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Preservation of stability and oscillation of EulerMaclaurin method for differential equation with piecewise constant arguments of alternately advanced and retarded type
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 165 (2015)
Abstract
This paper is devoted to stability and oscillation analysis of EulerMaclaurin 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 EulerMaclaurin method are obtained. We show that the EulerMaclaurin 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.
1 Introduction
The theory of differential equation with piecewise constant arguments (EPCA) was initiated in [1, 2], which provided a mathematical instrument to applied science [3, 4]. These systems have been under intensive investigation for the last twenty years. They describe hybrid dynamical systems and combine properties of both differential and difference equations. For example, applying the explicit linear multistep method to differential equation \(u'(t)=f(u(t))\), we have
where h is stepsize and \(u_{n}\) is approximation to \(u(t)\) at \(t_{n}\). By integration, we can see that the above difference equation is equivalent to the following EPCA:
so EPCA has a similar structure to the difference equation. In the present paper we shall consider the following EPCA:
where a, b and \(u_{0}\) are real constants, \(b\neq1\) for \(a=0\) and \([\cdot]\) denotes the greatest integer function. Differential equation of this form has stimulated considerable interest and has been studied by Cooke and Wiener [5], Jayasree and Deo [6], Wiener and Aftabizadeh [7]. In this type of equation the argument deviation \(\eta(t)=t2[(t+1)/2]\) is a piecewise linear period function with period 2. Also, \(\eta(t)\) is negative for \(t\in[2n1,2n)\) and positive for \(t\in[2n,2n+1)\). Thus (1) is advanced type on \([2n1,2n)\) and retarded type on \([2n,2n+1)\). Therefore (1) is EPCA of alternately advanced and retarded type.
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 Greenâ€™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 [13â€“19]. Numerical oscillation of Î¸methods and RungeKutta methods for equation \(x'(t)+ax(t)+a_{1}x([t1])=0\) 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 Î¸methods, RungeKutta methods, linear multistep method and Galerkin methods. However, to the best of our knowledge, very few results concerning EulerMaclaurin method were obtained (see [24]). The authors of [24] investigated the stability of EulerMaclaurin 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 EulerMaclaurin 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 SectionÂ 2, we propose some useful concepts and results for stability and oscillation of the exact solution. In SectionÂ 3, we obtain a discrete equation by applying the EulerMaclaurin method to (1), then the asymptotic stability, oscillation and nonoscillation of numerical method for (1) are considered. In SectionÂ 4, we discuss the preservation properties of EulerMaclaurin method. The conditions under which the analytical stability regions are contained in the numerical stability regions are obtained, and it is proved that the EulerMaclaurin method can preserve oscillation of the exact solution. In SectionÂ 5, we obtain a lot of connections between stability and oscillation. Finally, some numerical examples are reported in SectionÂ 6.
2 Stability and oscillation of exact solution
Definition 1
[12]
A solution of (1) on \([0, \infty)\) is a function \(u(t)\) which satisfies the following conditions:

(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 \(t=2n1\) for \(n \in\mathbf{N}\), where onesided derivatives exist,

