Abstract
This paper is concerned with the numerical properties of Runge-Kutta methods for the alternately of retarded and advanced equation . The stability region of Runge-Kutta 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 Runge-Kutta methods preserve the oscillations of the analytic solutions. Some numerical experiments are illustrated.
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1 Introduction
This paper deals with the numerical solution of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA)
where 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 is a piecewise linear periodic function with periodic 2. Also, is negative for and positive for . Therefore, (1.1) is of advanced type on and of retarded type on .
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 Runge-Kutta methods of delay differential equations with piecewise continuous arguments.
We consider the following equation:
where a, are constants and is the greatest integer function.
Definition 1.1 [9]
A solution of (1.2) on is a function that satisfies the conditions:
-
1.
is continuous on .
-
2.
The derivative exists at each point , with the possible exception of the points for , where one-sided derivatives exist.
-
3.
(1.2) is satisfied on each interval for .
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 . Then the initial value problem (1.2) has on a unique solution given by
where , .
Theorem 1.3 [9]
The solution of (1.2) is asymptotically stable () if and only if any one of the following hypotheses is satisfied:
-
1.
, or ;
-
2.
, ;
-
3.
, .
In the following, we give the definition of oscillation and non-oscillation.
Definition 1.4 A nontrivial solution of (1.2) is said to be oscillatory if there exists a sequence such that as and . Otherwise, it is called non-oscillatory. We say (1.2) is oscillatory if all nontrivial solutions of (1.2) are oscillatory. We say (1.2) is non-oscillatory if all nontrivial solutions of (1.2) are non-oscillatory.
Consider the difference equation
where , , , and its associated characteristic equation is
Definition 1.5 A nontrivial solution of (1.3) is said to be oscillatory if there exists a sequence such that as and . Otherwise, it is called non-oscillatory. Equation (1.3) is said to be oscillatory if all nontrivial solutions of Eq. (1.3) are oscillatory. Equation (1.3) is called non-oscillatory if all nontrivial solutions of Eq. (1.3) are non-oscillatory.
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 or .
2 Runge-Kutta methods
In this section we consider the adaptation of the Runge-Kutta methods . Let be a given step-size with an integer , and let the grid-points be defined by ().
For the Runge-Kutta methods, we always assume that and .
The adaptation of the Runge-Kutta methods to (1.2) leads to a numerical process of the following type:
where the matrix , vectors , , and is an approximation to at (). and are approximations to and respectively. Let , for , for . Then can be defined as according to Definition 1.1 ().
Let . Then (2.1) reduces to
where . Hence we have
where , is the stability function of the method.
We can obtain from (2.3)
3 Stability and oscillation of the Runge-Kutta methods
In this section we discuss stability and oscillation of the Runge-Kutta methods.
3.1 Numerical stability
Definition 3.1 The Runge-Kutta method is called asymptotically stable at if there exists a constant M such that defined by (2.1) tends to zero as for all () and any given .
Definition 3.2 The set of all points at which the Runge-Kutta method is asymptotically stable is called an asymptotic stability region denoted by S.
For any given Runge-Kutta method, , where and are polynomials. is a continuous function at the neighborhood of zero, and . So there are such that
which implies
Remark 3.3 It is known from [11] that is an increasing function in , and for , for . Hence we can take , for simplicity.
In the following, we always suppose .
It is easy to see from (2.4) and (2.5) that as if and only if as . Hence we have the following theorem.
Theorem 3.4 The Runge-Kutta method is asymptotically stable if any one of the following hypotheses is satisfied:
-
(i)
, ;
-
(ii)
or , ;
-
(iii)
, .
Proof In view of (2.5), the Runge-Kutta method is asymptotically stable if and only if
If , then (3.3) reduces to
which is equivalent to
If , 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.
is oscillatory;
-
2.
is oscillatory;
-
3.
, or .
Proof is not oscillatory if and only if
i.e.,
Hence for any
which is equivalent to
We obtain from (2.4) that is not oscillatory.
