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Numerical stability and oscillation of the RungeKutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type
Journal of Inequalities and Applications volume 2012, Article number: 290 (2012)
Abstract
This paper is concerned with the numerical properties of RungeKutta methods for the alternately of retarded and advanced equation \dot{x}(t)=ax(t)+{a}_{0}x(2[\frac{t+1}{2}]). The stability region of RungeKutta methods is determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained. A necessary and sufficient condition for the oscillation of the numerical solution is given. And it is proved that the RungeKutta methods preserve the oscillations of the analytic solutions. Some numerical experiments are illustrated.
1 Introduction
This paper deals with the numerical solution of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA)
where [\cdot ] is the greatest integer function. Differential equations of this form have stimulated considerable interest and have been studied by Cooker and Wiener [1], Jayasree and Deo [2], Wiener and Aftabizadeh [3]. In these equations the argument deviation T(t)=t2[\frac{t+1}{2}] is a piecewise linear periodic function with periodic 2. Also, T(t) is negative for 2n1\le t<2n and positive for 2n\le t<2n+1. Therefore, (1.1) is of advanced type on [2n1,2n) and of retarded type on (2n,2n+1).
EPCA describe hybrid dynamical systems, combine properties of both differential and difference equations and have applications in certain biomedical models in the work of Busenberg and Cooke [4]. For these equations of mixed type, the change of sign in the argument deviation leads not only to interesting periodic properties, but also to complications in the asymptotic and oscillatory behavior of solutions. Oscillatory, stability and periodic properties of the linear EPCA alternately of retarded and advanced form have been investigated in [1].
There are some papers concerning the stability of numerical solutions of delay differential equations with piecewise continuous arguments, such as [5–7]. Also, there have been results concerning oscillations of delay differential equations and delay difference equations, even including delay differential equations with piecewise continuous arguments [8]. But there is no paper concerned with the stability and oscillation of the numerical solutions of Eq. (1.1).
In this paper, we investigate the numerical properties, including the stability and oscillation, of RungeKutta methods of delay differential equations with piecewise continuous arguments.
We consider the following equation:
where a, {a}_{0} are constants and [\cdot ] is the greatest integer function.
Definition 1.1 [9]
A solution of (1.2) on [0,\mathrm{\infty}) is a function x(t) that satisfies the conditions:

1.
x(t) is continuous on [0,\mathrm{\infty}).

2.
The derivative \dot{x}(t) exists at each point t\in [0,\mathrm{\infty}), with the possible exception of the points t=2n1 for n\in \mathbb{N}, where onesided derivatives exist.

3.
(1.2) is satisfied on each interval [2n1,2n+1) for n\in \mathbb{N}.
In the following, we use these notations
The following theorems give existence and uniqueness of solutions and provide necessary and sufficient conditions for the asymptotic stability and the oscillation of all solutions of (1.2).
Theorem 1.2 [9]
Assume that a,{a}_{0},{x}_{0}\in \mathbb{R}. Then the initial value problem (1.2) has on [0,\mathrm{\infty}) a unique solution x(t) given by
where T(t)=t2[\frac{t+1}{2}], x(0)={x}_{0}.
Theorem 1.3 [9]
The solution x(t)=0 of (1.2) is asymptotically stable ({lim}_{t\to \mathrm{\infty}}x(t)=0) if and only if any one of the following hypotheses is satisfied:

1.
a<0, {a}_{0}>\frac{a({e}^{2a}+1)}{{({e}^{a}1)}^{2}} or {a}_{0}<a;

2.
a>0, \frac{a({e}^{2a}+1)}{{({e}^{a}1)}^{2}}<{a}_{0}<a;