(iii)
(1) is satisfied on each interval \([2n1,2n+1)\) for \(n \in\mathbf{N}\).
Theorem 1
[12]
Assume that a, b and \(u_{0} \in\mathbf{R}\), then (1) has on \([0,\infty)\) a unique solution \(u(t)\) given by
for \(a\neq0\) and
for \(a= 0\), where
Theorem 2
[12]
The solution \(u(t)=0\) of (1) is asymptotically stable (\(u(t) \to0\) as \(t \to\infty\)) if and only if any one of the following conditions is satisfied:
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 \(u(t_{k})u(t_{k1}) \leq0\). Otherwise, it is called nonoscillatory. We say (1) is oscillatory if all nontrivial solutions of (1) are oscillatory. We say (1) is nonoscillatory if all nontrivial solutions of (1) are nonoscillatory.
Theorem 3
[12]
A necessary and sufficient condition for all solutions of (1) to be oscillatory is any one of the following conditions is satisfied:
3 Stability and oscillation of EulerMaclaurin method
3.1 Background of EulerMaclaurin method
EulerMaclaurin 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 EulerMaclaurin method â€˜one of the most remarkable formulas of mathematicsâ€™. After that, it shows us how to change a finite sum for an integral. It works much like Taylorâ€™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 EulerMaclaurin formula. See, for example, Munroâ€™s paper [26]. In [27], the author indicated how the NewtonCotes quadrature formulas and various other quadrature formulas can be developed from special cases of the periodic EulerMaclaurin formula.
3.2 The discretization and convergence
Let q be a positive integer, assume that the function \(f(t)\) is at least \((2q+2)\)times continuously differentiable on \([c,d]\). We further assume that h evenly divides c and d, then Atkinsonâ€™s version of the EulerMaclaurin formula [28] is as follows:
where \(B_{j}\) denotes the jth Bernoulli number, D denotes the differentiation operator. The definition, property and application of the Bernoulli number can be found in [29â€“32]. For brevity, we omit them.
Let h be a given stepsize, \(m\geq1\) be a given integer and satisfy \(h=1/m\). Let the gridpoints \(t_{i}\) be defined by \(t_{i}=ih\) (\(i=0,1,2,\ldots\)). Applying (2) to (1), we have
where \(u_{i}\) and \(u_{i+1}\) are approximations to \(u(t)\) at \(t_{n}\) and \(t_{n+1}\), respectively, \(u_{i}^{(n)}\) is an approximation to \(u(2[(t+1)/2])\) at \(t_{n}\). Let us denote \(i=2km+l\), \(l=m,m+1,\ldots ,m2,m1\) for \(k \geq1\) and \(l=0,1,\ldots,m1\) for \(k=0\). Then \(u_{i}^{(n)}\) can be defined as \(u_{2km}\) according to DefinitionÂ 1. So we have
Denote
where \(\Omega(z)\) is called a stability function of the EulerMaclaurin method. Then (4) turns into
We also consider the iteration of difference scheme. Formula (5) leads to
for \(a\neq0\) and
for \(a=0\), where
To guarantee that \(G(m)\neq0\), we require that
where \(b\neq0\).
Lemma 1
[33]
Assume that \(f(t)\) has \((2n+3)\) rd continuous derivative on the interval \([t_{i},t_{i+1}]\), then we have
According to (2.4) in [24] and LemmaÂ 1, we obtain the following theorem for convergence.
Theorem 4
For any given \(n\in\mathbf{N}\), the EulerMaclaurin method is of order \(2n+2\).
Proof
Let \(km \leq i <(k+1)m1\), then from LemmaÂ 1 with \(f(t)=u'(t)\) we have
Put \(i=(k+1)m1\), then for any given \(0<\epsilon< h\), we get
let \(\epsilon\to0^{+}\) in (12), we can get that (11) holds for \(i=(k+1)m1\).
Setting \(u_{i}=u(t_{i})\) and \(u_{2km}=u(k)\), then from (3) and (11) we also find
therefore, the EulerMaclaurin method is of order \(2n+2\). The proof is complete.â€ƒâ–¡
3.3 Numerical stability
Definition 3
The EulerMaclaurin 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]
If \(z\leq1\), then
In the following theorem we consider numerical stability for (1).
Theorem 5
The EulerMaclaurin method is asymptotically stable if any one of the following conditions is satisfied:
Proof
From (6) and (7) we can easily see that \(u_{n} \rightarrow0\) as \(n \rightarrow\infty\) if and only if \(u_{2km} \rightarrow0\) as \(k \rightarrow\infty\). Therefore, the EulerMaclaurin method is asymptotically stable if and only if
from which we obtain
it is obvious from \(b\neq1\) for \(a=0\) and (10) that the denominators in (14) and (15) do not vanish. From the second item of (15) we can see that the third condition in (13) is obtained. Then we consider the sign of \(a+bb\Omega (z)^{m}\) in (15), thus we have the following two cases.
Case I. If \(a+bb\Omega(z)^{m}>0\), then the first inequality of (15) reduces to
If \(a>0\) then by LemmaÂ 2 we have \(\Omega(z)>1\), that is, \(\Omega(z)^{m} \neq1\), so the denominator of the first inequality in (16) does not vanish. In view of (16) we obtain \(b<a\), thus
which is the first condition in (13). If \(a<0\), then by LemmaÂ 2 we have \(0<\Omega(z)<1\), thus, the denominator of the first item in (16) does not vanish because of \(\Omega(z)^{m} \neq1\). By (16) we have \(b>a\). Due to
then by (16) we get
Case II. If \(a+bb\Omega(z)^{m}<0\), then the first inequality of (15) yields
Similar to Case I, we have
Consequently, by virtue of (15), (17), (18) and (19), the proof is complete.â€ƒâ–¡
3.4 Numerical oscillation
Theorem 6
The following statements are equivalent:

(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
When \(a\neq0\), to prove (i) and (ii) are equivalent, first of all, we show that the following two statements

(a)
\(\{u_{n}\}\) is not oscillatory,

(b)
\(\{u_{2km}\}\) is not oscillatory
are equivalent. Obviously, (a) implies (b). If (b) holds, we have \(\lambda>0\), where
or equivalently,
which gives
Then, for any \(l\in\{1,2,\ldots,m1\}\), we have
from the above inequality, we can get
and
We obtain from (6) that \(\{u_{n}\}\) is not oscillatory. So (a) and (b) are equivalent; in other words, (i) and (ii) are equivalent. Next, we will prove that (ii) and (iii) are equivalent. We know that \(\{ u_{2km}\}\) is oscillatory if and only if \(\lambda<0\), i.e.,
then we immediately obtain
so (ii) and (iii) are equivalent. When \(a= 0\), from (9) we only let \(\lambda=(1+b)/(1b)\) in the above process. Therefore the proof is finished.â€ƒâ–¡
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 EulerMaclaurin method is asymptotically stable is called the asymptotic stability region denoted by S.
In the following we will find which conditions lead to \(H \subseteq S\). For convenience, we divide H and S into three parts, respectively:
and
It is easily seen that \(H=H_{0} \cup H_{1} \cup H_{2}\), \(S=S_{0} \cup S_{1} \cup S_{2}\) and
Therefore, we can conclude that \(H \subseteq S\) is equivalent to \(H_{i} \subseteq S_{i}\) (\(i=0,1,2\)). In the following theorem, we establish some results for preservation of stability.
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
According to Theorems 2 and 5, we have that \(H_{1} \subseteq S_{1}\) if and only if
that is,
it is not difficult to know that the function \(g(x)=(x^{2}+1)/(x1)^{2}\) is increasing in \([0,1)\) and decreasing in \((1,\infty)\), so (20) leads to
that is,
as a consequence of LemmaÂ 3, we have n is even. The other case can be proved analogously.â€ƒâ–¡
Obviously, the next result is valid.
Theorem 8
For the EulerMaclaurin method with any \(n \in\mathbf{N}\), we have \(H_{0}=S_{0}\).
4.2 Preservation of oscillation
Definition 6
We say that the EulerMaclaurin 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 EulerMaclaurin method preserves the oscillation of (1) if and only if n is even.
Proof
In view of Theorems 3 and 6, the EulerMaclaurin method preserves the oscillation of (1) if and only if
If \(a>0\), then we have
Because the function \(\omega(x)=x/(x1)\) is decreasing, so from (21) we obtain
that is,
then by LemmaÂ 3, we get n is even. The case of \(a<0\) can be proved in the same way.â€ƒâ–¡
With a proof similar to that of TheoremÂ 9, the following theorem can be obtained.
Theorem 10
If \(a\neq0\), then the EulerMaclaurin method preserves the nonoscillation 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 EulerMaclaurin method preserves the oscillation and nonoscillation 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.
For simplicity, we define
and
A combination of Theorems 2, 3, 5 and 6 leads to the following three results.
Theorem 12
When \(a>0\), the exact solution of (1) is

