Preservation of stability and oscillation of Euler-Maclaurin method for differential equation with piecewise constant arguments of alternately advanced and retarded type
- Qi Wang^{1}Email author,
- Xiaoming Wang^{1},
- Yucheng Xie^{1} and
- Ling Chen^{1}
https://doi.org/10.1186/s13660-015-0685-5
© Wang et al. 2015
Received: 25 December 2014
Accepted: 4 May 2015
Published: 21 May 2015
Abstract
This paper is devoted to stability and oscillation analysis of Euler-Maclaurin method for differential equation with piecewise constant arguments \(u'(t)=au(t)+bu(2[(t+1)/2])\). The necessary and sufficient conditions under which the numerical stability regions contain the analytical stability regions are given. Moreover, the conditions of oscillation for the Euler-Maclaurin method are obtained. We show that the Euler-Maclaurin method preserves the oscillation of the exact solution. In addition, the connection between stability and oscillation are discussed theoretically and numerically. Finally, some numerical examples are also provided.
Keywords
Euler-Maclaurin method stability oscillation piecewise constant argumentsMSC
65L07 65L201 Introduction
There exists an extensive literature dealing with EPCA, for instance, the existence and uniqueness of the solution of a class of first order nonhomogeneous advanced impulsive EPCA were considered in [8], the stability property of first order EPCA of generalized type (EPCAG) was addressed in [9], oscillation of exact solution of EPCA with retarded and advanced arguments was discussed in [10]. In [11], the authors constructed Green’s function to the linear operator of boundary value EPCA and obtained some comparison results for the same differential equation. The general theory and basic results for EPCA have been thoroughly developed in the book of Wiener [12].
In contrast to the study on the qualitative behavior of EPCA, the research on the numerical solution of EPCA has become a hot issue recently. Numerical stability of many kinds of EPCA was addressed in [13–19]. Numerical oscillation of θ-methods and Runge-Kutta methods for equation \(x'(t)+ax(t)+a_{1}x([t-1])=0\) was investigated in [20, 21], respectively. Moreover, stability and oscillation of numerical solution for EPCA with \([t+1/2]\) and \(2[(t+1)/2]\) were considered in [22, 23], respectively. Numerical methods in the above mentioned papers involve θ-methods, Runge-Kutta methods, linear multistep method and Galerkin methods. However, to the best of our knowledge, very few results concerning Euler-Maclaurin method were obtained (see [24]). The authors of [24] investigated the stability of Euler-Maclaurin method for a linear neutral EPCA. Different from [24], in the present paper, we study the stability and oscillation of the numerical solution in the Euler-Maclaurin method for (1). Whether the numerical method preserves stability and oscillation and the connection between stability and oscillation are also investigated.
The rest of this paper is arranged as follows. In Section 2, we propose some useful concepts and results for stability and oscillation of the exact solution. In Section 3, we obtain a discrete equation by applying the Euler-Maclaurin method to (1), then the asymptotic stability, oscillation and non-oscillation of numerical method for (1) are considered. In Section 4, we discuss the preservation properties of Euler-Maclaurin method. The conditions under which the analytical stability regions are contained in the numerical stability regions are obtained, and it is proved that the Euler-Maclaurin method can preserve oscillation of the exact solution. In Section 5, we obtain a lot of connections between stability and oscillation. Finally, some numerical examples are reported in Section 6.
2 Stability and oscillation of exact solution
Definition 1
[12]
- (i)
\(u(t)\) is continuous on \([0, \infty)\),
- (ii)
the derivative \(u'(t)\) exists at each point t in \([0, \infty)\), with the possible exception of the points \(t=2n-1\) for \(n \in\mathbf{N}\), where one-sided derivatives exist,
- (iii)
(1) is satisfied on each interval \([2n-1,2n+1)\) for \(n \in\mathbf{N}\).
Theorem 1
[12]
Theorem 2
[12]
Definition 2
A nontrivial solution of (1) is said to be oscillatory if there exists a sequence \(\{t_{k}\}_{k=1}^{\infty}\) such that \(t_{k} \to\infty \) as \(k \to\infty\) and \(u(t_{k})u(t_{k-1}) \leq0\). Otherwise, it is called non-oscillatory. We say (1) is oscillatory if all nontrivial solutions of (1) are oscillatory. We say (1) is non-oscillatory if all nontrivial solutions of (1) are non-oscillatory.