3.
a=0, {a}_{0}<0.
In the following, we give the definition of oscillation and nonoscillation.
Definition 1.4 A nontrivial solution of (1.2) is said to be oscillatory if there exists a sequence {\{{t}_{k}\}}_{k=1}^{\mathrm{\infty}} such that {t}_{k}\to \mathrm{\infty} as k\to \mathrm{\infty} and x({t}_{k})x({t}_{k1})\le 0. Otherwise, it is called nonoscillatory. We say (1.2) is oscillatory if all nontrivial solutions of (1.2) are oscillatory. We say (1.2) is nonoscillatory if all nontrivial solutions of (1.2) are nonoscillatory.
Consider the difference equation
where k=1,2,\dots , {p}_{i}\in \mathbb{R}, i=1,2,\dots ,k, and its associated characteristic equation is
Definition 1.5 A nontrivial solution \{{a}_{n}\} of (1.3) is said to be oscillatory if there exists a sequence \{{n}_{k}\} such that {n}_{k}\to \mathrm{\infty} as k\to \mathrm{\infty} and {a}_{{n}_{k}}{a}_{{n}_{k}1}\le 0. Otherwise, it is called nonoscillatory. Equation (1.3) is said to be oscillatory if all nontrivial solutions of Eq. (1.3) are oscillatory. Equation (1.3) is called nonoscillatory if all nontrivial solutions of Eq. (1.3) are nonoscillatory.
Theorem 1.6 [10]
Eq. (1.3) is oscillatory if and only if the characteristic equation (1.4) has no positive roots.
Theorem 1.7 [9]
A necessary and sufficient condition for all solutions of Eq. (1.2) to be oscillatory is either {a}_{0}<\frac{a{e}^{a}}{{e}^{a}1} or {a}_{0}>\frac{a}{{e}^{a}1}.
2 RungeKutta methods
In this section we consider the adaptation of the RungeKutta methods (A,b,c). Let h=\frac{1}{m} be a given stepsize with an integer m\ge 1, and let the gridpoints {t}_{n} be defined by {t}_{n}=nh (n=0,1,2,\dots).
For the RungeKutta methods, we always assume that {b}_{1}+{b}_{2}+\cdots +{b}_{\nu}=1 and 0\le {c}_{1}\le {c}_{2}\le \cdots \le {c}_{\nu}\le 1.
The adaptation of the RungeKutta methods to (1.2) leads to a numerical process of the following type:
where the matrix A={({a}_{ij})}_{\nu \times \nu}, vectors b={({b}_{1},{b}_{2},\dots ,{b}_{\nu})}^{T}, c={({c}_{1},{c}_{2},\dots ,{c}_{\nu})}^{T}, and {x}_{n} is an approximation to x(t) at {t}_{n} (n=1,2,3,\dots). {y}_{i}^{(n)} and {z}_{i}^{(n)} are approximations to x({t}_{n}+{c}_{i}h) and x(2[\frac{{t}_{n}+{c}_{i}h}{2}]) respectively. Let n=2km+l, l=m,m+1,\dots ,m1 for k\ge 1, l=0,1,\dots ,m1 for k=0. Then {z}_{i}^{(2km+l)} can be defined as {x}_{2km} according to Definition 1.1 (i=1,\dots ,\nu).
Let {Y}^{(n)}={({y}_{1}^{(n)},{y}_{2}^{(n)},\dots ,{y}_{\nu}^{(n)})}^{T}. Then (2.1) reduces to
where e={(1,1,\dots ,1)}^{T}. Hence we have
where x=ha, R(x)=1+x{b}^{T}{(IxA)}^{1}e is the stability function of the method.
We can obtain from (2.3)
3 Stability and oscillation of the RungeKutta methods
In this section we discuss stability and oscillation of the RungeKutta methods.
3.1 Numerical stability
Definition 3.1 The RungeKutta method is called asymptotically stable at (a,{a}_{0}) if there exists a constant M such that {x}_{n} defined by (2.1) tends to zero as n\to \mathrm{\infty} for all h=\frac{1}{m} (m>M) and any given {x}_{0}.
Definition 3.2 The set of all points (a,{a}_{0}) at which the RungeKutta method is asymptotically stable is called an asymptotic stability region denoted by S.
For any given RungeKutta method, R(x)=\frac{P(x)}{Q(x)}, where P(x) and Q(x) are polynomials. R(x) is a continuous function at the neighborhood of zero, and R(0)={R}^{\prime}(0)=1. So there are {\delta}_{1}<0<{\delta}_{2} such that
which implies
Remark 3.3 It is known from [11] that R(x) is an increasing function in [1,1], and 1<R(x)<\mathrm{\infty} for 0<x\le 1, 0<R(x)<1 for 1\le x<0. Hence we can take {\delta}_{1}=1, {\delta}_{2}=1 for simplicity.
In the following, we always suppose h<\frac{1}{a}.
It is easy to see from (2.4) and (2.5) that {x}_{n}\to 0 as n\to \mathrm{\infty} if and only if {x}_{2km}\to 0 as k\to \mathrm{\infty}. Hence we have the following theorem.
Theorem 3.4 The RungeKutta method is asymptotically stable if any one of the following hypotheses is satisfied:

(i)
\frac{a({R}^{2m}(x)+1)}{{({R}^{m}(x)1)}^{2}}<{a}_{0}<a, a>0;

(ii)
\frac{a({R}^{2m}(x)+1)}{{({R}^{m}(x)1)}^{2}}<{a}_{0} or {a}_{0}<a, a<0;

(iii)
{a}_{0}<0, a=0.
Proof In view of (2.5), the RungeKutta method is asymptotically stable if and only if
If (a+{a}_{0}){a}_{0}{R}^{m}(x)>0, then (3.3) reduces to
which is equivalent to
If a+{a}_{0}{a}_{0}{R}^{m}(x)<0, then (3.3) reduces to
which is equivalent to
By virtue of (3.4) and (3.5), the theorem is proved. □
3.2 Numerical oscillations
Theorem 3.5 The following statements are equivalent:

1.
\{{x}_{n}\} is oscillatory;

2.
\{{x}_{2km}\} is oscillatory;

3.
{a}_{0}<\frac{a{R}^{m}(x)}{{R}^{m}(x)1}, or {a}_{0}>\frac{a}{{R}^{m}(x)1}.
Proof \{{x}_{2km}\} is not oscillatory if and only if
i.e.,
Hence for any l=1,2,\dots ,m1
which is equivalent to
We obtain from (2.4) that \{{x}_{n}\} is not oscillatory.
Moreover, \{{x}_{2km}\} is oscillatory if and only if
which is equivalent to
□
4 Preservation of stability and oscillations
In this section, we investigate the conditions under which the analytic stability region is contained in the numerical stability region and the conditions under which the numerical solution and the analytic solution are oscillatory simultaneously. We also study the stability and oscillation of the RungeKutta method with the stability function which is given by the (r,s)Padé approximation to {e}^{z}.
In order to do this, the following lemmas and corollaries will be useful to determine conditions.
The (r,s)Padé approximation to {e}^{z} is given by
where
with error
It is the unique rational approximation to {e}^{z} of order r+s, such that the degree of a numerator and a denominator are r and s respectively.
If R(z) is the (r,s)Padé approximation to {e}^{z}, then

(i)
there are s bounded star sectors in the righthalf plane, each containing a pole of R(z);

(ii)
there are r bounded white sectors in the lefthalf plane, each containing a zero of R(z);

(iii)
all sectors are symmetric with respect to the real axis.
Suppose R(z) is the (r,s)Padé approximation to {e}^{z}. Then

1.
x>0
R(x)<{e}^{x} for all x>0 if and only if s is even.
R(x)>{e}^{x} for all 0<x<\xi if and only if s is odd.

2.
x<0
R(x)>{e}^{x} for all x<0 if and only if r is even.
R(x)<{e}^{x} for all \eta <x<0 if and only if r is odd.
Where \xi >1 is a real zero of {Q}_{s}(z) and \eta <1 is a real zero of {P}_{r}(z).
4.1 Preservation of stability
We introduce the set H consisting of all pairs (a,{a}_{0})\in {\mathbb{R}}^{2} at which the RungeKutta method is asymptotically stable. In the following we investigate the conditions which lead to H\subseteq S. For convenience, we divide the region H into three parts
and in the similar way we denote
It is easy to see 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.
Theorem 4.4 Suppose that the stability function R(x) of the RungeKutta method is given by the (r,s)Padé approximation to {e}^{x}. Then {H}_{1}\subseteq {S}_{1} if and only if r is odd and {H}_{2}\subseteq {S}_{2} if and only if s is even.
Proof In view of Theorem 1.3 and Theorem 3.4, we have that H\subseteq S if and only if
which is equivalent to
since f(x)=\frac{{x}^{2}+1}{{(x1)}^{2}} is increasing in [0,1) and decreasing in (1,\mathrm{\infty}).
According to Corollary 4.3, the proof is complete. □
Theorem 4.5 For all RungeKutta methods, we have {H}_{0}={S}_{0}.
Corollary 4.6 For the Astable higher order RungeKutta methods, it is easy to see from Theorem 4.4 that

