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Asymptotical stability of RungeKutta for a class of impulsive differential equations
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 39 (2016)
Abstract
The aim of this paper is to study asymptotical stability of RungeKutta methods for a class of linear impulsive differential equations with piecewise continuous arguments. New results about the asymptotical stability region of RungeKutta methods for these equations are obtained by the theory of the PadÃ© approximation. Finally, some numerical examples are given to illustrate the theoretical results.
1 Introduction
In the past two decades, the theory of impulsive differential equations has been developed rapidly [1â€“3]. Such equations consist of differential equations with impulse effects and emerge in modeling of realworld 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 [4â€“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 [10â€“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.
In this paper, we study the asymptotical stability of RungeKutta methods for the following scalar equation:
where p, q, r, and \(x_{0}\) are real constants and \([\cdot]\) denotes the greatest integer function. This paper can be seen not only as the extension of asymptotically stable theory of differential equations with piecewise arguments in [14], but also as the extension of asymptotically stable theory of impulsive ordinary differential equations in [4].
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 RungeKutta 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
If a function \(x:\mathbb{R}^{+} \rightarrow\mathbb{R}\) satisfies the following conditions, it is said to be a solution of (1.1):

(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 lefthand 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 onesided derivatives exist,

(4)
\(x(n)\) satisfies \(\Delta x(n)=rx(n^{})\) for \(n\in\mathbb{Z}^{+}\).
Definition 2.2
The zero solution of (1.1) is said to be asymptotically stable, if
where \(x(t)\) is the solution of (1.1) for any initial data \(x_{0}\).
By [15], p.179, Theorem 3, and [15], p.183, Theorem 9, we immediately obtain the following theorem.
Theorem 2.3
When \(p\neq0\), on \(t\in[0,\infty)\), (1.1) has a unique solution
where \(\{t\}\) is the fractional part of t and
When \(p=0\), on \(t\in[0,\infty)\), (1.1) also has a unique solution
Consequently, when \(p\neq0\), the zero solution of (1.1) is asymptotically stable if and only if
when \(p=0\), the zero solution of (1.1) is asymptotically stable if and only if
We introduce the sets \(H_{0}\) and H consisting of \((p,q,r)\in \mathbb {R}^{3}\) which satisfy the condition (2.2) and (2.3), respectively, i.e.
and we divide the region H into four parts as \(p+q\neq0\):
3 RungeKutta methods for (1.1)
Consider the RungeKutta methods for equation (1.1):
where \(h=\frac{\tau}{m}\), \(m\geq1\), m is an integer, h is the stepsize, v is referred to as the number of stages. The weights are \(b_{i}\), the abscissas are \(c_{i}=\sum_{j=1}^{v}a_{ij}\) and the matrix \(A=[a_{i,j}]_{i,j=1}^{v}\) will be denoted by \((A,b,c)\).
Definition 3.1
Process (3.1) for equation (1.1) is called asymptotically stable at \((p,q,r)\) if and only if for all \(m \geq M\) and \(h=\frac{1}{m}\)