Moreover, 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 Runge-Kutta method with the stability function which is given by the -Padé approximation to .
In order to do this, the following lemmas and corollaries will be useful to determine conditions.
The -Padé approximation to is given by
where
with error
It is the unique rational approximation to of order , such that the degree of a numerator and a denominator are r and s respectively.
If is the -Padé approximation to , then
-
(i)
there are s bounded star sectors in the right-half plane, each containing a pole of ;
-
(ii)
there are r bounded white sectors in the left-half plane, each containing a zero of ;
-
(iii)
all sectors are symmetric with respect to the real axis.
Suppose is the -Padé approximation to . Then
-
1.
for all if and only if s is even.
for all if and only if s is odd.
-
2.
for all if and only if r is even.
for all if and only if r is odd.
Where is a real zero of and is a real zero of .
4.1 Preservation of stability
We introduce the set H consisting of all pairs at which the Runge-Kutta method is asymptotically stable. In the following we investigate the conditions which lead to . For convenience, we divide the region H into three parts
and in the similar way we denote
It is easy to see that , and
Therefore, we can conclude that is equivalent to , .
Theorem 4.4 Suppose that the stability function of the Runge-Kutta method is given by the -Padé approximation to . Then if and only if r is odd and if and only if s is even.
Proof In view of Theorem 1.3 and Theorem 3.4, we have that if and only if
which is equivalent to
since is increasing in and decreasing in .
According to Corollary 4.3, the proof is complete. □
Theorem 4.5 For all Runge-Kutta methods, we have .
Corollary 4.6 For the A-stable higher order Runge-Kutta methods, it is easy to see from Theorem 4.4 that
-
1.
For the ν-stage Radau IA and IIA methods, if and only if ν is even;
-
2.
For the ν-stage Lobatto IIIA and IIIB methods, if and only if ν is even and if and only if ν is odd;
-
3.
For the ν-stage Gauss-Legerdre and Lobatto IIIC methods, if and only if ν is odd and if and only if ν is even.
It is known that all ν-stage explicit Runge-Kutta methods with possess the stability function (see [12])
which is the -Padé approximation to .
Theorem 4.7 For the ν-stage explicit Runge-Kutta methods with , if and only if ν is odd.
4.2 Preservation of oscillations
Definition 4.8 We call that the Runge-Kutta methods preserve oscillations of Eq. (1.2) if (1.2) oscillates, which implies that there is an such that (2.4) oscillates for .
Owing to Theorem 1.7 and Theorem 3.5, the Runge-Kutta method preserves the oscillation of (1.2) if and only if
Theorem 4.9 Suppose that the stability function is given by the -Padé approximation to , then the Runge-Kutta method preserves the oscillation of (1.2) if and only if
and
Proof The proof is completed by (4.3) and noting that is decreasing in . □
Corollary 4.10
-
1.
The ν-stage Gauss-Legendre and Lobatto IIIC methods preserve the oscillation of Eq. (1.2) if and only if ν is odd.
-
2.
The ν-stage Lobatto IIIA and IIIB methods preserve the oscillation of Eq. (1.2) if and only if ν is even.
-
3.
The ν-stage Radau IA and IIA methods preserve the oscillation of Eq. (1.2) if ν is odd for and if ν is even for .
Theorem 4.11 The ν-stage explicit Runge-Kutta methods with preserve the oscillation of Eq. (1.2) with if ν is odd.
Remark 4.12 We consider
where M, N are positive integers and .
We can obtain the same results about the stability and oscillation as Eq. (1.2), i.e.,
-
(i)
The Runge-Kutta method preserves the asymptotic stability of Eq. (4.4) if .
-
(ii)
The Runge-Kutta method preserves the oscillation of Eq. (4.4) if
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 Runge-Kutta methods for the differential equations with piecewise continuous arguments alternately of retarded and advanced type. J Inequal Appl 2012, 290 (2012). https://doi.org/10.1186/1029-242X-2012-290
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DOI: https://doi.org/10.1186/1029-242X-2012-290