Theorem 3
[12]
3 Stability and oscillation of Euler-Maclaurin method
3.1 Background of Euler-Maclaurin method
Euler-Maclaurin method is an important tool of numerical analysis which was discovered independently and almost simultaneously by Euler and Maclaurin in the first half of the eighteenth century. Rota [25] called Euler-Maclaurin method ‘one of the most remarkable formulas of mathematics’. After that, it shows us how to change a finite sum for an integral. It works much like Taylor’s formula: The equation involves an infinite series that may be truncated at any point, leaving an error term that can be bounded.
It is well known that the trapezoidal rule can be derived from the Euler-Maclaurin formula. See, for example, Munro’s paper [26]. In [27], the author indicated how the Newton-Cotes quadrature formulas and various other quadrature formulas can be developed from special cases of the periodic Euler-Maclaurin formula.
3.2 The discretization and convergence
Lemma 1
[33]
According to (2.4) in [24] and Lemma 1, we obtain the following theorem for convergence.
Theorem 4
For any given \(n\in\mathbf{N}\), the Euler-Maclaurin method is of order \(2n+2\).
Proof
3.3 Numerical stability
Definition 3
The Euler-Maclaurin method is called asymptotically stable at \((a,b)\) if there exists a constant M such that \(u_{n}\) defined by (3) tends to zero as \(n \rightarrow\infty\) for all \(h=1/m\) and any given \(u_{0}\).
In the rest of this paper, we always assume \(M>|a|\), which implies that \(|z|<1\) for the stepsize \(h=1/m\) with \(m\geq M\). The following lemmas play an essential role in proving the main theorem.
Lemma 2
[24]
If \(|z|\leq1\), then \(\phi(z)\geq1/2\) for \(z>0\) and \(\phi(z)\geq1\) for \(z \leq0\).
Lemma 3
[24]
In the following theorem we consider numerical stability for (1).
Theorem 5
Proof
3.4 Numerical oscillation
Theorem 6
- (i)
\(\{u_{n}\}\) is oscillatory,
- (ii)
\(\{u_{2km}\}\) is oscillatory,
- (iii)
\(b<-a\Omega(z)^{m}/(\Omega(z)^{m}-1)\) or \(b>a/(\Omega(z)^{m}-1)\) for \(a\neq0\) and \(b<-1\) or \(b>1\) for \(a=0\).
Proof
- (a)
\(\{u_{n}\}\) is not oscillatory,
- (b)
\(\{u_{2km}\}\) is not oscillatory
4 Preservation of stability and oscillation
For one equation, generally speaking, the exact solution and the numerical solution may have the same or different stability and oscillatory properties. It is known to us that the numerical method which can preserve the corresponding properties of original problem is useful and practical. Therefore, it is necessary to study the conditions under which the numerical solution and the exact solution have the same stability and oscillatory properties.
In this part, we discuss the conditions under which the analytical stability regions are contained in the numerical stability regions and the conditions under which the numerical solution and the exact solution are oscillatory at the same time.
4.1 Preservation of stability
Definition 4
The set of all points \((a,b)\) at which (1) is asymptotically stable is called the asymptotic stability region denoted by H.
Definition 5
The set of all points \((a,b)\) at which the Euler-Maclaurin method is asymptotically stable is called the asymptotic stability region denoted by S.
Theorem 7
\(H_{1} \subseteq S_{1}\) if and only if n is even, \(H_{2} \subseteq S_{2}\) if and only if n is odd.
Proof
Obviously, the next result is valid.
Theorem 8
For the Euler-Maclaurin method with any \(n \in\mathbf{N}\), we have \(H_{0}=S_{0}\).
4.2 Preservation of oscillation
Definition 6
We say that the Euler-Maclaurin method preserves oscillation of (1) if (1) oscillates, which implies that there is \(h_{0}\) such that (5) oscillates for \(h< h_{0}\).
The following theorem states the condition that the numerical method preserves the oscillation of (1).
Theorem 9
If \(a\neq0\), then the Euler-Maclaurin method preserves the oscillation of (1) if and only if n is even.
Proof
With a proof similar to that of Theorem 9, the following theorem can be obtained.
Theorem 10
If \(a\neq0\), then the Euler-Maclaurin method preserves the non-oscillation of (1) if and only if n is odd.
According to Theorems 3 and 6, we can easily get the following result for the case of \(a= 0\).
Theorem 11
If \(a= 0\), then the Euler-Maclaurin method preserves the oscillation and non-oscillation of (1) for any \(n \in\mathbf{N}\).