1.
\(IzA\) 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
When \(p\neq0\), the method (3.1) is asymptotically stable if and only if
where \(R(z)=1+zb^{T}(IzA)^{1}e\), \(e=(1, 1, \ldots, 1)^{T}\) is a vector of v dimension. On the other hand, when \(p=0\), the consistent RungeKutta method (3.1) is asymptotically stable if and only if
Proof
The RungeKutta method (3.1) can be written as
which implies that
which also implies that
Hence when \(p\neq0\), the method (3.1) is asymptotically stable if and only if \((1+r)(R(z)^{m}+\frac{q}{p} (R(z)^{m}1))<1\).
On the other hand, the RungeKutta method being consistent implies \(\sum_{i=1}^{v}b_{i}=1\), which also implies
Consequently, when \(p=0\), the consistent RungeKutta method (3.1) is asymptotically stable if and only if
â€ƒâ–¡
Lemma 3.4
If the same RungeKutta method applied to the following test equation:
is convergent (in the following, we always assume this condition holds), then we have \(\lim_{m\rightarrow\infty, mh=1}R(z)^{m}=\lim_{m\rightarrow\infty,mh=1}R(hp)^{m}=\mathrm{e}^{p}\), where \(R(z)=1+zb^{T}(IzA)^{1}e\) is the stability function.
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 \(\lim_{m\rightarrow\infty,mh=1}R(z)^{m}=\mathrm{e}^{p}\).â€ƒâ–¡
Lemma 3.5
The \((j,k)\)PadÃ© approximation to \(\mathrm{e}^{z}\) is given by
where
with error
It is the unique rational approximation to \(\mathrm{e}^{z}\) of order \(j+k\), such that the degrees of numerator and denominator are j and k, respectively.
It is easy to prove the following lemma. Therefore, the proof is omitted.
Lemma 3.6
Assume \(p\neq0\), \(p+q\neq0\), and \(f(x)=x+\frac{q}{p} (x1)\). Then \(f(x)\) is decreasing for \(x<\frac{q}{p+q}\). On the other hand, \(f(x)\) is increasing for \(x>\frac{q}{p+q}\).
The set S of all \((p,q,r)\) at which the scheme (3.1) is asymptotically stable is called the asymptotical stability region,
Theorem 3.7
Assume \(p\neq0\), \(p+q\neq0\), and the stability function of the RungeKutta method \(R(z)\) is the \((j,k)\)PadÃ© approximation to \(\mathrm{e}^{z}\), \(z=hp\). Then for \(h=\frac{1}{m}\), m being an integer,

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,
where \(h_{1}=\sup\{x:R(hp)^{\frac{1}{h}}<\frac{q}{p+q}\textit{ for all }h\in(0,x)\}\), \(h_{2}=\sup\{x:R(hp)^{\frac{1}{h}}>\frac{q}{p+q}\textit{ for all }h\in(0,x)\}\), Î· is a real zero of \(Q_{k}(z)\), and Î¶ is a real zero of \(P_{j}(z)\). (See Table 1.)
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 \(h\leq\min\{h_{1},\frac{\varsigma}{p}\}\).
âŸ¸) By Lemma 3.5 and j is even, we can obtain \(\mathrm{e}^{z}\leq R(z)\), which implies that \(\mathrm{e}^{p}\leq R(z)^{m}\leq\frac{q}{p+q}\) for \(m\geq M\). Consequently, by Lemma 3.6, we have
which implies \(H_{1}\subseteq S\).
âŸ¹) \(H_{1}\subseteq S\) 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.â€ƒâ–¡
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
In this subsection, the special case of (1.1) when \(q=0\) is studied. Equation (1.1) is changed as
where p, r, and \(x_{0}\).
Theorem 2.3 is changed to the following result.
Theorem 4.1
On \(t\in[0,\infty)\), (4.1) has a unique solution
where \(\{t\}\) is the fractional part of t and
Consequently, the zero solution of (1.1) is asymptotically stable if and only if
The asymptotical stability sets \(H_{0}\), H, \(H_{2}\), and \(H_{4}\) are changed as follows, respectively:
Obviously, \(H_{1}\) and \(H_{3}\) are empty as \(q=0\), so it is not considered.
We also consider the RungeKutta methods for (4.1):
where \(h=\frac{1}{m}\), \(m\geq1\), m is an integer, h is the stepsize. The set \(S_{1}\) of all \((p,r)\) at which the scheme (4.3) is asymptotically stable is called the asymptotical stability region,
From Theorem 3.7, we immediately obtain the following results.
Theorem 4.2
Assume the stability function for the RungeKutta method is \(R(z)\), the \((j,k)\)PadÃ© approximation to \(\mathrm{e}^{z}\), \(z=ph\). Then for \(h=\frac{1}{m}\), m being an integer,

