq0\).
Consequently, \(x_{k,0}=x_{k+1,0}=0\), which implies \(\bar{x}_{k}(t)\equiv 0\), which is a contradiction to part (1) of the theorem. Hence, \(\bar {x}_{k}(t)\) has at most one zero in \([k,k+1]\), which implies \(\bar{x}(t)\) has at most one zero in \([k,k+1]\) for any integer k. □
Theorem 4.6
- (1)
\(\bar{x}(\bar{t})=0\); moreover, \(t-\bar{t}=O(h)\) (\(\theta\neq \frac{1}{2}\)), \(t-\bar{t}=O(h^{2})\) (\(\theta=\frac{1}{2}\));
- (2)\(\bar{x}(t)\) intersects the axis of abscissas at t̄, that is, \(\bar{x}(\bar{t})=0\),
- (i)
\(\bar{x}(k+(l+1)h)\bar{x}(k+(l-1)h)<0\) for \(\bar{t}=k+lh\),
- (ii)
\(\bar{x}(k+(l+1)h)\bar{x}(k+lh)<0\) for \(\bar{t}=k+(l+\mu)h\) (\(0<\mu<1\)).
- (i)
Proof
5 Numerical experiments
Example 5.1
Preservation of oscillation for six methods
Gauss-Legendre | Radau IA, IIA | Lobatto IIIA, IIIB | Lobatto IIIC | |
---|---|---|---|---|
(v,v) | (v − 1,v) | (v − 1,v − 1) | (v − 2,v) | |
a<0 | v is odd | v is odd | v is even | v is odd |
a>0 | v is odd | v is even | v is even | v is odd |
Preservation of nonoscillation for six methods
Gauss-Legendre | Radau IA, IIA | Lobatto IIIA, IIIB | Lobatto IIIC | |
---|---|---|---|---|
(v,v) | (v − 1,v) | (v − 1,v − 1) | (v − 2,v) | |
a<0 | v is even | v is even | v is odd | v is even |
a>0 | v is even | v is odd | v is odd | v is even |
Example 5.2
The errors of the first zero between numerical solutions and exact solution of ( 5.2 )
m | Explicit Euler | Trapezoidal rule | Implicit Euler | |||
---|---|---|---|---|---|---|
AE | RE | AE | RE | AE | RE | |
100 | 0.0020 | 0.0025 | 9.1599e − 06 | 1.1672e − 05 | 0.0020 | 0.0025 |
200 | 9.9290e − 04 | 0.0013 | 2.2900e − 06 | 2.9180e − 06 | 9.9369e − 04 | 0.0013 |
400 | 4.9657e − 04 | 6.3277e − 04 | 5.7249e − 07 | 7.2951e − 07 | 4.9673e − 04 | 6.3296e − 04 |
800 | 2.4833e − 04 | 3.1644e − 04 | 1.4312e − 07 | 1.8238e − 07 | 2.4832e − 04 | 3.1643e − 04 |
1,600 | 1.2418e − 04 | 1.5824e − 04 | 3.3076e − 08 | 4.2148e − 08 | 1.2421e − 04 | 1.5828e − 04 |
3,200 | 6.2108e − 05 | 7.9142e − 05 | 8.2691e − 09 | 1.0537e − 08 | 6.2109e − 05 | 7.9144e − 05 |
Ratio | 1.9992 | 1.9992 | 4.0654 | 4.0654 | 2.0009 | 2.0009 |
The errors of the third zero between numerical solutions and exact solution of ( 5.2 )
m | Explicit Euler | Trapezoidal rule | Implicit Euler | |||
---|---|---|---|---|---|---|
AE | RE | AE | RE | AE | RE | |
100 | 0.0020 | 7.1229e − 04 | 9.1599e − 06 | 3.2893e − 06 | 0.0020 | 7.1532e − 04 |
200 | 9.9290e − 04 | 3.5655e − 04 | 2.2900e − 06 | 8.2233e − 07 | 9.9369e − 04 | 3.5683e − 04 |
400 | 4.9657e − 04 | 1.7832e − 04 | 5.1768e − 07 | 1.8590e − 07 | 4.9673e − 04 | 1.7837e − 04 |
800 | 2.4833e − 04 | 8.9173e − 05 | 6.8482e − 08 | 2.4592e − 08 | 2.4832e − 04 | 8.9172e − 05 |
1,600 | 1.2418e − 04 | 4.4594e − 05 | 3.3076e − 08 | 1.1878e − 08 | 1.2421e − 04 | 4.4604e − 05 |
3,200 | 6.2108e − 05 | 2.2303e − 05 | 5.6725e − 09 | 2.0370e − 09 | 6.2109e − 05 | 2.2303e − 05 |
Ratio | 1.9992 | 1.9992 | 4.7769 | 4.7769 | 2.0009 | 2.0009 |
Declarations
Acknowledgements
I would like to thank the referees for their helpful comments and suggestions. This work is supported by the Natural Science Foundations of Hebei Province A2015501130, the Research Project of Higher School Science and Technology in Hebei province ZD2015211,the Fundamental Research Funds for Central Universities N152304007 and the Youth Science Foundations of Heilongjiang Province QC2016001.
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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