5 The connection between stability and oscillation
Stability and oscillation are two significant properties in the research of differential equation, so it is necessary to study the connection between them. In this section, the connection between stability and oscillation for the exact solution and the numerical solution will be discussed, respectively.
Theorem 12
- (i)
oscillatory and unstable if \(b \in(-\infty, V_{3})\) or \(b \in (V_{1}, +\infty)\),
- (ii)
oscillatory and asymptotically stable if \(b \in(V_{3}, V_{2})\),
- (iii)
non-oscillatory and asymptotically stable if \(b \in(V_{2}, -a)\),
- (iv)
non-oscillatory and unstable if \(b \in(-a,V_{1})\),
- (i)
oscillatory and asymptotically stable if \(b \in(-\infty, V_{2})\) or \(b \in(V_{3}, +\infty)\),
- (ii)
non-oscillatory and asymptotically stable if \(b \in(V_{2}, -a)\),
- (iii)
non-oscillatory and unstable if \(b \in(-a, V_{1})\),
- (iv)
oscillatory and unstable if \(b \in(V_{1}, V_{3})\).
Theorem 13
- (i)
oscillatory and unstable if \(b \in(-\infty, V_{3}(m))\) or \(b \in(V_{1}(m), +\infty)\),
- (ii)
oscillatory and asymptotically stable if \(b \in(V_{3}(m), V_{2}(m))\),
- (iii)
non-oscillatory and asymptotically stable if \(b \in (V_{2}(m), -a)\),
- (iv)
non-oscillatory and unstable if \(b \in(-a,V_{1}(m))\),
- (i)
oscillatory and asymptotically stable if \(b \in(-\infty, V_{2}(m))\) or \(b \in(V_{3}(m), +\infty)\),
- (ii)
non-oscillatory and asymptotically stable if \(b \in (V_{2}(m), -a)\),
- (iii)
non-oscillatory and unstable if \(b \in(-a, V_{1}(m))\),
- (iv)
oscillatory and unstable if \(b \in(V_{1}(m), V_{3}(m))\).
Theorem 14
- (i)
oscillatory and asymptotically stable if \(b \in(-\infty, -1)\),
- (ii)
non-oscillatory and asymptotically stable if \(b \in(-1, 0)\),
- (iii)
non-oscillatory and unstable if \(b \in(0, 1)\),
- (iv)
oscillatory and unstable if \(b \in(1,+\infty)\).
6 Numerical experiments
In order to give a numerical illustration to the results in the paper, we present some examples made by applying MATLAB 7.0.
( 22 ) | ( 23 ) | |||
---|---|---|---|---|
AE | RE | AE | RE | |
m = 2 | 5.9270e−04 | 3.0000e−03 | 2.1142e−08 | 1.8000e−03 |
m = 3 | 4.9566e−05 | 2.4792e−04 | 1.8071e−09 | 1.5564e−04 |
m = 5 | 2.2540e−06 | 1.1274e−05 | 8.3186e−11 | 7.1644e−06 |
m = 10 | 3.4836e−08 | 1.7424e−07 | 1.2924e−12 | 1.1131e−07 |
m = 20 | 5.4283e−10 | 2.7151e−09 | 2.0165e−14 | 1.7367e−09 |
m = 40 | 8.4785e−12 | 4.2407e−11 | 3.1361e−16 | 2.7010e−11 |
Ratio | 64.0243 | 64.0248 | 64.2996 | 64.2984 |
We further investigate the connection between stability and oscillation from (22) to (30). Take (25) as an example. Let us set \(m=30\) and \(n=2\) in Figure 4, then we calculate that \(V_{3} \approx3.3308\) and \(V_{3}(m) \approx3.3308\). Clearly, \(b=3.5 \in(V_{3},+\infty)\) and \(b=3.5 \in(V_{3}(m),+\infty)\). Thus, the exact solution and the numerical solution of (25) are both oscillatory and asymptotically stable. That is to say, the connection between stability and oscillation is in agreement with Theorems 12 and 13. For (22)-(24), (26)-(30), we can test them analogously (see Figures 1-3, 5-9).
Declarations
Acknowledgements
The reviewer’s valuable suggestions are greatly acknowledged. The authors would like to thank Professors Mingzhu Liu, Minghui Song and Zhanwen Yang for their helpful comments and constructive suggestions. This work is supported by the National Natural Science Foundation of China (No. 11201084) and the State Scholarship Fund grant [2013]3018 from China Scholarship Council.
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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