1.
\(H_{5}\subseteq S_{1}\) for an arbitrary consistent RungeKutta 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,
where \(h_{3}=\sup\{x:R(hp)>0\textit{ for all }h\in(0,x)\}\). (See Table 2.)
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
In this subsection, the special case of (1.1) when \(r=0\) is studied. Equation (1.1) is changed as
where we introduced p, q, and \(x_{0}\).
Theorem 2.3 is changed to the following result.
Theorem 4.3
On \(t\in[0,\infty)\), (4.4) has a unique solution
where \(\{t\}\) is the fractional part of t and
Consequently, when \(p\neq0\), the zero solution of (1.1) is asymptotically stable if and only if
Obviously, when \(p=0\), the zero solution of (1.1) is asymptotically stable if and only if \(1+q<1\).
The asymptotical stability sets H, \(H_{1}\), and \(H_{4}\) are changed as follows, respectively:
Obviously, the sets \(H_{2}\) and \(H_{3}\) are empty when \(r=0\), so it is not considered.
We also consider the RungeKutta methods for (4.4):
where \(h=\frac{1}{m}\), \(m\geq1\), m is an integer, h is the stepsize, v is referred to as the number of stages. The following set \(S_{2}\) of all \((p,q)\) at which the scheme (3.1) is asymptotically stable is called the asymptotical stability region,
From Theorem 3.7, we immediately obtain the following results.
Theorem 4.4
Assume the stability function RungeKutta method is \(R(z)\), the \((j,k)\)PadÃ© approximation to \(\mathrm{e}^{z}\), \(z=ph\). Then for \(h=\frac{1}{m}\), m being an integer,

1.
\(H_{9}\subseteq S_{2}\) for arbitrary consistent RungeKutta 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.)
The results obtained in this subsection are consistent with the results of Liu et al. in [14].
5 Numerical experiments
First of all, we consider the following equation:
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.
By Theorem 3.7 (case 4 of the theorem), we can see that explicit Euler method for (5.1) (see Figure 1) is asymptotically stable for arbitrary \(h=\frac{1}{m}\), m is an integer. (Obviously, \(R(hp)^{\frac{1}{h}}=(1+hp)^{\frac{1}{h}}=(1+\frac{1}{m} )^{m}\geq 2>\frac{3}{2} =\frac{q}{p+q}\), which implies \(h\leq h_{2}\) holds for arbitrary \(h=\frac{1}{m}\), m being an integer.)
We all know that the implicit Euler method for the test equation \(y'(t)=\lambda y(t)\), \(\Re(\lambda)<0\), is asymptotically stable for arbitrary step size \(h>0\). But this does not hold for arbitrary step size \(h>0\) when the implicit Euler method is adopted for (5.1) (see Figure 2).
By Theorem 3.7 and Table 1, we can also see that the 2stage Lobatto IIIC method and the 3stage Lobatto IIIA method are all asymptotically stable (see Figures 3 and 4) for h small enough.
Consider the following equation:
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.
By Theorem 3.7 (case 3 of the theorem), we can also see that the implicit Euler method for (5.1) (see Figure 5) is asymptotically stable for \(h=\frac{1}{m}\), m being an integer and \(m\geq6\). (Obviously, \(\frac{\eta}{p}=1\) and \(R(hp)^{\frac{1}{h}}=(\frac{1}{ 1hp})^{\frac{1}{h}}=(1\frac{1}{m})^{m}< 3=\frac{q}{p+q}\), which implies \(h\leq\min\{h_{1},\frac{\eta}{p}\}\) holds for \(h=\frac{1}{m}\), m being an integer and \(m\geq6\).)
Similarly, we can also see that the 2stage Lobatto IIIA method for (5.1) (see Figure 6) is asymptotically stable for \(h=\frac{1}{m}\), m being an integer and \(m\geq2\). (Obviously, \(\frac{\eta}{p}=2\) and \(R(hp)^{\frac{1}{h}}=(\frac{1+\frac {hp}{2}}{1\frac{hp}{2}})^{\frac{1}{h}}=(1+\frac{2}{2m1})^{m}< 3=\frac{q}{ p+q}\), which implies \(h\leq\min\{h_{1},\frac{\eta}{p}\}\) holds for \(h=\frac{1}{m}\), m being an integer and \(m\geq2\).)
By Theorem 3.7 and Table 1, we also see that the 2stage Lobatto IIIB and the 3stage Lobatto IIIC (see Figures 7 and 8) are all asymptotically stable for h small enough.
We can see that the methods conserve their orders of convergence from Table 4 and Table 5.
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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.
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Zhang, GL. Asymptotical stability of RungeKutta for a class of impulsive differential equations. Adv Differ Equ 2016, 39 (2016). https://doi.org/10.1186/s136620150740z
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DOI: https://doi.org/10.1186/s136620150740z
Keywords
 asymptotical stability
 RungeKutta methods
 impulsive differential equations
 piecewise constant arguments
 stability region