(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)
nonoscillatory and asymptotically stable if \(b \in(V_{2}, a)\),

(iv)
nonoscillatory and unstable if \(b \in(a,V_{1})\),
when \(a<0\), the exact solution of (1) is

(i)
oscillatory and asymptotically stable if \(b \in(\infty, V_{2})\) or \(b \in(V_{3}, +\infty)\),

(ii)
nonoscillatory and asymptotically stable if \(b \in(V_{2}, a)\),

(iii)
nonoscillatory and unstable if \(b \in(a, V_{1})\),

(iv)
oscillatory and unstable if \(b \in(V_{1}, V_{3})\).
Theorem 13
When \(a>0\), the numerical solution of (1) is

(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)
nonoscillatory and asymptotically stable if \(b \in (V_{2}(m), a)\),

(iv)
nonoscillatory and unstable if \(b \in(a,V_{1}(m))\),
when \(a<0\), the numerical solution of (1) is

(i)
oscillatory and asymptotically stable if \(b \in(\infty, V_{2}(m))\) or \(b \in(V_{3}(m), +\infty)\),

(ii)
nonoscillatory and asymptotically stable if \(b \in (V_{2}(m), a)\),

(iii)
nonoscillatory and unstable if \(b \in(a, V_{1}(m))\),

(iv)
oscillatory and unstable if \(b \in(V_{1}(m), V_{3}(m))\).
Theorem 14
When \(a=0\), the exact solution and the numerical solution of (1) both are

(i)
oscillatory and asymptotically stable if \(b \in(\infty, 1)\),

(ii)
nonoscillatory and asymptotically stable if \(b \in(1, 0)\),

(iii)
nonoscillatory 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.
The first part of this section is devoted to examining the convergence and the stability of the EulerMaclaurin method. Consider the following three problems:
From TheoremÂ 2 and the definitions of \(H_{i}\) (\(i=0,1,2\)), it is easy to see that the coefficients in (22), (23) and (24) satisfy \((2.5,3.1)\in H_{1}\), \((1.8,2.2)\in H_{2}\) and \((0,6.6)\in H_{0}\), respectively. We shall use the EulerMaclaurin method with the stepsize \(h=1/m\) to get the numerical solution at \(t=10\), where the exact solutions are \(u(10)\approx 0.1999\), \(u(10)\approx1.1611 \times10^{5}\) and \(u(10)\approx0.2172\) for (22), (23) and (24), respectively. In TableÂ 1 we list the absolute errors (AE) and the relative errors (RE) between the numerical solution and the exact solution at \(t=10\) and the ratio of the errors of the case \(m=20\) over that of \(m=40\). We can see from this table that the EulerMaclaurin method with \(n=2\) is of orderÂ 6, which is consistent with TheoremÂ 4.
For (22) and (23), it is easy to verify that condition (10) holds true. In FiguresÂ 13, we plot the numerical solution with different parameters for (22), (23) and (24), respectively. We can see from these figures that the numerical solutions all are stable.
The second part of this section is devoted to examining the oscillation and the connection between stability and oscillation. Consider the following problems:
It is not difficult to test that condition (10) holds true for (25), (26), (28) and (29). As to (25)(30), the exact solutions of (25), (26) and (27) are oscillatory; the exact solutions of (28), (29) and (30) are nonoscillatory according to TheoremÂ 3. In FiguresÂ 49, we draw the figures of the exact solutions and the numerical solutions, respectively. As shown in these figures, the numerical solutions of (25), (26) and (27) are oscillatory; the numerical solutions of (28), (29) and (30) are nonoscillatory, which coincides with TheoremÂ 6.
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 FigureÂ 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 FiguresÂ 13, 59).
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Acknowledgements
The reviewerâ€™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.
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Wang, Q., Wang, X., Xie, Y. et al. Preservation of stability and oscillation of EulerMaclaurin method for differential equation with piecewise constant arguments of alternately advanced and retarded type. J Inequal Appl 2015, 165 (2015). https://doi.org/10.1186/s1366001506855
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DOI: https://doi.org/10.1186/s1366001506855