1.
For the νstage Radau IA and IIA methods, H\subseteq S if and only if ν is even;

2.
For the νstage Lobatto IIIA and IIIB methods, {H}_{1}\subseteq {S}_{1} if and only if ν is even and {H}_{2}\subseteq {S}_{2} if and only if ν is odd;

3.
For the νstage GaussLegerdre and Lobatto IIIC methods, {H}_{1}\subseteq {S}_{1} if and only if ν is odd and {H}_{2}\subseteq {S}_{2} if and only if ν is even.
It is known that all νstage explicit RungeKutta methods with p=\nu =1,2,3,4 possess the stability function (see [12])
which is the (\nu ,0)Padé approximation to {e}^{x}.
Theorem 4.7 For the νstage explicit RungeKutta methods with p=\nu =1,2,3,4, H\subseteq S if and only if ν is odd.
4.2 Preservation of oscillations
Definition 4.8 We call that the RungeKutta methods preserve oscillations of Eq. (1.2) if (1.2) oscillates, which implies that there is an {h}_{0} such that (2.4) oscillates for h<{h}_{0}.
Owing to Theorem 1.7 and Theorem 3.5, the RungeKutta method preserves the oscillation of (1.2) if and only if
Theorem 4.9 Suppose that the stability function R(x) is given by the (r,s)Padé approximation to {e}^{x}, then the RungeKutta method preserves the oscillation of (1.2) if and only if
and
Proof The proof is completed by (4.3) and noting that \frac{x}{x1} is decreasing in [0,1). □
Corollary 4.10

1.
The νstage GaussLegendre and Lobatto IIIC methods preserve the oscillation of Eq. (1.2) if and only if ν is odd.

2.
The νstage Lobatto IIIA and IIIB methods preserve the oscillation of Eq. (1.2) if and only if ν is even.

3.
The νstage Radau IA and IIA methods preserve the oscillation of Eq. (1.2) if ν is odd for a>0 and if ν is even for a<0.
Theorem 4.11 The νstage explicit RungeKutta methods with p=\nu =1,2,3,4 preserve the oscillation of Eq. (1.2) with a<0 if ν is odd.
Remark 4.12 We consider
where M, N are positive integers and M=2N.
We can obtain the same results about the stability and oscillation as Eq. (1.2), i.e.,

(i)
The RungeKutta method preserves the asymptotic stability of Eq. (4.4) if R(x)\le {e}^{x}.

(ii)
The RungeKutta method preserves the oscillation of Eq. (4.4) if
R(x)\ge {e}^{x}\phantom{\rule{1em}{0ex}}\text{for}a0,\phantom{\rule{2em}{0ex}}R(x)\le {e}^{x}\phantom{\rule{1em}{0ex}}\text{for}a0.
Hence the Corollary 4.6, 4.10 and Theorem 4.7, 4.11 hold.
5 Numerical experiments
In this section, we give some examples to illustrate the conclusions in the paper. To illustrate the stability, we consider the following two problems:
In Figure 1 to Figure 4, we draw the numerical solutions for Eq. (5.1) and Eq. (5.2) respectively. It is easy to see that the numerical solutions are asymptotically stable.
To illustrate the oscillation, we consider the following two problems:
In Figure 5 to Figure 7, we draw the numerical solutions for Eq. (5.3) and Eq. (5.4) respectively. It is easy to see that the numerical solutions are oscillatory.
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The financial support from the National Natural Science Foundation of China (No.11071050) is gratefully acknowledged.
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Song, M., Liu, M. Numerical stability and oscillation of the RungeKutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type. J Inequal Appl 2012, 290 (2012). https://doi.org/10.1186/1029242X2012290
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DOI: https://doi.org/10.1186/1029242X2012290